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\begin{document}

\title{A cross-border market model\\with limited transmission capacities}
\author{Cassandra Milbradt \footnote{Humboldt-Universität zu Berlin, Germany. E-mail: cassandra.milbradt@gmail.com}}
\date{}

\maketitle

\vspace{-1cm}

\begin{abstract}
We develop a cross-border market model between two countries in which the transmission capacities that enable transactions between market participants of different countries are limited. Starting from two so-called reduced-form representations of national limit order book dynamics, we allow incoming market orders to be matched with standing volumes of the foreign market, resulting in cross-border trades. We introduce a microscopic model that consists of two bid and ask price processes, four queue length processes that describe the number of unexecuted limit orders at the best bid and ask prices, and a capacity process. The latter counts the net number of executed cross-border trades over time. Since the transmission capacities in our model are limited, our model alternates between regimes in which cross-border trades are possible and regimes in which incoming market orders can only be matched against limit orders of the same origin. If the size of an individual order converges to zero while the order arrival rate tends to infinity, we derive a continuous-time limit approximation of our microscopic market dynamics. In the limit the volume dynamics is approximated by a four-dimensional, non-negative semimartingale with a non-trivial martingale part and reflections to the interior of $\mathbb{R}^4_+$ on hits of the (cumulative) queue length process of the axes. The approximation of the capacity dynamics turns out to be a bounded continuous process of finite variation. Since the tick size in our model is constant, the approximation of the bid (resp. ask) price dynamics is a two-dimensional pure jump process with jump times equal to those of the volume approximation. The usefulness of the ability to transact across borders is illustrated through a simulation study of price evolution.
\end{abstract}

\keywords{multiple limit order books, market microstructure, functional convergence theorems, scaling limit, regime switching process}

\raggedbottom

\section{Introduction}\label{CB:sec:motSU}

Limit order books (LOBs) are a standard tool for price formation in modern financial markets. They are records of unexecuted buy and sell orders awaiting execution.
While the recent financial mathematical literature is concerned with limit order book models for a single market, real-world opportunities to trade on multiple markets simultaneously require more complex model structures. For example, the introduction of the integrated European intraday electricity market ``Single Intraday Coupling'' (SIDC) has created the need to describe the effects of coupling multiple markets, i.e., when cross-border interactions are allowed between market participants of different countries. Due to limited transmission capacities in the SIDC, market participants could be (temporarily) prohibited from trading across borders. In this work, we introduce realistic microscopic cross-border market dynamics between two countries which are based on limit order books. In particular, we limit the total number of cross-border trades. Unfortunately, the resulting microscopic market dynamics are too complex to get a good understanding of the effects of coupling multiple markets. For this reason, we introduce suitable scaling constants to our microscopic model. If the size of an individual order converges to zero while the order arrival rate tends to infinity, we show that our model can be approximated by a tractable, continuous-time regime switching process. Using this continuous-time approximation, we investigate the impacts of coupling two markets on price evolution in a detailed simulation study.\par
At any given point in time, a limit order book depicts the number of unexecuted buy and sell orders at different price levels (cf. Figure \ref{CB:Fig:orderBook}). The highest price a potential buyer is willing to pay is called the best bid price, whereas the best ask price is the smallest price of all placed sell orders.
Incoming limit orders can be placed at many different price levels, while incoming market orders are matched against standing limit orders according to a set of priority rules.

\begin{figure}[H]
\centering
\includegraphics[scale=0.65]{images/OrderBook2.png}
\caption{\small Illustration of the state of a limit order book model.}
\label{CB:Fig:orderBook}
\end{figure}

For purely financial markets the financial mathematical literature already provides powerful tools for analyzing limit order books. One popular approach to study limit order books is based on an event-by-event description of the order flow as done in e.g., \cite{L03, CST10, HP17, HorstKreherCLT, HorstKreherDiffusion, HorstKreherFluid, GD18, CL13, KY18}. The derived stochastic systems typically yield realistic models as they preserve the discrete nature of the dynamics at high frequencies but turn out to be computationally challenging. To overcome the drawbacks of these models, some researchers deal with continuum approximations of the order book, describing it through its time-dependent density satisfying either certain partial differential equations as in \cite{LL07, CGGK09, CMP11, BCMW13} or certain stochastic partial differential equations as in \cite{MTW16, ContMueller}.\par
Combining these two approaches, one can introduce suitable scaling constants to the microscopic order book dynamics and study its scaling behavior when the number of orders gets large while each of them is of negligible size. The scaling limit can then either be described through a system of (partial) differential equations (in the ``fluid'' limit, where random fluctuations vanish), through a system of stochastic (partial) differential equations (in the ``diffusion'' limit, where random fluctuations dominate), or through a mixture thereof.  Deriving a deterministic high-frequency limit for microscopic limit order book models guarantees that the scaling limit approximation stays tractable in view of practical applications. Such an approach is pursued by Horst and Paulsen \cite{HP17}, Horst and Kreher \cite{HorstKreherFluid}, and Gao and Deng \cite{GD18}. However, the absence of arbitrage considerations encourages price approximations by diffusion processes. As discussed in Cont and de Larrard \cite{CL21}, depending on the market and/or stock of interest either a fluid or a diffusive volume approximation seems to be appropriated. Horst and Kreher \cite{HorstKreherDiffusion} studied the approximation of microscopic order book dynamics by both diffusive price and volume processes in the scaling limit. However, their consideration of a diffusive infinite dimensional volume process is not suitable for practical applications, as e.g., the uniqueness of a solution to the established infinite dimensional stochastic differential equation is in general not guaranteed. For this reason, in Kreher and Milbradt \cite{KM20} we have studied diffusive price approximations coupled with infinite dimensional fluid type volume approximations. This model yields realistic price approximations while the approximations of the infinite dimensional volumes are still tractable. The authors in \cite{CL21} guaranteed that their diffusive volume approximation stays tractable considering only the standing volumes at the top of the book and hence reducing the state space of the limit order book to a finite-dimensional space. Moreover, the price dynamics are implicitly determined by the volume dynamics. As the tick size is constant, prices in the high-frequency limit must be approximated by pure jump processes and not by diffusion type processes.\par

In our work, we introduce a first model of a cross-border market between two countries based on limit order books. Our model is a further development of the one considered in Cont and de Larrard \cite{CL21} in which the order flow directly effects the price evolution. The authors studied a reduced-form representation of a limit order book, i.e., in which the order book dynamics are given by the best bid and ask prices as well as the number of orders at the best bid respectively ask price (cf. Figure \ref{CB:Fig:redOrderBook}).

\begin{figure}[H]
\centering
\includegraphics[scale=0.85]{images/OrderBookReduced.png}
\caption{\small Illustration of a limit order book model with a reduced-state space.}
\label{CB:Fig:redOrderBook}
\end{figure}

With this reduction of the state space, a diffusion type limit for the queue lengths has been derived. In more detail, under heavy traffic conditions the authors prove that the bid and ask queue lengths are given in the high-frequency limit by a planar Brownian motion in the first quadrant with reflections to the interior at the boundaries. At the same time, the price processes are implicitly determined by the volume dynamics and turn out to be pure jump processes whose jump times equal to those of the corresponding volume processes.\par

We extend their model in multiple ways. First, we analyze the reduced-form re-presentations of two limit order book models over time. Second, we allow market orders to be matched with standing volumes of the limit order book of the foreign country which leads to cross-border trades. Finally, motivated by the limitation of transmission capacities in the SIDC, we limit the total number of cross-border trades. This might lead to structural changes in the trading behavior and market matching mechanism as market orders can only be matched with domestic standing volumes if all capacities are occupied. For this reason, we need to keep track of the origin of each incoming order and introduce a capacity process that counts the net number of executed cross-border trades over time. \par
Our microscopic model is therefore described by two best bid and ask queue size processes, two best bid and ask price processes, and a two-sided capacity process. Both, the price and capacity dynamics are determined implicitly by the queue size dynamics as follows: if one queue would become negative due to an incoming market order, the corresponding queue is set to zero and the remaining order size is depleted from the corresponding queue in the foreign market as long as both, enough standing volume and transmission capacity remain. This leads to a cross-border trade. If the cumulative best bid or ask queue would be depleted by an incoming market order, all queues are reinitialized by random variables (representing the depth of the books) and the price processes change by one tick. Each time a cross-border trade has been executed, the capacity process is updated. Now, starting in a so-called \textit{active} regime in which cross-border trading is possible, we switch to a so-called \textit{inactive} regime if the capacity process hits one of its boundaries (and hence the total number of cross-border trades has been executed). Then, in the inactive regime, market participants can only execute market orders against limit orders of the same origin. While in the active regime the best bid and ask prices of both national limit order books coincide, they become different in the inactive regime. With a simple trick of an efficient allocation of capacities, it is possible to switch back to an active regime. Based on this microscopic cross-border market model, by introducing appropriate scaling constants, we establish a high-frequency approximation in which the limit approximation is given by a continuous-time regime switching processes. In the high-frequency limit, the volume dynamics is approximated by a four-dimensional, non-negative semimartingale with a non-trivial martingale part and reflections to the interior of $\mathbb{R}^4_+$ on hits of the (cumulative) queue length process of the axes. The approximation of the bid (resp. ask) price dynamics is a two-dimensional pure jump process with jump times equal to those of the volume approximation. The approximation of the capacity dynamics turns out to be a bounded continuous process of finite variation. The evolution of our cross-border market dynamics is depicted in Figure \ref{CB:fig:scalLimit}. We discuss the behavior of our cross-border market model in different market situations through a detailed simulation study. Moreover, we study the effects of coupling two markets on price evolution by comparing the mean number of price changes and the mean bid price ranges  in simulated active and simulated inactive dynamics.

\begin{figure}[H]
	\centering
	\includegraphics[scale=0.3]{images/Simulation-1-balancedOB.png}
	\caption{The cross-border market model based on two LOBs: the queue size processes at the best bid (top left) and best ask price (top right), the bid price processes (bottom left), and the capacity process (bottom right). The white and gray areas illustrate the different regimes in which cross-border trades are possible (white) or prohibited (gray).}
	\label{CB:fig:scalLimit}
\end{figure}

\subsection{Model dynamics: empirical evidence}\label{CB:sec:empEv}

In this subsection, we motivate that the order flow, the trading behavior in neighboring countries, and the limitation of transmission capacities all have a considerable effect on the evolution of limit order books in a cross-border market and hence, should be included to any reasonable model. We analyze order book data\footnote{\label{CB:fn:SIDC}The data set is publicly available at \href{https://www.epexspot.com/}{EPEX SPOT}. We consider the German and Austrian market area and hour products for different delivery times.} (from March 05, 2020), trade book data\footref{CB:fn:SIDC} (from June 30, 2020), and capacity flow data\footnote{The data set is publicly available at the at \href{https://www.omie.es/en/market-results/monthly/continuous-intradaily-market/capacity-occupation-import-export?scope=monthly&year=2021&month=1&country=3}{OMIE} and focuses on France and Spain.} (from January 2021).\newline

\textit{Incorporation of the order flow and rise in trading liquidity:} empirical studies (cf. e.g. \cite{KP17, HPW16}) suggest that the incorporation of the market microstructure is crucial in any realistic model of intraday electricity markets with continuous trading. In \cite{KP17} updated weather forecasts, trade volume, and the demand quote are identified as the main price-driving factors in the German intraday market. Similarly, it is shown in \cite{HPW16} that not only fundamentals, but also trading behavior are important determinants of the liquidity available in intraday electricity markets. \par
Moreover, the coupling of multiple intraday electricity markets has increased trading volumes compared to the time before the introduction of the SIDC (cf. Figure \ref{CB:fig:SIDCmarketData1}). At the same time, the overall number of trades in the SIDC is steadily growing and has increased more than fivefold (from around $3.5$ million to $18$ million) since the launch of the SIDC in June 2018 (cf. Figure \ref{CB:fig:SIDCmarketData2}). That might be explained, on the one hand, by the $2$nd and $3$rd go-live waves in November 2019 and September 2021, and, on the other hand, by the growing acceptance of the SIDC which is supported by e.g., the rising share of renewable energy in the European energy mix. We note that the $4$th and $5$th go-live waves are already planed coupling Greece and Slovakia with the other SIDC countries. Going forward, we expect this trend toward more liquid intraday electricity markets to continue as we anticipate a rising share of renewables in the European energy mix, an expansion of grid capacities, and a merging of products with different delivery durations into a single market.

\begin{figure}[H]
    \centering
    \begin{subfigure}{0.45 \textwidth}
        \includegraphics[scale = 0.7]{images/IncreasingLiquidity2ndGoLiveWave.PNG}
        \caption{Trading volume (in thousand MWh) before and after the $2$nd go-live wave of the SIDC in the corresponding European countries.}
            \label{CB:fig:SIDCmarketData1}
    \end{subfigure}\hspace{0.5cm}
    \begin{subfigure}{0.45\textwidth}
        \includegraphics[scale = 0.52]{images/IncreasingLiquidityOverall.png}
        \caption{Overall number of trades (in million) per quarter of the SIDC since the launch in June 2018.}
            \label{CB:fig:SIDCmarketData2}
    \end{subfigure}
    \caption{Rise in trading liquidity due to the launch of the SIDC.}
\end{figure}

At the same time, the main share of order and trading volume over the trading session appears in the last few hours before closing. This can be seen in Figure \ref{CB:fig:SIDCmarketData3} in which we depict the cumulative trading volumes and execution prices of the trade book (from March 05, 2020) for expiry $2$\,pm over time. The increase of liquidity also yields relatively small bid-ask spreads compared to other electricity markets with continuous trading. In Figure \ref{CB:fig:SIDCmarketData4}, we depict the best bid and ask prices (from June 30, 2020) for expiry $2$\,pm over the last three hours before closing. It shows that bid-ask spreads are most of the time smaller than one euro. We expect the spreads to become smaller if the liquidity in the SIDC increases. In summary, even if these markets are still very illiquid compared to many financial markets, the development of the past years shows a clear trend toward significantly more liquid intraday electricity markets.

\begin{figure}[H]
    \centering
    \begin{subfigure}{0.45 \textwidth}
        \includegraphics[scale = 0.45]{images/ExecutedTrades.png}
        \caption{Cumulative trading volume (blue area) and execution prices (orange line).}
            \label{CB:fig:SIDCmarketData3}
    \end{subfigure}\hspace{0.5cm}
    \begin{subfigure}{0.45\textwidth}
        \includegraphics[scale = 0.45]{images/BidAskSpreadMean.png}
        \caption{Mean bid-ask spreads per minute (in the last three hours before closing).}
            \label{CB:fig:SIDCmarketData4}
    \end{subfigure}
    \caption{Executed trades and bid-ask spreads in the German continuous intraday electricity market for expiry $2$\,pm.}
\end{figure}

In the recent financial mathematical literature, a standard approach is to approximate the typically intractable microscopic limit order book dynamics by well-studied, tractable continuous-time scaling limits if the number of order events is large while each individual order is of negligible size. This might be a reasonable modeling assumption for highly liquid financial markets. Models for intraday electricity markets with continuous trading should include the market microstructure, i.e., each individual market and
limit order. To obtain a good understanding of these markets, we will also study its high-frequency behavior after carefully formulating and incorporating scaling assumptions. The continuous increase in trading liquidity in the SIDC gives hope that the resulting limit approximations will gain plausibility in the future.
 \newline

\textit{Influence of the trading behavior in the neighboring countries:} coupling multiple national limit order books and the resulting possibility to match incoming market orders with standing volumes of foreign limit order books influence the trading behaviors. To see that, we compare the transaction flow between the German and Austrian market area. In Figure \ref{CB:fig:Sankey}, we depict the number of transactions in the German respectively Austrian market area with focus on the buy and sell areas of transactions in form of two Sankey diagrams. In each diagram, the buy areas are depicted on the left and the sell areas on the right axis. ``XBID'' refers to products in which one part of a trade was made by a trading member of a different exchange (as opposed to EPEX SPOT). We observe that for both, the German and Austrian market area, the other country is the main importer of electricity in comparison to other countries.  At the same time, Austria exports four times more electricity to Germany than it domestically transacts. We therefore expect Austria's trading activities to be heavily dependent on the German trading behavior. In contrast, Germany exports comparably few electricity to other countries and most of its transactions are made domestically.

 \begin{figure}[H]
\centering
\begin{subfigure}[b]{0.49 \textwidth}
\centering
\includegraphics[width = \textwidth]{images/SankeyDE-200630-DT19.png}
\caption{Germany}
\label{CB:fig:SankeyDE}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49 \textwidth}
\centering
\includegraphics[width = \textwidth]{images/SankeyAT-200630-DT19.png}
\caption{Austria}
\label{CB:fig:SankeyAT}
\end{subfigure}
\caption{Sankey diagram depicting the number of SIDC trades for delivery time $7$\,pm relative to the different buy and sell market areas. Buy areas are depicted on the left and sell areas on the right axis.}
\label{CB:fig:Sankey}
\end{figure}

Studying the transaction flow over time reveals the same picture. In Figure \ref{CB:fig:transFlow}, we depict the cumulative trading volumes and execution prices relative to the execution time.  We observe that the cumulative quantity in the German market area is much higher (over five times higher) than in the Austrian market area. The dashed gray lines in Figure \ref{CB:fig:transFlow} denote the last transaction between Austria and Germany which coincides with the last cross-border trade in both market areas.

\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.49 \textwidth}
\centering
\includegraphics[width = \textwidth]{images/TF-DE19-newScala-2.png}
\caption{Germany}
\label{CB:fig:transFlowDE}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49 \textwidth}
\centering
\includegraphics[width = \textwidth]{images/TF-AT19-newScala-2.png}
\caption{Austria}
\label{CB:fig:transFlowAT}
\end{subfigure}
\caption{Transactions for the German and Austrian market area for delivery time $7$\,pm. The $y$-axis displays the executed trades: prices (left) and accumulated volumes (right). The time stamps are presented on the $x$-axis. The dotted gray line depicts the last executed trade between Germany and Austria.}
\label{CB:fig:transFlow}
\end{figure}

 In both pictures, we observe that the execution prices rapidly change after the last transaction between Germany and Austria. The behavior of the prices in both market areas indicates that traders from Germany could buy electricity for a smaller price while traders from Austria could sell electricity for a higher price due to the coupling.  Additionally, in both pictures the execution prices recover after short time to their old levels. The rapid change of the execution prices in both market areas indicates that the coupling has a high influence on the trading behavior in this market situation. We further observe a decrease in the number of transactions in the Austrian market area after the last trade with Germany. This effect is hardly observable in the German market area. The higher liquidity in the German intraday market may be a reasonable explanation for that.\newline

\textit{Occupation of transmission capacities:} to move toward a fully integrated European electricity market, the major obstacle to overcome is the limitation of transmission capacities. If national markets decouple during a trading session due to the full occupation of capacity, cross-border trading between them is prohibited. Different energy generation in different European countries, such as e.g., the utilization of nuclear or renewable energy resources, yields huge price differences in the national order books and hence motivate participants to buy or sell energy on the foreign market for the best price. The rapid change of the prices in Figure \ref{CB:fig:transFlow} after the last cross-border trade between Germany and Austria indicates a full occupation of the transmission capacity in our considered data set.  Moreover, the available transmission capacities are typically occupied at the end of a trading session: Figure \ref{CB:fig:SIDCmarketData5} shows  that in most continuous intraday electricity markets in January 2021, the transmission capacities between France and Spain are occupied at the end of the trading session.

\begin{figure}[H]
    \centering
    \includegraphics[scale = 0.7]{images/CapacityOccupation-SpainFrance-202101.png}
    \caption{Available transmission capacity (yellow) and its occupation in direction Spain to France and vice versa in January 2021.}
\label{CB:fig:SIDCmarketData5}
\end{figure}

Transmission system operators expand the existing grid capacities. However, the different utilization of resources and the rising share of renewables in the European energy mix will probably still yield an occupation of transmission capacities in the future. Hence, incorporating a limitation of cross-border trading is necessary to develop an understanding of the SIDC market dynamics.

\subsection{Structure of this paper}

The remainder of paper is structured as follows: in Section \ref{CB:sec:disTimeMM} we specify the discrete-time cross-border market dynamics $S^{(n)}$ and state conditions that guarantee its convergence to a continuous-time limit. Moreover, we introduce the active market dynamics $\tilde{S}^{(n)}$ (resp. the inactive market dynamics $\dbtilde{S}^{(n)}$) describing the evolution of the cross-border market dynamics when cross-border trades are possible (resp. prohibited). Thereafter, in Section \ref{CB:sec:anaAD}, we analyze the active dynamics and derive a functional convergence result for it. In Section \ref{CB:sec:anaID}, we provide an overview for analyzing the inactive market dynamics and state the corresponding limit result. Finally, in Section \ref{CB:sec:mainResult}, with help of the convergence results for $\tilde{S}^{(n)}$ and $\dbtilde{S}^{(n)},$ we present our main result and prove that the cross-border market dynamics $S^{(n)}$ converges weakly in the Skorokhod topology to a continuous-time regime switching process. In Section \ref{CB:sec:simStudyCB}, we present different market situations of our model through simulations and study the impact of coupling two markets on price evolution. Many technical proofs are stated in the appendix. \newline

\textbf{Notations.} We denote by $\mathbb{R}_+ := [0,\infty)$ and $\mathbb{R}_- := (-\infty, 0]$ the positive and negative real half-line, respectively. Moreover, for each $x\in \mathbb{R}^d,$ $d\geq 1,$ we denote by $\|x\|^2 := \sum_{i=1}^d x_i^2$ the euclidean norm in $\mathbb{R}^d,$ and for $\omega \in D([0,T], \mathbb{R}^d),$ $\|\omega\|_{\infty} := \sup_{t\in [0,T]}\|\omega(t)\|$ the sup norm. Furthermore, for stochastic processes $X$ and $Y$ we write $X \simeq Y$ if they have the same finite-dimensional distributions. In what follows, we encounter projections onto a single or onto multiple coordinates of a function $\omega \in D([0,T], \mathbb{R}^k)$ or a vector $x \in \mathbb{R}^k.$ Therefore, let $\pi^{(k)}_j : D([0,T], \mathbb{R}^k) \rightarrow D([0,T], \mathbb{R}),$ $k \geq 1$, denote the $j$-th projection map, i.e., for $\omega = (\omega_1, \cdots, \omega_k) \in D([0,T], \mathbb{R}^k),$ we have that
$$\pi^{(k)}_j \omega = \omega_j \in D([0,T], \mathbb{R}), \quad \text{ for } 1 \leq j \leq k.$$
With a little abuse of notation, we also write $\pi^{(k)}_jx = x_j,$ for $x = (x_1, \cdots, x_k) \in \mathbb{R}^k$, $1 \leq j \leq k.$ Further, let us denote by $\pi^{(k)}_{i,j}: D([0,T], \mathbb{R}^k) \rightarrow D([0,T], \mathbb{R}^2)$, $k \geq 2,$ the $(i,j)$-th projection map, i.e.,  $\pi^{(k)}_{i,j} \omega = (\omega_i, \omega_j) \in D([0,T], \mathbb{R}^2)$, $1\leq i, j \leq k.$ Again, for $x \in \mathbb{R}^k$, we write $\pi^{(k)}_{i,j} x = (x_i,x_j) \in \mathbb{R}^2.$ Finally, we introduce the following short hand notations: $\pi_j := \pi^{(4)}_j$ and $\pi_{i,j} := \pi^{(4)}_{i,j}$ for $1 \leq i,j \leq 4.$ Moreover, $\pi_F := \pi_{1,2},$ $\pi_G := \pi_{3,4},$ $\pi_b := \pi_{1,3},$ and $\pi_a := \pi_{2,4}$. These projections will be used to determine the standing volumes of one country or of the bid/ask queues from the four-dimensional queue size dynamics.

\section{The microscopic market dynamics}\label{CB:sec:disTimeMM}

Let us fix some finite time horizon $T > 0,$ the tick size $\delta > 0,$ and the space $E:= \mathbb{R}^2 \times \mathbb{R}^4_+ \times \mathbb{R}.$ Let us consider two neighboring countries $F$ (``France'') and $G$ (``Germany''). Each of these countries has a national limit order book through which it can trade its goods domestically. Moreover, as long as enough transmission capacities remain, market orders can also be matched against the standing volumes of the foreign limit order book ($G$ is the foreign country for orders with origin $F$ and vise versa).  In the following, we describe the cross-border market dynamics by an extension of the \textit{reduced-form representation} of a limit order book model introduced by Cont and de Larrard \cite{CL21} to two possibly interacting limit order books. In more detail, we describe the random evolution of a sequence of cross-border market models through a sequence of $E$-valued stochastic processes $S^{(n)} := (S^{(n)}(t))_{t\in [0,T]}$ with
\begin{equation*}
S^{(n)}(t) =\left(B^{(n)}(t), Q^{(n)}(t), C^{(n)}(t)\right),
\end{equation*}
where for each $n\in \mathbb{N},$ the $\mathbb{R}^2$-valued process $B^{(n)} := (B^{F,(n)}, B^{G,(n)})$ specifies the dynamics of the best bid prices in $F$ and $G$, the $\mathbb{R}^4_+$-valued process $Q^{(n)} := (Q^{b,F,(n)}, Q^{a,F,(n)},$ $Q^{b,G,(n)}, Q^{a,G,(n)})$ specifies the dynamics of standing volumes at the best bid/ask price in $F$ and $G,$ and the $\mathbb{R}$-valued process $C^{(n)}$ describes the (scaled) net number of cross-border trades. We will refer to $C^{(n)}$ as the capacity process of the cross-border market model $S^{(n)}$. In the following, for all $n\in \mathbb{N}$, the bid price process $B^{(n)}$ takes values in the subspace $(\delta \mathbb{Z})^2 \subset \mathbb{R}^2,$ where $\delta\mathbb{Z} := \{j\delta: j\in \mathbb{Z}\}$ denotes the $\delta$-grid. Moreover, the best ask price process $A^{(n)} := (A^{F,(n)}, A^{G,(n)})$ is implicitly given by the best bid price process as we set $A^{I,(n)} = B^{I,(n)} + \delta$ for $I = F,G$ and all $n\in \mathbb{N}.$  Throughout, we assume that all random variables are defined on some common, complete probability space $(\Omega, \mathcal{F}, \ensuremath\mathbb{P})$.

\begin{Rmk}[Constant spread condition in a reduced-form model]\label{CB:rmk:askPrice}
	Motivated by the reduced-form representation of a limit order book model (cf. Cont and de Larrard \cite{CL13, CL21}), we assume that the spread is of fixed size $\delta > 0$ for all $t \in [0,T]$ and $n \in \mathbb{N}$. Keeping the spread fixed over time is crucial as we only concentrate on modeling limit order placements outside the spread. However, this is not so unrealistic as a closing of a spread greater than one tick happens very quickly in liquid markets (cf. e.g. the empirical study in \cite{CL21}). A frequently used assumption in recent literature dealing with high-frequency approximations of microscopic limit order book dynamics is that the tick size converges to zero as $n\rightarrow \infty$ (cf. e.g. \cite{HorstKreherFluid, HorstKreherDiffusion, KM20}). However, for the reduced-form model, this would yield a zero spread as well as constant price processes in the high-frequency limit. For this reason, we also fix the tick size $\delta > 0$ for all $n\in \mathbb{N}$.
\end{Rmk}

The cross-border market dynamics change due to arriving market orders, limit orders, and cancellations at the best bid and ask queues in $F$ respectively $G.$ For simplicity, we assume the time intervals between two consecutive order arrivals to be of equal length $\Delta t^{(n)} > 0$. This assumption is also frequently used in other recent literature on limit order book models (cf. e.g. \cite{HorstKreherFluid, HorstKreherDiffusion}). Note that extensions to randomly spaced arrival times are possible as has been done in \cite{BHQ17,HP17, CL21} but this will not be in the focus of our analysis. Then, there are $T_n := \lfloor T/\Delta t^{(n)}\rfloor$ such events taking place at times
\[\t_k := k \Delta t^{(n)}, \quad k = 1,\cdots, T_n.\]
We assume that $\Delta t^{(n)}$ goes to zero as $n\rightarrow \infty$, i.e., the number of orders increases as $n\rightarrow \infty.$ Furthermore, we introduce the average size of a limit order placement, which we denote by $\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} > 0.$ As $\Delta t^{(n)},$ it is assumed to tend to zero as $n\rightarrow \infty$.  In the following, we specify the sequence of order book models for which we establish a scaling limit when the average limit order size tends to zero while the number of order events tends to infinity.

\subsection{The initial state}

In the $n$-th model, the initial state of the cross-border market model is given by (positive) best bid prices $B^{F,(n)}_0,$ $B^{G,(n)}_0 \in \delta \mathbb{Z},$ non-negative queue sizes $Q^{(n)}_0 := (Q^{b,F,(n)}_0,$ $Q^{a,F,(n)}_0, Q^{b,G,(n)}_0,$ $Q^{a,G,(n)}_0) \in (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4,$ and initially occupied capacity $C^{(n)}_0 \in \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{Z}.$ For simplicity, we choose $C^{(n)}_0 = 0$ for all $n\in \mathbb{N}$. Since we are interested in studying the cross-border market dynamics when the transmission capacities are limited, we introduce $\kappa_- > 0$ denoting the total quantity of transmission capacity in direction $F$ to $G$ (i.e. \textit{exports} from $F$) and $\kappa_+ > 0$ denoting the total quantity in direction $G$ to $F$ (i.e. \textit{imports} to $F$). Since $C^{(n)}_0 = 0$, we therefore assume that at time $t = 0$ cross-border trading is possible. Moreover, if cross-border trading is allowed, it is natural to assume that the bid prices of $F$ and $G$ coincide, so we further assume that $B^{F,(n)}_0 = B^{G,(n)}_0$ for all $n\in \mathbb{N}$. Hence, at time $t = 0$, the state of the cross-border market is deterministic for all $n\in \mathbb{N}$ and is denoted by
\[S^{(n)}_0 := \left(\left(B^{F,(n)}_0, B^{F,(n)}_0\right), \left(Q^{b,F,(n)}_0, Q^{a,F,(n)}_0, Q^{b,G,(n)}_0, Q^{a,G,(n)}_0\right), 0\right) \in E.\]
In order to prove a convergence result for the microscopic model to a high-frequency limit, we need to state the following convergence assumptions on the initial values.

\begin{Ass}[Convergence of the initial state]\label{CB:ass:initialCon}
	There exist $B^{F}_0 \in \delta \mathbb{Z}$ and $Q_0 := (Q^{b,F}_0, Q^{a,F}_0,$ $Q^{b,G}_0, Q^{a,G}_0) \in (0,\infty)^4$ such that $B^{F,(n)}_0 \rightarrow B^F_0$ and $Q^{(n)}_0 \rightarrow Q_0$ as $n\rightarrow \infty.$ In the following, we denote $S_0 := ((B^F_0, B^F_0), ( Q^{b,F}_0, Q^{a,F}_0, Q^{b,G}_0, Q^{a,G}_0), 0)$ $\in E$.
\end{Ass}

Note that under this assumption, neither the initial cumulative best bid queue nor the initial cumulative best ask queue is zero. This prevents an occurrence of a price changing event at time $t= 0.$

\subsection{Event types, order sizes, and the depth of the limit order books}

The cross-border market dynamics change by incoming order events. In order to determine their effects on the state of the cross-border market dynamics $S^{(n)}$, we introduce two sequences of random variables $(\phi^{(n)}_k)_{k=1, \cdots, T_n}$ and $(\psi^{(n)}_k)_{k=1,\cdots, T_n}$ determining the \textit{type} and the  \textit{origin} of an incoming order, i.e., for all $k= 1,\cdots, T_n,$
$$\phi^{(n)}_k \in \{b,a\}, \quad \psi^{(n)}_k \in \{F,G\},$$
and $\phi^{(n)}_k = b$ (resp. $\phi^{(n)}_k = a$) if the $k$-th incoming order event affects the bid side (resp. the ask side) of a limit order book and $\psi^{(n)}_k = F$ (resp. $\psi^{(n)}_k = G$) if the $k$-th incoming order has origin $F$ (resp. origin $G$). Dependent on the evaluation of the random vector $(\phi^{(n)}_k, \psi^{(n)}_k) \in \{b,a\} \times \{F,G\}$, we observe four different order events which change the state of $S^{(n)}$. For $k = 1, \cdots, T_n$ the following two order events corresponds to possible transactions in direction $F$ to $G$ (i.e. \textit{exports} from $F$):
\begin{enumerate}[leftmargin = 1.5cm]
 \setlength\itemsep{0em}
	\item[$\textbf{(b,F)}$] A market sell / limit buy order with origin $F$ arrives.
	\item[$\textbf{(a,G)}$] A market buy / limit sell order with origin $G$ arrives.
\end{enumerate}
The next two order events corresponds to possible transactions in direction $G$ to $F$ (i.e. \textit{imports} to $F$):
\begin{enumerate}[leftmargin = 1.5cm]
 \setlength\itemsep{0em}
	\item[$\textbf{(b,G)}$]  A market sell / limit buy order with origin $G$ arrives.
	\item[$\textbf{(a,F)}$]  A market buy / limit sell order with origin $F$ arrives.
\end{enumerate}
In the following, we refer to the evaluation of the random vector $(\phi^{(n)}_k, \psi^{(n)}_k)$ as the \textit{type} of the $k$-th order. Further, we denote by $\mathcal{I}^{Ex} := \{(b,F), (a,G)\}$ and $\mathcal{I}^{Im} := \{(a,F), (b,G)\}$ the order types corresponding to possible exports from $F$ or imports to $F$, respectively. \par

Next, let us introduce the random sequence $(V^{(n)}_k)_{k = 1,\cdots, T_n}$ representing the sizes of incoming orders and let us denote by
$$V^{i,I,(n)}_k := V^{(n)}_k \textup{$\mathds{1}$}_{\left\{\left(\phi^{(n)}_k, \psi^{(n)}_k\right) = (i, I)\right\}}$$
the order sizes of incoming orders of type $(i,I) \in \{b,a\} \times \{F,G\},$ $k = 1, \cdots, T_n.$  Throughout, we assume that $V^{(n)}_k \in \{-\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}, \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\}$ for all $k = 1, \cdots, T_n$ and $n\in \mathbb{N}.$ Note that $V^{(n)}_k = \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}$ if the $k$-th order is a limit order placement and $V^{(n)}_k = -\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}$ if the $k$-th order is a market order or an order cancellation at the best bid or ask price.\par

In order to derive a heavy traffic approximation for the cross-border market model $S^{(n)}$, we need to state further assumptions to our model. First, we present an assumption on the mean and covariance structure of $(V^{i,I,(n)}_k, (i,I) \in \{b,a\} \times \{F,G\})_{k=1,\cdots, T_n}$. In particular, we allow a dependence structure corresponding to strong (or uniform) mixing of the incoming orders. Combined with the right scaling relation between $\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}$ and $\Delta t^{(n)}$, this assumption guarantees that the partial sums of the order sizes verify a certain version of Donsker's theorem for a dependent sequence of random variables (cf. \cite[Theorem 19.1]{B99}). Note, that this dependence condition can be replaced by a $\rho$-mixing condition as discussed in \cite[Theorem 19.2]{B99} and is therefore stronger than the $\alpha$-mixing condition.

\begin{Ass}[Sequence of order sizes]\label{CB:ass:prob}
 For all $n \in \mathbb{N}$, $(V^{i,I,(n)}_k, (i,I) \in \{b,a\} \times \{F,G\})_{k = 1,\cdots, T_n}$ is a stationary, uniform mixing array of random variables with $V^{(n)}_k \in \{-\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}, \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\}$. Moreover,
 \begin{enumerate}
 	\item[i)] there exist $\mu^{i,I,(n)}\in \mathbb{R}$, $\sigma^{i,I,(n)} \in \mathbb{R}_+$ for all $(i,I) \in \{b,a\} \times \{F,G\}$ such that
	\begin{align*}
	\ensuremath\mathbb{E}\left[V^{i,I,(n)}_1\right] &= \left(\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\right)^2 \mu^{i,I,(n)},\\
	\ensuremath\mathrm{Var}\left(V^{i,I,(n)}_1\right) + 2 \sum^{T_n}_{k = 2}\ensuremath\mathrm{Cov}&\left(V^{i,I,(n)}_1, V^{i,I,(n)}_k\right) = \left(\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\right)^2 \left(\sigma^{i,I,(n)}\right)^2
	\end{align*}
	as well as $\sigma^{(i,I),(j,J),(n)} \in \mathbb{R}$ for all $(i,I), (j,J) \in \{b,a\} \times \{F,G\}$ with $(i,I) \neq (j,J)$ such that
	\begin{align*}
	&2 \,\ensuremath\mathrm{Cov}\left(V^{i,I,(n)}_1, V^{j,J,(n)}_1\right)\\
	& \hspace{-0.3cm}+ 2 \sum_{k=2}^{T_n} \left(\ensuremath\mathrm{Cov}\left(V^{i,I,(n)}_1, V^{j,J,(n)}_k\right)	 + \ensuremath\mathrm{Cov}\left(V^{i,I,(n)}_k, V^{j,J,(n)}_1\right)\right) = \left(\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\right)^2\sigma^{(i,I), (j,J),(n)}.
	\end{align*}
	\item[ii)] For all $(i,I) \in \{b,a\} \times \{F,G\}$, there exist $\mu^{i,I} \in \mathbb{R}$, $\sigma^{i,I} \in \mathbb{R}_+$ and for all $(i,I), (j,J) \in \{b,a\} \times \{F,G\}$ with $(i,I) \neq (j,J)$ there exist $\sigma^{(i,I), (j,J)} \in \mathbb{R}$ such that as $n \rightarrow \infty,$ we have
	 $$\left(\mu^{i,I,(n)}, \sigma^{i,I,(n)}\right) \rightarrow \left(\mu^{i,I}, \sigma^{i,I}\right) \quad \text{ and } \quad \sigma^{(i,I),(j,J),(n)} \rightarrow \sigma^{(i,I),(j,J)}.$$
	 \item[iii)] For all $(i,I), (j,J) \in \{b,a\} \times \{F,G\}$ with $(i,I) \neq (j,J),$ the corresponding correlation coefficient
	 $$\rho^{(i,I),(j,J)} := \frac{\sigma^{(i,I),(j,J)}}{\sigma^{(i,I)} \sigma^{(j,J)}} \quad \text{ satisfies } \quad \left|\rho^{(i,I),(j,J)}\right|< 1.$$
\end{enumerate}
\end{Ass}

The dependence structure given in the above assumption either requires that the sequence of order sizes become independent as $n \rightarrow \infty$ or asks for $m$-dependence of the sequence of order sizes for all $n\in \mathbb{N}$ large enough, for any $m \in \mathbb{N}$ fixed, i.e., $(V^{(n)}_1, \cdots, V^{(n)}_k)$ and $(V^{(n)}_{k+j}, \cdots, V^{(n)}_{k+j+l})$ are independent for all $k, l \in \mathbb{N}$ for large $n$ whenever $j > m$. This dependence structure is frequently referred in the literature as weak dependence. The third statement in the above assumption requires that the partial sums of the order sizes corresponding to different order types converge to limits with different underlying Brownian motions, i.e., a perfect correlation between their scaling limits is prohibited. \par

The next assumption introduces the critical scaling assumption under which the partial sums of the order sizes verify a functional central limit theorem. A similar assumption is stated in e.g. \cite{HorstKreherDiffusion, CL13}.

\begin{Ass}[Relation between the scaling parameters]\label{CB:ass:scaling}
	There exists a constant $C > 0$ such that
	$$\lim_{n \rightarrow \infty} \frac{\Delta t^{(n)}}{\left(\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\right)^2} = C.$$
	In what follows, we assume that $C = 1$. Note, that each other constant would simply lead to further constants in the scaling limit.
\end{Ass}

In order to describe the dynamics of the queue size process $Q^{(n)}$, we need to characterize its value after price changes. In the following, we allow the sizes of the queue lengths after a price change to depend on the current state of the queue length process. This is motivated by pegged limit orders which are typically observed in electronic trading platforms. Let $(\tau^{(n)}_k)_{k\geq 1}$ be the sequence of stopping times at which we observe price changes in our cross-border market dynamics $S^{(n)}.$

\begin{Ass}[Size of the order queues after price changes]\label{CB:ass:R} $ $
	\begin{enumerate}
	\item[i)] For all $n \in \mathbb{N}$, there exist independent sequences of iid random variables $(\epsilon^{+,(n)}_k)_{k\geq 1}$ and $(\epsilon^{-,(n)}_k)_{k\geq 1}$ with values in $(0,\infty)^4$, where $\epsilon^{+,(n)}_1 \sim f^+_n$ and $\epsilon^{-,(n)}_1 \sim f^-_n$ for some probability distributions $(f^+_n)_{n\in \mathbb{N}}, (f^-_n)_{n\in \mathbb{N}}$. Moreover, there exists a function $\Phi^{(n)} : \mathbb{R}^4_+ \times \mathbb{R}^4_+ \rightarrow (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4$ such that
	\[\exists\, \alpha\, > 0, \, \forall\, (x,y) \in \mathbb{R}^4_+ \times \mathbb{R}^4_+, \, \forall \, j = 1,2,3,4, \quad \pi_j\Phi^{(n)}(x,y) \geq \alpha \pi_j y.\]
	Furthermore, the queue sizes after the $k$-th price change are equal to $R^{+,(n)}_k$ or $R^{-,(n)}_k,$ depending on whether the $k$-th price change is a price increase or decrease, where for all $k\geq 1,$ we set
\begin{align}\label{CB:def:R-n}
\begin{aligned}
    R^{+,(n)}_k &:= \Phi^{(n)}\left(Q^{(n)}(\tau^{(n)}_k-), \epsilon^{+,(n)}_k\right),\\
     R^{-,(n)}_k &:= \Phi^{(n)}\left(Q^{(n)}(\tau^{(n)}_k-), \epsilon^{-,(n)}_k\right).
    \end{aligned}
\end{align}
	\item[ii)]  Additionally, there exist probability distributions $f^+$ and $f^-$ on $(0,\infty)^4$ and \\$\Phi \in C^2\left(\mathbb{R}^4_+ \times \mathbb{R}^4_+, (0,\infty)^4 \right)$, such that
		\begin{align*}
		(f^+_n, f^-_n) \Rightarrow (f^+, f^-) \quad \text{ and } \quad \|\Phi^{(n)} - \Phi\|_{\infty} \rightarrow 0, \quad \text{ as } n \rightarrow \infty.
		\end{align*}
	\end{enumerate}
\end{Ass}

\begin{Rmk}
	With minor modifications, we could also allow in the subsequent analysis that the distributions determining the queue sizes after price changes depend on the origin of the queue whose depletion caused the price change. For notational reasons, we will not include this to our model.
\end{Rmk}

In the next subsections, for all $n\in \mathbb{N}$, we describe the dynamics of the cross-border market model $S^{(n)}$ as follows: if there are enough transmission capacities remain, the national LOBs are \textit{coupled}, i.e., incoming market orders can be matched against the standing volumes of the national \textit{and} foreign limit order book. As  the transmission capacities in both directions are limited by $\kappa_-, \kappa_+ >0$, it may happen that the national LOBs \textit{decouple}, i.e., market orders can only be matched against limit orders with the same origin. Hence, our cross-border market model switches between the following two regimes:

\begin{itemize}
 \setlength\itemsep{0em}
    \item the \textit{active} regime in which the LOBs of $F$ and $G$ are coupled, and
    \item the \textit{inactive} regime in which the LOBs of $F$ and $G$ are decoupled.
\end{itemize}
In order to describe how $S^{(n)}$ behaves during its different regimes, we introduce the active dynamics $\tilde{S}^{(n)}$ and inactive dynamics $\dbtilde{S}^{(n)}$ describing the evolution of the two national LOBs as if we were in the active respectively inactive regime for the whole trading period.

\subsection{Description of the active dynamics}\label{CB:sec:defAD}

In this subsection, we introduce for each $n\in \mathbb{N}$ the active dynamics given by the piecewise constant interpolation
\[\tilde{S}^{(n)}(t) = \sum_{k=0}^{T_n}\tilde{S}^{(n)}_k \textup{$\mathds{1}$}_{\left[\t_k, \t_{k+1}\right)}(t), \quad t \in [0,T],\]
of the $E$-valued random variables
\[\tilde{S}^{(n)}_k := \left(\tilde{B}^{(n)}_k, \tilde{Q}^{(n)}_k, \tilde{C}^{(n)}_k\right), \quad k \in \mathbb{N}_0,\]
where $\tilde{B}^{(n)}_k$ denotes the bid prices of $F$ and $G,$ $\tilde{Q}^{(n)}_k$ denotes the sizes of the best bid respectively ask queues in $F$ and $G,$ and $\tilde{C}^{(n)}_k$ denotes the net number of cross-border trades after $k$ order events. Note that $(\tilde{C}^{(n)}_k)_{k = 0,\cdots, T_n}$ describes net number of cross-border trades in an unlimited setting, i.e., as if $\kappa_- = \kappa_+ = \infty.$ Since the national LOBs are coupled in the active dynamics, we may suppose that $\tilde{B}^{F,(n)}_k = \tilde{B}^{G,(n)}_k$ for all $k= 0,\cdots, T_n$ and $n\in \mathbb{N}$. For the same reason, we have chosen the initial best bid prices to coincide in Assumption \ref{CB:ass:initialCon}. Moreover, we allow a national order queue to be equal to zero as long as the corresponding order queue in the foreign market is strictly larger than zero. Keeping this in mind, we can summarize the national order books in a shared order book (cf. Figure \ref{CB:fig:cbMarket}).

\begin{figure}[H]
\centering
\begin{tikzpicture}
            \fill[CMorange]
                (5,-0.25) rectangle (6, 0.25)
                (6.5, -0.25) rectangle (7.5, 0.25)
                (9, 0) rectangle (10,0.5)
                (10.5, 0) rectangle (11.5, 0.5);
            \fill[CMblue]
                (5, 0.75) rectangle (6, 1.75)
                (6.5,0.75) rectangle (7.5, 1.25)
                (9, 0.5) rectangle (10, 1.5)
                (10.5, 0.5) rectangle (11.5, 1);
        \draw
            (8.75,0) -- (11.75,0)
            (4.75,-0.25) -- (7.75, -0.25)
            (4.75, 0.75) -- (7.75, 0.75)
            (9.5, -0.1) -- (9.5, 0.1)
            (11, -0.1) -- (11, 0.1)
            (5.5, -0.35) -- (5.5, -0.15)
            (7, -0.35) -- (7, -0.15)
            (5.5, 0.65) -- (5.5, 0.85)
            (7, 0.65) -- (7, 0.85)
            (5, -0.25) rectangle (6, 0.25)
            (6.5, -0.25) rectangle (7.5, 0.25)
            (5, 0.75) rectangle (6, 1.75)
            (6.5, 0.75) rectangle (7.5, 1.25)
            (9, 0) rectangle (10,0.5)
            (10.5, 0) rectangle (11.5, 0.5)
            (9, 0.5) rectangle (10, 1.5)
            (10.5, 0.5) rectangle (11.5, 1);
        \draw[->, thick]
            (8, 1.5) -- (8.6, 0.9);
        \draw[->, thick]
            (8, 0) -- (8.6, 0.6);
        \node at (9.5, -0.3) { \tiny B };
        \node at (11, -0.3) { \tiny B$+\delta$ };
        \node at (5.5, -0.55) { \tiny B };
        \node at (7, -0.55) { \tiny B$+\delta$ };
        \node at (5.5, 0.45) { \tiny B };
        \node at (7, 0.45) { \tiny B$+\delta$ };
        \node[white] at (9.5, 1) { \tiny Q\textsuperscript{b,G} };
        \node[white] at (11, 0.75) { \tiny Q\textsuperscript{a,G} };
        \node[white] at (9.5, 0.25) { \tiny Q\textsuperscript{b,F} };
        \node[white] at (11, 0.25) { \tiny Q\textsuperscript{a,F} };
        \node[white] at (5.5, 1.25) { \tiny Q\textsuperscript{b,G} };
        \node[white] at (5.5, 0) { \tiny Q\textsuperscript{b,F} };
        \node[white] at (7, 0) { \tiny Q\textsuperscript{a,F} };
        \node[white] at (7, 1) { \tiny Q\textsuperscript{a,G} };
\end{tikzpicture}
\caption{Summary of the national order books in a shared order book, provided the national markets are coupled.}
\label{CB:fig:cbMarket}
\end{figure}

Next, we specify how incoming orders change the state of the active dynamics. Therefore, let us denote by $(\tilde{\tau}^{(n)}_l)_{l\geq 1}$ the sequences of stopping times at which we observe a price change in $(\tilde{S}^{(n)}_k)_{k= 0,\cdots, T_n}.$ Further, we introduce the sequences of random variables representing the order sizes after a price change, for $l\geq 1,$ by
$$\tilde{R}^{+,(n)}_l := \Phi^{(n)}\left(\tilde{Q}^{(n)}_{\lfloor\tilde{\tau}^{(n)}_l-/\Delta t^{(n)}\rfloor}, \epsilon^{+,(n)}_l\right), \quad \tilde{R}^{-,(n)}_l := \Phi^{(n)}\left(\tilde{Q}^{(n)}_{\lfloor\tilde{\tau}^{(n)}_l-/\Delta t^{(n)}\rfloor}, \epsilon^{-,(n)}_l\right).$$
Let $l(k)$ denote the number of price changes after $k$ order events in $(\tilde{S}^{(n)}_k)_{k=0,\cdots,T_n}$. Then, $(\tilde{S}^{(n)}_k)_{k = 0,\cdots, T_n}$ evolves as follows: let $\tilde{S}^{(n)}_0 \in (\delta\mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\mathbb{Z} \subset E$ with $\tilde{B}^{F,(n)}_0 = \tilde{B}^{G,(n)}_0$ be the deterministic initial state and denote by $\tilde{Q}^{i,(n)}_k := \tilde{Q}^{i,F,(n)}_k + \tilde{Q}^{i,G,(n)}_k$ the cumulative queue of type $i = b,a$ after $k \geq 1$ order events. If the $k$-th incoming order is of type $(b,F),$ then
    \begin{equation}\label{CB:def:AD1}
    \begin{split}
       &\hspace{-0.2cm}\tilde{S}^{(n)}_{k} = \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (V^{(n)}_{k}, 0, 0, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F,(n)}_{k-1} \geq - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, +  \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, V^{(n)}_{k}, 0), V^{(n)}_{k}\right)\right) \textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F,(n)}_{k-1} < - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} - (\delta, \delta), \tilde{R}^{-,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F,(n)}_{k-1} = -V^{(n)}_k\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} = 0\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} - (\delta, \delta), \tilde{R}^{-,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1} + V^{(n)}_{k}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F,(n)}_{k-1} = 0\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} = -V^{(n)}_{k}\right\}}.
        \end{split}
        \end{equation}
If the $k$-th incoming order is of type $(a,F),$ then
     \begin{equation}\label{CB:def:AD2}
    \begin{split}
        &\hspace{-0.2cm}\tilde{S}^{(n)}_{k} = \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, V^{(n)}_{k}, 0, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} \geq - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, 0,  V^{(n)}_{k}), -V^{(n)}_{k}\right)\right) \textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} < - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} + (\delta, \delta), \tilde{R}^{+,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} = -V^{(n)}_k\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} = 0\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} + (\delta, \delta), \tilde{R}^{+,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1} - V^{(n)}_{k}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} = 0\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} = -V^{(n)}_{k}\right\}}.
   \end{split}
   \end{equation}
If the $k$-th incoming order is of type $(b,G),$ then
    \begin{equation}\label{CB:def:AD3}
    \begin{split}
       &\hspace{-0.2cm}\tilde{S}^{(n)}_{k} = \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, V^{(n)}_{k}, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} \geq - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (V^{(n)}_{k}, 0, 0, 0), -V^{(n)}_{k}\right)\right) \textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} < - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} - (\delta, \delta), \tilde{R}^{-,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} = -V^{(n)}_k\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F,(n)}_{k-1} = 0\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} - (\delta, \delta), \tilde{R}^{-,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1} - V^{(n)}_{k}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,G,(n)}_{k-1} = 0\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{b,F),(n)}_{k-1} = -V^{(n)}_{k}\right\}}.
    \end{split}
    \end{equation}
If the $k$-th incoming order is of type $(a,G),$ then
     \begin{align}\label{CB:def:AD4}
    \begin{aligned}
        &\hspace{-0.2cm}\tilde{S}^{(n)}_{k} = \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, 0, V^{(n)}_{k}), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} \geq - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{S}^{(n)}_{k-1} + \left((0, 0), (0, V^{(n)}_{k}, 0, 0), V^{(n)}_{k}\right)\right) \textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} < - V^{(n)}_{k}\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,(n)}_{k-1} > -V^{(n)}_{k}\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} + (\delta, \delta), \tilde{R}^{+,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} = -V^{(n)}_k\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} = 0\right\}}\\
        &\, + \left(\tilde{B}^{(n)}_{k-1} + (\delta, \delta), \tilde{R}^{+,(n)}_{l(k)}, \tilde{C}^{(n)}_{k-1} + V^{(n)}_{k}\right)\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,G,(n)}_{k-1} = 0\right\}}\textup{$\mathds{1}$}_{\left\{\tilde{Q}^{a,F,(n)}_{k-1} = -V^{(n)}_{k}\right\}}.
    \end{aligned}
    \end{align}

\begin{Rmk}Let the $k$-th incoming order be of type $(i,I)\in \{b,a\} \times \{F,G\}.$ Let $J = \{F,G\} \setminus I$ be the index corresponding to the neighboring country of the $k$-th order. As described in \eqref{CB:def:AD1}-\eqref{CB:def:AD4}, the active dynamics change as follows:
    \begin{enumerate}
        \item[i)] Let the $k$-th incoming order be a limit order, i.e., $V^{i,I,(n)}_k = \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}.$ Then, its size is added to the national queue $\tilde{Q}^{i,I,(n)}_{k-1}$, i.e., $\tilde{Q}^{i,I,(n)}_k = \tilde{Q}^{i,I,(n)}_{k-1} + V^{i,I,(n)}_k.$
        \item[ii)] Let the $k$-th incoming order be a market order, i.e., $V^{i,I,(n)}_k = -\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}$:
        \begin{itemize}
        		\item If the incoming order can be matched against the national queue $\tilde{Q}^{i,I,(n)}_{k-1}$ while its cumulative queue $\tilde{Q}^{i,(n)}_{k-1}$ would not be depleted by the size of the incoming order, we reduce the national queue $\tilde{Q}^{i,I,(n)}_{k-1}$ by the size of the incoming market order, i.e., $\tilde{Q}^{i,I,(n)}_k = \tilde{Q}^{i,I,(n)}_{k-1} + V^{i,I,(n)}_k$. In particular, we allow that $\tilde{Q}^{i,I,(n)}_{k-1} + V^{i,I,(n)}_k = 0$ provided that the corresponding cumulative queue is strictly greater than zero.
        		 \item If the national queue satisfies $\tilde{Q}^{i,I,(n)}_{k-1} = 0$ while its cumulative queue $\tilde{Q}^{i,(n)}_{k-1}$ would not be depleted by the size of the incoming order, we reduce the foreign queue $\tilde{Q}^{i,J,(n)}_{k-1}$ by the size of the incoming market order, i.e., $\tilde{Q}^{i,J,(n)}_k = \tilde{Q}^{i,J,(n)}_{k-1} + V^{i,I,(n)}_k.$ This yields a cross-border trade and therefore changes the state of the capacity process by $\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}.$
        \item If its cumulative queue $\tilde{Q}^{i,(n)}_{k-1}$ would be depleted by the size of the incoming order, all order queues are reinitialized by either the random variable $\tilde{R}^{+,(n)}_{l(k)}$ or $\tilde{R}^{-,(n)}_{l(k)}$ and both national bid price processes change by one tick in the same direction. Depending on whether the national queue $\tilde{Q}^{i,I,(n)}_{k-1} = 0$ or not, we also change the state of the capacity process by $\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}.$
\end{itemize}
    \end{enumerate}
\end{Rmk}

The price processes in \eqref{CB:def:AD1}--\eqref{CB:def:AD4} have increments of maximum length $\delta > 0$ and hence describe the evolution of the prices of limit order books which contain no gaps (empty levels). If there were gaps in the order books, this would result in jumps of more than one tick in the price dynamics. For convenience, we ignore the feature of price jumps larger than one tick. Note that the assumption that price changing events increase respectively decrease the prices only by a single tick is not unrealistic. It has been show in an empirical study (cf. e.g. \cite{FGLMS04}), that around $85 \%$ of the sell market orders which lead to price changes match exactly the size of the standing volumes at the best ask price. \par
Moreover, as described in \eqref{CB:def:AD1}-\eqref{CB:def:AD4}, the dynamics of the two-sided capacity process over time equals
\begin{equation}\label{CB:def:capacity1}
\tilde{C}^{(n)}_k = M^{b,G,(n)}_k + M^{a,F,(n)}_k - M^{b,F,(n)}_k - M^{a,G,(n)}_k,
\end{equation}
where for each $(i,I) \in \{b,a\} \times \{F,G\}$ and $M^{i,I,(n)}_0 = 0$ we define by
\begin{equation}\label{CB:def:sNoCB}
M_k^{i,I,(n)}:= \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \sum_{j=1}^k \textup{$\mathds{1}$}_{\left\{\tilde{Q}^{i,I,(n)}_{j-1} = 0\right\}} \textup{$\mathds{1}$}_{\left\{V^{i,I,(n)}_j = -\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\right\}}, \quad k = 1,\cdots, T_n
\end{equation}
the (scaled) number of cross-border trades triggered by events of type $(i,I).$  In the following, we set $M^{(n)}_k := \left(M^{b,F,(n)}_k, M^{a,F,(n)}_k, M^{b,G,(n)}_k, M^{a,G,(n)}_k\right)$ for $k = 0,\cdots, T_n.$

\begin{Rmk}
    The definition of the capacity process corresponds to an efficient cross-border trading assumption and therefore describes the net number of cross-border trades. In more detail, we understand cross-border trading to be \textit{efficient} if the following is satisfied:
\begin{enumerate}
        \item[i)] The goods are actual transmitted at the end of the trading session.
        \item[ii)] Only the difference of imports and exports are transmitted cross-border while all other goods are distributed through domestic trading.
    \end{enumerate}
The idea of efficient cross-border trading is presented in Figure \ref{CB:fig:effTrading}.
\end{Rmk}

\begin{figure}[H]
\centering
    \begin{tikzpicture}
        \fill[CMorange] (0.4,-0.6) rectangle (4.5, 2.7);
        \fill[CMblue] (4, -0.6) rectangle (7.6, 2.7);
        \draw[dashed, ->]
            (1.4, 0.4) -- (6.6, 0.4) node[fill=white, draw, ellipse, pos=.6, sloped] {\tiny $x$};
        \draw[dashed, <-]
            (1.4, 2) -- (6.6, 2) node[fill=white, draw, ellipse, pos=.5, sloped] {\tiny $y$};
        \draw[very thick, ->]
            (1.4, 0) -- (6.6, 0) node[fill=white, draw, ellipse, pos=.4] {\tiny $x-y$};
        \draw[very thick, ->]
            (1, 0.4) -- (1, 1.6);
        \draw[very thick, <-]
            (7, 0.4) -- (7, 1.6);
         \node at (1,0) { $F_1$ };
         \node at (1,2) { $F_2$ };
         \node at (7,0) { $G_1$ };
         \node at (7,2) { $G_2$ };
         \node[fill=white, draw, ellipse] at (7,1) {\tiny $y$};
         \node[fill=white, draw, ellipse] at (1, 1) {\tiny $y$};
    \end{tikzpicture}
	\caption{Efficient cross-border trading between different traders $F_1, F_2$ from $F$ and $G_1, G_2$ from $G$ and order sizes $x,y > 0$ with $x > y$. Only the difference $x-y$ is actually transmitted cross-border.}
	\label{CB:fig:effTrading}
\end{figure}

\subsection{Description of the inactive dynamics}\label{CB:sec:defID}

In this subsection, we introduce for each $n\in \mathbb{N}$ the inactive dynamics given by the piecewise constant interpolation
\[\dbtilde{S}^{(n)}(t) = \sum_{k=0}^{T_n} \dbtilde{S}^{(n)}_k \textup{$\mathds{1}$}_{\left[\t_k, \t_{k+1}\right)}(t), \quad t\in [0,T],\]
of the $E$-valued random variables
\[\dbtilde{S}^{(n)}_k := \left(\dbtilde{B}^{(n)}_k, \dbtilde{Q}^{(n)}_k, \dbtilde{C}^{(n)}_k\right), \quad k\in \mathbb{N}_0,\]
where $\dbtilde{B}^{(n)}_k$ denotes the bid prices of $F$ and $G,$ $\dbtilde{Q}^{(n)}_k$ denotes the sizes of the best bid respectively ask queues in $F$ and $G$, and $\dbtilde{C}^{(n)}_k$ denotes the net number of cross-border trades after $k$ order events. Since the national order books are decoupled in the inactive dynamics, we generally differentiate the best bid prices of $F$ and $G$. Hence, we cannot summarize the national LOBs in a shared order book and the depletion of a single national order queue already causes a price change in the corresponding national LOB. Finally, since no cross-border trades are possible due to the fact that the order books are decoupled, the capacity process $(\dbtilde{C}^{(n)}_k)_{k= 0,\cdots,T_n}$ is constant for the whole trading period, i.e., $\dbtilde{C}^{(n)}_k = \dbtilde{C}^{(n)}_0$ for all $k = 0,\cdots, T_n$ and $n\in \mathbb{N}$. \par
Next, let us specify how incoming order events change the state of the inactive dynamics. Therefore, let us denote by $(\dbtilde{\tau}^{(n)}_l)_{l\geq 1}$ the sequences of stopping times at which we observe a price change at either the best bid price of $F$ or $G$ in $(\dbtilde{S}^{(n)}_k)_{k=0,\cdots,T_n}.$ Further, we introduce the sequences of random variables representing the order sizes after a price change, for $l\geq 1,$ by
\begin{equation*}
    \dbtilde{R}^{+,(n)}_l := \Phi^{(n)}\left(\dbtilde{Q}^{(n)}_{\lfloor\dbtilde{\tau}^{(n)}_l-/\Delta t^{(n)}\rfloor}, \epsilon^{+,(n)}_l\right), \quad \dbtilde{R}^{-,(n)}_l := \Phi^{(n)}\left(\dbtilde{Q}^{(n)}_{\lfloor\dbtilde{\tau}^{(n)}_l-/\Delta t^{(n)}\rfloor}, \epsilon^{-,(n)}_l\right).
\end{equation*}
Again, let $l(k)$ denote the number of price changes after $k$ order events in $(\dbtilde{S}^{(n)}_k)_{k = 0,\cdots, T_n}$. Then, $(\dbtilde{S}^{(n)}_k)_{k=0,\cdots,T_n}$ evolves as follows: let $\dbtilde{S}^{(n)}_0 \in (\delta Z)^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\mathbb{Z} \subset E$ be the deterministic initial state. If the $k$-th incoming order is of type $(b,F),$ then
    \begin{equation}\label{CB:def:ID1}
    \begin{split}
        \dbtilde{S}^{(n)}_{k} =& \left(\dbtilde{S}^{(n)}_{k-1} + \left((0, 0), (V^{(n)}_{k}, 0, 0, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{b,F,(n)}_{k-1} > - V^{(n)}_{k}\right\}}\\
        &+ \left(\dbtilde{B}^{(n)}_{k-1} - (\delta, 0), \pi_{F} \dbtilde{R}^{-,(n)}_{l(k)}, \pi_{G}\dbtilde{Q}^{(n)}_{k-1}, \dbtilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{b,F,(n)}_{k-1} \leq  -V^{(n)}_{k}\right\}}.
    \end{split}
    \end{equation}
If the $k$-th incoming order is of type $(a,F),$ then
    \begin{equation}\label{CB:def:ID2}
    \begin{split}
        \dbtilde{S}^{(n)}_{k} =& \left(\dbtilde{S}^{(n)}_{k-1} + \left((0, 0), (0, V^{(n)}_{k}, 0, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{a,F,(n)}_{k-1} > - V^{(n)}_{k}\right\}}\\
        &+ \left(\dbtilde{B}^{(n)}_{k-1} + (\delta, 0), \pi_{F}\dbtilde{R}^{+,(n)}_{l(k)}, \pi_{G}\dbtilde{Q}^{(n)}_{k-1}, \dbtilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{a,F,(n)}_{k-1} \leq -V^{(n)}_{k} \right\}}.
    \end{split}
    \end{equation}
If the $k$-th incoming order is of type $(b,G),$ then
    \begin{equation}\label{CB:def:ID3}
    \begin{split}
        \dbtilde{S}^{(n)}_{k} =& \left(\dbtilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, V^{(n)}_{k}, 0), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{b,G,(n)}_{k-1} > - V^{(n)}_{k}\right\}}\\
        &+ \left(\dbtilde{B}^{(n)}_{k-1} - (0, \delta), \pi_{F}\dbtilde{Q}^{(n)}_{k-1}, \pi_{G}\dbtilde{R}^{-,(n)}_{l(k)}, \dbtilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{b,G,(n)}_{k-1} \leq -V^{(n)}_{k}\right\}}.
    \end{split}
    \end{equation}
If the $k$-th incoming order is of type $(a,G),$ then
     \begin{equation}\label{CB:def:ID4}
    \begin{split}
        \dbtilde{S}^{(n)}_{k} =& \left(\dbtilde{S}^{(n)}_{k-1} + \left((0, 0), (0, 0, 0, V^{(n)}_{k}), 0\right)\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{a,G,(n)}_{k-1} > - V^{(n)}_{k}\right\}}\\
        &+ \left(\dbtilde{B}^{(n)}_{k-1} + (0, \delta), \pi_{F}\dbtilde{Q}^{(n)}_{k-1}, \pi_{G}\dbtilde{R}^{+,(n)}_{l(k)}, \dbtilde{C}^{(n)}_{k-1}\right)\textup{$\mathds{1}$}_{\left\{\dbtilde{Q}^{a,G,(n)}_{k-1} \leq - V^{(n)}_{k}\right\}}.
    \end{split}
    \end{equation}

\begin{Rmk}
    The above description of inactive dynamics is a straight-forward extension of the discrete-time dynamics in Cont and de Larrard \cite{CL21} to two non-interacting LOBs.
\end{Rmk}

\subsection{The cross-border market dynamics as a regime switching process} \label{CB:sec:CBMD}

In this subsection, we finally introduce the microscopic dynamics of our cross-border market model $S^{(n)}$ which can be interpreted as a regime switching process switching between the active and inactive regimes.

\begin{Rmk}[Preliminary considerations]\label{CB:rmk:preCons}
	$ $
	\begin{itemize}
		\item[i)] While the capacity process $\tilde{C}^{(n)}$ is unbounded and changes over time, $\dbtilde{C}^{(n)}$ stays constant. The bounded capacity process $C^{(n)}$ will take its values in $[-\kappa_-, \kappa_+]$ and its dynamics will develop similarly to $\tilde{C}^{(n)}$ during active regimes and will stay constant and equals either $\kappa_-$ or $\kappa_+$ during inactive regimes. Moreover, we will use the unbounded capacity process $\tilde{C}^{(n)}$ as an indicator to switch from an active to an inactive regime.
		\item[ii)] The bid price processes of the national LOBs of the active dynamics $\tilde{S}^{(n)}$ coincide. In contrast, the price processes of the inactive dynamics $\dbtilde{S}^{(n)}$ develop as non-identical pure-jump processes. In order to switch back to an active regime, the price processes during inactive regimes have to coincide.
		\item[iii)] It is not clear what happens if the capacity process $C^{(n)}$ is equal to one of its boundary values $\{-\kappa_-, \kappa_+\}$ while the best bid prices coincide as the limit order books might be coupled or decoupled. This situation can be interpreted as a \textit{partially active} regime in which cross-border trades are possible but only in one direction. Instead of modeling a third regime, we include the partially active regimes into the active or inactive regimes. This is possible since the active respectively inactive dynamics develop equally as long as no order queue has been depleted. In a partially active regime, depending on the type of the incoming market order, we might stay in the current regime or switch to the next regime. Note that before switching from an inactive to an active regime, we always are in a partially active regime (which will be included to the inactive regimes). In contrast, we might directly switch from an active to an inactive regime without previously being in a partially active regime (cf. Figure \ref{CB:fig:ideaCBM}).
	\end{itemize}
\end{Rmk}

\begin{figure}[H]
     \centering
     \begin{tikzpicture}
            \fill[color = CMblue!20!white]
                (0.1,0) rectangle (2,3)
                (7, 0) rectangle (9.7,3);
            \fill[color = CMorange!20!white]
                (2,0) rectangle (7, 3)
                (9.7, 0) rectangle (10.7, 3);
            \draw[pattern=north west lines]
                (1.5, 0) rectangle (2,3);
            \draw[pattern=north east lines]
                (3.5, 0) rectangle (4,3)
                (6,0) rectangle (7,3);
            \draw[very thick, ->]
                (0,0) -- (11.3,0);
            \draw[very thick]
            		(0.05,0)--(0.05,3)
                (0.1, 0) -- (0.1, 3)
                (2,0) -- (2,3)
                (7, 0) -- (7,3)
                (9.7, 0) -- (9.7,3);
            \draw[dashed]
                (1.5, 0) -- (1.5,3)
                (3.5, 0) -- (3.5, 3)
                (4,0) -- (4,3)
                (6,0) -- (6,3);
            \draw
                (0.05,3) -- (10.7, 3);
            \draw[thick, decorate, decoration={brace,amplitude=6pt}] (2, -0.1) -- (0.1, -0.1) node[midway, below, yshift=-6pt,]{\tiny active regime};
            \draw[thick, decorate, decoration={brace,amplitude=6pt}] (7, -0.1) -- (2, -0.1) node[midway, below,yshift=-6pt,]{\tiny inactive regime};
            \draw[thick, decorate, decoration={brace,amplitude=6pt}] (9.7, -0.1) -- (7, -0.1) node[midway, below,yshift=-6pt,]{\tiny active regime};
             \draw[thick, decorate, decoration={brace,amplitude=6pt}] (10.7, -0.1) -- (9.7, -0.1) node[midway, below,yshift=-6pt,]{\tiny inactive regime};
            \node at (0, -0.3) {\tiny 0};
            \node at (11.3, -0.3) {\tiny time };
     \end{tikzpicture}
     \caption{Incorporation of the partially active regime. The dynamics of the cross-border market develop in the partially active regime as in the active (patterned, turquoise) or inactive regime (patterned, orange).}
     \label{CB:fig:ideaCBM}
\end{figure}

A short consideration of the definition of the active dynamics in \eqref{CB:def:AD1}-\eqref{CB:def:AD4} reveals, that a single LOB is allowed to be ``empty'', i.e., might have no standing volumes at the best bid and ask queues, provided that the order queues of the foreign LOB are both strictly larger than zero. Then, we may run into problems if we switch to the next inactive regime as we cannot uniquely determine the direction of the next price change for the empty LOB. In order to bypass this issue, we assume that the probability space $(\Omega, \mathcal{F}, \ensuremath\mathbb{P})$ is rich enough to support a sequence of iid Bernoulli random variables $(U^{(n)}_k)_{k,n \geq 1}$, where $U^{(n)}_k \in \{-1, 1\}$. \par
Next, let us introduce short-hand notations for the order type indicator random variables at time $t \in [0,T]$ by
\[\textup{$\mathds{1}$}^{(n)}_{(i,I)}(t) := \textup{$\mathds{1}$}_{(i,I)}\left(\phi^{(n)}_{\lfloor t/\tn \rfloor}, \psi^{(n)}_{\lfloor t/\tn \rfloor}\right) \hspace{0.3cm} \text{and} \hspace{0.3cm} \textup{$\mathds{1}$}^{(n)}(t) := \left(\textup{$\mathds{1}$}^{(n)}_{(b,F)}, \textup{$\mathds{1}$}^{(n)}_{(a,F)}, \textup{$\mathds{1}$}^{(n)}_{(b,G)}, \textup{$\mathds{1}$}^{(n)}_{(a,G)}\right)(t).\]
In order to describe the state of the cross-border market model at the start of an inactive regime, we introduce the a sequence of processes $\tilde{Z}^{(n)}_k := (\tilde{Z}^{F,(n)}_k, \tilde{Z}^{G,(n)}_k)$ taking values in $\{-1,0,1\}^2$ by
\begin{equation}\label{CB:def:Zn}
\begin{split}
    \tilde{Z}_k^{I,(n)}(t) := \begin{cases}
    0, &\text{ if } \pi_IQ^{(n)}(t-) + V^{(n)}_{\lfloor t/\tn \rfloor}\left(\textup{$\mathds{1}$}^{(n)}_{(b,I)}(t), \textup{$\mathds{1}$}^{(n)}_{(a,I)}(t)\right) \in (0,\infty)^2\\
    -1, &\text{ if } \pi_IQ^{(n)}(t-) + V^{(n)}_{\lfloor t/\tn \rfloor}\left(\textup{$\mathds{1}$}^{(n)}_{(b,I)}(t), \textup{$\mathds{1}$}^{(n)}_{(a,I)}(t)\right) \in \mathbb{R}_- \times (0,\infty)\\
    1, &\text{ if } \pi_IQ^{(n)}(t-) + V^{(n)}_{\lfloor t/\tn \rfloor}\left(\textup{$\mathds{1}$}^{(n)}_{(b,I)}(t), \textup{$\mathds{1}$}^{(n)}_{(a,I)}(t)\right) \in (0,\infty)\times \mathbb{R}_-\\
    U^{(n)}_k, &\text{ if } \pi_IQ^{(n)}(t-) + V^{(n)}_{\lfloor t/\tn \rfloor}\left(\textup{$\mathds{1}$}^{(n)}_{(b,I)}(t), \textup{$\mathds{1}$}^{(n)}_{(a,I)}(t)\right) \in \mathbb{R}_-^2
    \end{cases}
\end{split}
\end{equation}
for all $t\in [0,T]$ and $I = F,G$. Moreover, we introduce the function
\[h: D([0,T], \mathbb{R}^4) \rightarrow D([0,T], \mathbb{R}^2), \quad h: \omega \mapsto \left(\pi_1\omega + \pi_3 \omega, \pi_2\omega +\pi_4\omega\right).\]
Note that for all $t \in [0,T]$ and $Q^{(n)}(t)$ denoting the sizes of the best bid and ask queues of $F$ respectively $G$ at time $\lfloor t/\tn \rfloor$, $h(Q^{(n)})(t)$ describes the sizes of the cumulative best bid and ask queues at time $\lfloor t/\tn \rfloor.$

Recall that we denote by $(R^{+,(n)}_l)_{l\geq 1},$ $(R^{-,(n)}_l)_{l\geq 1}$ introduced in \eqref{CB:def:R-n} the order sizes after a price increase respectively decrease, by $(\tau^{(n)}_l)_{l\geq 1}$ the sequence of stopping times at which we observe a price change in $S^{(n)},$ and by $l^{(n)}(t)$ the number of price changes in $S^{(n)}$ in $[0,t],$ for all $t\in [0,T].$ Last, for all $s\in [0,T]$, let us introduce the short-hand notations $\tilde{S}^{(n),s}$ and $\dbtilde{S}^{(n),s}$ denoting the active respectively inactive dynamics starting in the state $S^{(n)}(s) \in E.$  \par

With all this preparation done, we are finally able to introduce the cross-border market dynamics as a regime switching process.

\begin{Def}[Cross-border market dynamics $S^{(n)}$]\label{CB:def:Sn}
    Let $n\in \mathbb{N},$ $S^{(n)}_0 \in (\delta \mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}\mathbb{N})^4 \times \{0\}$ with $B^{F,(n)}_0 = B^{G,(n)}_0,$  $\kappa_+, \kappa_- > 0,$ $\rho^{(n)}_0 := 0,$ and $k\geq 1.$ The cross-border market dynamics are given as follows:
    \begin{itemize}[leftmargin = 0.5cm]
	 \setlength\itemsep{0em}
        \item The active regime: for $0 \leq t < \sigma^{(n)}_k - \rho^{(n)}_{k-1},$ we set
        $$S^{(n)}\left(t+ \rho^{(n)}_{k-1}\right) := \tilde{S}^{(n), \rho^{(n)}_{k-1}}(t),$$
        where $\sigma^{(n)}_k := \sigma^{Im,(n)}_k \wedge \sigma^{Ex,(n)}_k$ determines the start of the next inactive regime and
        \begin{align*}
                \sigma^{Im,(n)}_k &:= \inf\Big\{t \geq \rho^{(n)}_{k-1}: C^{(n)}(t) = \kappa_+ \text{ and } \exists \,(i,I) \in \mathcal{I}^{Im} \text{ with } h(Q^{(n)})(t-) + \\
                &\hspace{1.3cm} V^{(n)}_{\lfloor t/\tn \rfloor}h(\textup{$\mathds{1}$}^{(n)})(t) \in (0,\infty)^2 \text{ and } Q^{i,I,(n)}(t-)+ V^{i,I,(n)}_{\lfloor t/\tn \rfloor} \leq 0\Big\} \wedge T,\\
                \sigma^{Ex,(n)}_k &:= \inf\Big\{t \geq \rho^{(n)}_{k-1}: C^{(n)}(t) = -\kappa_- \text{ and } \exists \, (i,I) \in \mathcal{I}^{Ex} \text{ with } h(Q^{(n)})(t-) +\\
                &\hspace{1.3cm} V^{(n)}_{\lfloor t/\tn \rfloor}h(\textup{$\mathds{1}$}^{(n)})(t) \in (0,\infty)^2 \text{ and } Q^{i,I,(n)}(t-)+ V^{i,I,(n)}_{\lfloor t/\tn \rfloor} \leq 0\Big\} \wedge T.
        \end{align*}
        \item Starting value of the next inactive regime: we set $C^{(n)}(\sigma^{(n)}_k) := \tilde{C}^{(n), \rho^{(n)}_{k-1}}(\sigma^{(n)}_k).$ Further, for $I  = F,G$ and $\tilde{Z}^{I,(n)}_k(\sigma^{(n)}_k) \in \{-1,0,1\}$ being introduced in \eqref{CB:def:Zn}, we set
        \begin{align*}
            \pi_IQ^{(n)}(\sigma^{(n)}_k) &:= \pi_I\tilde{Q}^{(n), \rho^{(n)}_{k-1}}(\sigma^{(n)}_k) \textup{$\mathds{1}$}_{\left\{\tilde{Z}_k^{I,(n)}(\sigma^{(n)}_k)= 0\right\}}  + \pi_IR^{+,(n)}_{l^{(n)}(\sigma^{(n)}_k-)+1} \textup{$\mathds{1}$}_{\left\{\tilde{Z}_k^{I,(n)}(\sigma^{(n)}_k )= 1\right\}}\\
            &\qquad +\pi_I R^{-,(n)}_{l^{(n)}(\sigma^{(n)}_k-)+1}\textup{$\mathds{1}$}_{\left\{\tilde{Z}_k^{I,(n)}(\sigma^{(n)}_k) = -1\right\}}
        \end{align*}
        and
        \begin{align*}
            B^{I,(n)}(\sigma^{(n)}_k) := B^{I,(n)}(\sigma^{(n)}_k-) + \delta\left(\textup{$\mathds{1}$}_{\left\{\tilde{Z}_k^{I,(n)}(\sigma^{(n)}_k) = 1\right\}} - \textup{$\mathds{1}$}_{\left\{\tilde{Z}_k^{I,(n)}(\sigma^{(n)}_k) = -1\right\}}\right).
        \end{align*}
        \item The inactive regime: for $0 \leq t < \rho^{(n)}_k - \sigma^{(n)}_k,$ we set
        $$S^{(n)}\left(t + \sigma^{(n)}_k\right) := \dbtilde{S}^{(n), \sigma^{(n)}_k}(t),$$
        where $\rho^{(n)}_k := \rho^{Im,(n)}_k\textup{$\mathds{1}$}_{\left\{C^{(n)}(\sigma^{(n)}_k) = -\kappa_-\right\}} + \rho^{Ex,(n)}_k\textup{$\mathds{1}$}_{\left\{C^{(n)}(\sigma^{(n)}_k) = \kappa_+\right\}}$ determines the start of the next active regime and
        \begin{align*}
            \rho^{Im,(n)}_k &:= \inf\Big\{t \geq \sigma^{(n)}_k: \left|B^{F,(n)}(t) - B^{G,(n)}(t)\right| = 0 \text{ and }\\
            &\hspace{3cm} \exists \, (i,I) \in \mathcal{I}^{Im} \text{ with } Q^{i,I,(n)}(t-) + V^{i,I,(n)}_{\lfloor t/\tn \rfloor} \leq 0\Big\}\wedge T,\\
            \rho^{Ex,(n)}_k &:= \inf\Big\{t \geq \sigma^{(n)}_k: \left|B^{F,(n)}(t) - B^{G,(n)}(t)\right|  = 0 \text{ and }\\
            &\hspace{3cm} \exists \, (i,I) \in \mathcal{I}^{Ex} \text{ with } Q^{i,I,(n)}(t-) + V^{i,I,(n)}_{\lfloor t/\tn \rfloor} \leq 0\Big\}\wedge T.
        \end{align*}
        \item Starting value of the next active regime: we set  $C^{(n)}(\rho^{(n)}_k) := C^{(n)}(\rho^{(n)}_k -),$ $B^{(n)}(\rho^{(n)}_k) := B^{(n)}(\rho^{(n)}_k-),$ and for $(i,I) \in \{b,a\}\times \{F,G\},$
        \[Q^{i,I,(n)}(\rho^{(n)}_k) := Q^{i,I,(n)}(\rho^{(n)}_k-) + V^{i,I,(n)}_{\lfloor \rho^{(n)}_k/\Delta t^{(n)}\rfloor}.\]
    \end{itemize}
\end{Def}

\begin{Rmk}$ $
    \begin{enumerate}
        \item[i)] The starting times of the inactive regimes being determined by the stopping times $(\sigma^{(n)}_k)_{k\geq 1}$ may look surprisingly complex. We wish to switch to an inactive regime if the capacity process hits one of its boundary values. Since we also include the partially active regime partly into the active regime (cf. Remark \ref{CB:rmk:preCons}), the stopping times have to modified appropriately. Note, that these stopping times simplify in the high-frequency limit (cf. Theorem \ref{CB:res:mainTheorem} below) in which the partially active regimes in the active regimes disappear.
        \item[ii)] The start of next active regime depends on the direction in which the transmission capacity has been occupied. After the national best bid prices coincide again and a national order queue would be depleted by an incoming market order allowing for cross-border trades in the non-occupied direction, we immediately switch back to an active regime and couple the national LOBs. We note, that this indeed prevents that the national bid prices diverge again (cf. the definition of the starting value of the next active regime).
    \end{enumerate}
\end{Rmk}

\begin{Rmk}[Distribution of order sizes might depend on the type of the regime]
	In our model, a regime switch only changes the order matching mechanism (cf. Definition \ref{CB:def:Sn}). However, it might be of interest to also allow a change in the trading behavior. This can be incorporated to our model by slight generalizations of Assumptions \ref{CB:ass:prob} and \ref{CB:ass:R}: one might assume that the distribution of the order sizes as well as of the queue sizes after price changes depend on the type of the current regime. For sake of notation, we will not include this to our model.
\end{Rmk}

\section{Analysis of the active dynamics}\label{CB:sec:anaAD}

In this section, we analyze the active market dynamics $\tilde{S}^{(n)} = (\tilde{S}^{(n)}(t))_{t\in [0,T]}$ and derive its heavy traffic approximation. Recall, that
$$\tilde{S}^{(n)}(t) = \tilde{S}^{(n)}_k \quad \text{ for } t \in \left[\t_k, \t_{k+1}\right),$$
where the discrete-time dynamics are defined in equations \eqref{CB:def:AD1}-\eqref{CB:def:AD4}. Let $\tilde{S}^{(n)}_0 := (\tilde{B}^{(n)}_0, \tilde{Q}^{(n)}_0, \tilde{C}^{(n)}_0) \in (\delta \mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{Z}$ be the initial value of $\tilde{S}^{(n)}$ with $\tilde{B}^{F,(n)}_0 = \tilde{B}^{G,(n)}_0.$ In order to derive a limit theorem for the active dynamics, we need to introduce the so-called net order flow process $X^{(n)}$ by
\begin{align}\label{CB:def:netorderflow}
X^{(n)} := \left(X^{b,F,(n)}, X^{a,F,(n)}, X^{b,G,(n)}, X^{a,G,(n)}\right),
\end{align}
where for $(i,I) \in \{b,a\} \times \{F,G\}$ and $t \in [0,T]$ we have
$$X^{i,I,(n)}(t): = \sum_{k=1}^{T_n} X^{i,I,(n)}_k \textup{$\mathds{1}$}_{\left[\t_k, \t_{k+1}\right)}(t) \quad \text{and} \quad X^{i,I,(n)}_k := \sum^k_{j=1} V_j^{i,I,(n)}.$$

\begin{Prop}[Functional central limit theorem for the net order flow process]\label{CB:res:fCLT}
	Let Assumption \ref{CB:ass:prob} and \ref{CB:ass:scaling} be satisfied. Then, the net order flow process $X^{(n)}$ converges weakly in the Skorokhod topology on $D([0,T], \mathbb{R}^4)$ to a four-dimensional linear Brownian motion, i.e.,
	\begin{align}\label{CB:eq:convBM}
	X^{(n)} \Rightarrow X := \left(\Sigma^{1/2} B(t) + t \mu\right)_{t\geq 0},
	\end{align}
where $B$ is a standard four-dimensional Brownian motion and
\begin{align*}
\mu := \left(\begin{array}{c}
\mu^{b,F}\\
\mu^{a,F}\\
\mu^{b,G}\\
\mu^{a,G}\end{array}\right),\quad
\Sigma := \left(\begin{array}{cccc}
\left(\sigma^{b,F}\right)^2 & \sigma^{(b,F),(a,F)} & \sigma^{(b,F),(b,G)} & \sigma^{(b,F),(a,G)}\\
\sigma^{(b,F),(a,F)} & \left(\sigma^{a,F}\right)^2 & \sigma^{(b,F),(a,G)} & \sigma^{(a,F),(a,G)}\\
\sigma^{(b,F),(b,G)} & \sigma^{(b,F),(a,G)} & \left(\sigma^{b,G}\right)^2 & \sigma^{(b,G),(a,G)}\\
\sigma^{(b,F),(a,G)} & \sigma^{(a,F),(a,G)} & \sigma^{(b,G),(a,G)} &\left(\sigma^{a,G}\right)^2
\end{array}\right).
\end{align*}
\end{Prop}

The proof is given in Appendix \ref{CB:app:fCLT}. Based on this limit theorem for the net order flow process, we establish convergence results for the queue size process $\tilde{Q}^{(n)},$ the capacity process $\tilde{C}^{(n)},$ and the price process $\tilde{B}^{(n)}.$
To this end, in the subsequent subsections, we rewrite the queue size, the price, and the capacity process in terms of the net order flow process $X^{(n)}.$

\subsection{The queue size process as a regulated process}\label{CB:sec:regProQ}

In the following, we rewrite the queue size process $\tilde{Q}^{(n)}$ in terms of transformations of the net order flow $X^{(n)}$ and the sequences of random variables $\tilde{R}^{+,(n)}:= (\tilde{R}^{+,(n)}_k)_{k\geq 1}$ and $\tilde{R}^{-,(n)}:= (\tilde{R}^{-,(n)}_k)_{k\geq 1}$ corresponding to the order sizes after price changes. \par
Let us introduce a function $g: D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2_+)$ which describes the evolution of the queue size process $\pi_i \tilde{Q},$ $i = b,a,$ corresponding to one side of the shared order book provided that the cumulative queue size process of the corresponding side stays strictly larger than zero (i.e. no price change has been observed).

\newpage

\begin{Def}\label{CB:def:g}
Let $\omega \in D([0,T], \mathbb{R}^2)$. We inductively define $g(\omega) \in D([0,T], \mathbb{R}^2_+)$ as follows:
\begin{itemize}
	\item For $t < \hat{\tau}_1$, let $g(\omega)(t) = g_1(\omega)(t) := \omega(t)$, where
	$$\hat{\tau}_1 := \hat{\tau}_1(\omega) := \inf\{t \geq 0: \exists \, i \in \{1,2\} \text{ with } \pi^{(2)}_i \omega(t) \leq 0\} \wedge T$$
	defines the first hitting time of $\omega$ of the axes $\{(0,y): y>0\} \cup \{(x,0):x\geq 0\}.$
	\item Let $k \geq 2.$ If $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \notin \mathbb{R}^2_-$, then for $\hat{\tau}_{k-1}\leq  t < \hat{\tau}_k$, we set $g(\omega)(t) = g_k(\omega)(t)$, where
	\begin{align*}
	g_k(\omega)(t) := g_{k-1}(\omega)(t) &+ \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0\right\}} \sup_{s\leq t}\left\{-\pi^{(2)}_1 g_{k-1}(\omega)(s), 0\right\} (1, -1)\\
	&+ \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_2 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0\right\}} \sup_{s\leq t} \left\{-\pi^{(2)}_2 g_{k-1}(\omega)(s), 0\right\} (-1, 1)
	\end{align*}
 and
	\begin{align*}
	\hat{\tau}_k := \hat{\tau}_k(\omega) := \inf\Big\{t \geq \hat{\tau}_{k-1} :& \, \pi^{(2)}_1 g_k(\omega)(t)\textup{$\mathds{1}$}_{\left\{\pi^{(2)}_2 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0\right\}}\\
	&\quad +  \pi^{(2)}_2 g_k(\omega)(t)\textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0\right\}} \leq 0\Big\} \wedge T.
	\end{align*}
In contrast, if $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \in \mathbb{R}^2_-$, then $g(\omega)(t) = (0,0)$  for all $t\in [\hat{\tau}_{k-1}, T]$.
	\item For all $t \geq \hat{\tau}_{\infty} := \lim_{k\rightarrow \infty} \hat{\tau}_k$, we set $g(\omega)(t) = (0,0).$
\end{itemize}
\end{Def}

\begin{Rmk}[Modifications for smaller time domains]
	We also want to apply the function $g$ to an element $\omega \in D([0,t], \mathbb{R}^2),$ where $t < T.$ With a little abuse of notation, we write $g(\omega) \in D([0,t], \mathbb{R}^2_+)$ without further comment. Here, $g(\omega) \in D([0,t], \mathbb{R}^2)$ is defined as in Definition \ref{CB:def:g}, where $T$ is replaced by $t$. In the same way, we apply the subsequently defined functions to elements of $D([0,t], \mathbb{R}^2)$ and $D([0,t], \mathbb{R}^4)$, respectively, for $t< T.$
\end{Rmk}

As desired, the function $g$ describes the dynamics of the queue size process $\pi_i\tilde{Q}^{(n)},$ $i = b,a,$ corresponding to one side of the shared order book between consecutive price changes if we plug in the corresponding components of the net order flow process $\pi_{i}X^{(n)},$ i.e., for each $t \in [0,T]$ such that no price changing event appears during $[0,t],$ we can write
\[\pi_i\tilde{Q}^{(n)}(s) = g\left(\pi_i \tilde{Q}^{(n)}_0 + \pi_i X^{(n)}\right)(s) \quad \text{ for all } s \in [0,t].\]

Setting $g(\omega) \equiv 0$ on $[\hat{\tau}_k, T]$ if $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \notin \mathbb{R}^2_-$ but $g_k(\omega)(\hat{\tau}_k) \in \mathbb{R}^2_-$ ensures that the function $g$ is defined on the whole time interval $[0,T].$ In order to guarantee its well-definedness for all $\omega \in D([0,T],\mathbb{R}^2)$, we further set $g(\omega)\equiv 0$ on $[\hat{\tau}_{\infty}, T].$  Next, let us introduce the function
\begin{align}\label{CB:def:h1}
    h_1: D([0,T],\mathbb{R}^2) \rightarrow D([0,T],\mathbb{R}), \quad h_1(\omega) = \pi^{(2)}_1 \omega + \pi^{(2)}_2 \omega.
\end{align}
With a little abuse of notation, we will write $h_1(x) = x_1 + x_2 \in \mathbb{R}$ for $x = (x_1, x_2) \in \mathbb{R}^2.$ Furthermore, we recall the definition of the function
\begin{align}\label{CB:def:h2}
\begin{aligned}
    h: D([0,T], \mathbb{R}^4) &\rightarrow D([0,T], \mathbb{R}^2)\\
    h(\omega) = \left(h_1(\pi_{b} \omega), h_1(\pi_{a}\omega)\right) &=  \left(\pi_1\omega+\pi_3\omega, \pi_2\omega+\pi_4\omega\right).
\end{aligned}
\end{align}
We note that the process $h(\tilde{Q}^{(n)})$ describes the evolution of the cumulative queue size process of $\tilde{S}^{(n)}$ over time. The following lemma states that the sum of the components of $g(\omega)$ can be identified with the sum of the components of $\omega$ as long as $g(\omega)$ does not hit the origin.

\begin{Lem}\label{CB:res:sumCompg}
    Let $\omega \in D([0,T], \mathbb{R}^2)$, $\ell_t(x) := \sup_{s\leq t}\{-x(s), 0\}$ the reflection at zero of $x \in D([0,T],\mathbb{R}),$ and $\tau(\omega) := \inf\{t \geq 0: g(\omega)(t) = (0,0)\} \wedge T.$ Then,
    $$\left(h_1 \circ g\right)(\omega)(t) = h_1(\omega)(t) + \ell_t\left(h_1(\omega)\right) \quad \text{ for } t \in [0, \tau(\omega)].$$
\end{Lem}

The proof is postponed to Appendix \ref{CB:app:auxRes}. The following corollary is an immediate consequence of the previous lemma and relates the first time at which both components of $g(\omega)$ hit zero to the first time $h_1(\omega)$ hits zero.

\begin{Cor}\label{CB:res:sumStTime}
    Let $\omega \in D([0,T], \mathbb{R}^2)$. Then, it holds
    $$\tau(\omega) := \inf\{t \geq 0: g(\omega)(t) = (0, 0)\} \wedge T = \inf\{t \geq 0: h_1(\omega)(t) \leq 0\} \wedge T.$$
\end{Cor}

In order to characterize the continuity set of the function $g$, we endow the space $D([0,T], \mathbb{R}^2)$ with the Skorokhod topology (cf. Billingsley \cite{B99}). Moreover, let us introduce the function space $C_0'([0,T], \mathbb{R}^2 \setminus \{(0,0)\})$ containing all continuous functions $\omega \in C([0,T], \mathbb{R}^2)$ avoiding the origin and whose components cross the axes each time they touch them (cf. equation \eqref{CB:def:C'} for details).  Then, the following lemma characterizes the continuity set of $g.$

\begin{Lem}[Continuity of $g$]\label{CB:res:contg}
Let $\omega_0 \in D([0,T], \mathbb{R}^2)$ be such that
    \begin{itemize}
    		\item[i)] $\omega_0 \in C_0'([0,T], \mathbb{R}^2 \setminus \{(0,0)\})$ and $h_1(\omega_0) \in C'_0([0,T],\mathbb{R}),$
    		\item[ii)] there exists a finite, $\mathbb{N}$-valued $k_0 := k(\omega_0)$ such that
    \begin{align}\label{CB:ass:finitek0}
        g(\omega_0)(t) = g_{k_0}(\omega_0)(t) \quad \text{ for all } t \in [0,\tau(\omega_0)),
    \end{align}
    		\item[iii)] the maps $\hat{\tau}_{j-1}: D([0,T], \mathbb{R}^2) \rightarrow [0,T],$ $2 \leq j \leq k_0$, introduced in Definition \ref{CB:def:g} are continuous at $\omega_0$.
    		\end{itemize}
    Then, the function $g: D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2_+)$ is continuous at $\omega_0.$
\end{Lem}

The proof is stated in Appendix \ref{CB:app:contg}. Next, we introduce a function that maps c\`adl\`ag, $\mathbb{R}^4$-valued functions onto the space of c\`adl\`ag functions with values in $\mathbb{R}^4_+$. For $\omega \in D([0,T], \mathbb{R}^4),$ let
\begin{align}\label{CB:def:G}
\begin{aligned}
	 &G : D([0,T], \mathbb{R}^4) \rightarrow D([0,T], \mathbb{R}^4_+),\\
	G(\omega) :=& \Big(\pi^{(2)}_1g(\pi_{b}\omega), \pi^{(2)}_1 g(\pi_{a}\omega), \pi^{(2)}_2 g(\pi_{b}\omega), \pi^{(2)}_2g(\pi_{a}\omega) \Big).
\end{aligned}
\end{align}

We observe that the function $G$ describes the dynamics of the queue size process $\tilde{Q}^{(n)}$ corresponding to the shared order book between consecutive price changes if we plug in the net order flow process $X^{(n)}$, i.e.,  for each $t \in [0,T]$ such that no price changing event appears during $[0,t],$ we can write
\[\tilde{Q}^{(n)}(s) = G\left(\tilde{Q}^{(n)}_0 + X^{(n)}\right)(s), \quad \text{ for all } s \in [0,t].\]

Note, the continuity set of $G$ can be directly deduced from Lemma \ref{CB:res:contg}.

\begin{Cor}[Continuity of $G$] \label{CB:res:contG}
    Let $\omega_0 \in D([0,T], \mathbb{R}^4)$ be such that for each $i = b,a,$ the conditions i)-iii) of Lemma \ref{CB:res:contg} are satisfied for the projection $\pi_i \omega_0$. Then, the function $G: D([0,T], \mathbb{R}^4) \rightarrow D([0,T], \mathbb{R}^4_+)$ is continuous at $\omega_0.$
\end{Cor}

Applying Corollary \ref{CB:res:sumStTime}, we further conclude that
\begin{align*}
    \tilde{\tau}_1(\omega):=& \inf\{t \geq 0: \exists\, i \in \{1,2\} \text{ with } (\pi^{(2)}_i \circ h\circ G)(\omega)(t) = 0\} \wedge T\\
    =& \inf \{t \geq 0: \exists \, i \in \{1,2\} \text{ with } (\pi^{(2)}_i \circ h)(\omega)(t) \leq 0\} \wedge T,
\end{align*}
i.e., the first time $(h \circ G)(\omega)$ hits the axes equals the first time $h(\omega)$ hits the axes while the latter one may also cross the axes (for appropriate $\omega \in D([0,T], \mathbb{R}^4)$). Next, based on the definition of the function $G$, we introduce another function that can be used to construct the queue size process $\tilde{Q}^{(n)}$ from the net order flow process $X^{(n)}$ and the random sequences $\tilde{R}^{+,(n)}$ and $\tilde{R}^{-,(n)}$. In particular, this function takes the possibility of price changes into account leading to reinitializations of the queue size process $\tilde{Q}^{(n)}$ at new positions inside $\mathbb{R}_+^4$.

\begin{Def}\label{CB:def:PsiQ1}
Let $\omega \in D([0,T], \mathbb{R}^4)$ and let $R = (R_n)_{n\geq 1}, \tilde{R} = (\tilde{R}_n)_{n\geq 1} \in (\mathbb{R}^4_+)^{\mathbb{N}}$.  We define $\tilde{\Psi}^{Q}(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4_+)$ as follows:
\begin{itemize}[leftmargin=0.5cm]
	\item For $t < \tilde{\tau}_1$, let $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})(t) = G(\omega)(t),$
	where
	\begin{align*}
	\tilde{\tau}_1 := \tilde{\tau}_1(\omega) := \inf\Big\{ t \geq 0: \, \exists\, i \, \in \{1,2\} \text{ with }& (\pi^{(2)}_i \circ h)(\omega)(t) \leq 0\Big\} \wedge T
	\end{align*}
	is the first hitting time of $h(\omega)$ of the axes $\{(0,y):y > 0\} \cup \{(x,0):x\geq 0\}$.
	\item We set
	\begin{align*}
	\tilde{\Psi}^{Q}&(\omega, R, \tilde{R})(\tilde{\tau}_1) \\
	&= \begin{cases}
	\tilde{R}_1 &: (\pi^{(2)}_1 \circ h)(\omega)(\tilde{\tau}_1) \leq 0\\
	R_1 &:(\pi^{(2)}_1 \circ h)(\omega)(\tilde{\tau}_1) > 0 \text{ and } (\pi^{(2)}_2 \circ h)(\omega)(\tilde{\tau}_1) \leq 0\\
	\tilde{\Psi}^Q(\omega, R, \tilde{R})(\tilde{\tau}_1-) &: \text{otherwise}
	\end{cases}.
	\end{align*}
	\item Let $l \geq 1$.
	\begin{itemize}[leftmargin=0.5cm]
		\item[$\circ$] For $0 \leq t < \tilde{\tau}_{l+1} - \tilde{\tau}_l$, we have
		$$\tilde{\Psi}^{Q}(\omega, R, \tilde{R})(\tilde{\tau}_l +t) = G\Big(\tilde{\varphi}_l(\omega, R, \tilde{R})\Big)(t),$$
		where
		\begin{align*}
		\tilde{\varphi}_l(\omega, R,\tilde{R}) := \tilde{\Psi}^Q(\omega, R,\tilde{R})(\tilde{\tau}_l) + \omega(\cdot + \tilde{\tau}_l) - \omega(\tilde{\tau}_l)
		\end{align*}
		and
		\begin{align*}
		\tilde{\tau}_{l+1} &:= \tilde{\tau}_{l+1}(\omega, R, \tilde{R})\\
		&:= \inf\Big\{ t \geq \tilde{\tau}_{l}:\,\exists\, i \in \{1,2\}  \text{ with } (\pi^{(2)}_i \circ h)\Big(\tilde{\varphi}_l(\omega, R, \tilde{R})\Big)(t-\tilde{\tau}_l) \leq 0\Big\} \wedge T.
		\end{align*}
		\item[$\circ$] We set
		\begin{align*}
		\tilde{\Psi}^{Q}&(\omega, R, \tilde{R})(\tilde{\tau}_{l+1})\\
		&= \begin{cases}
		\tilde{R}_{l+1} &: (\pi^{(2)}_1 \circ h)\left(\tilde{\varphi}_l(\omega, R, \tilde{R})\right)(\tilde{\tau}_{l+1}-\tilde{\tau}_l) \leq 0\\
		R_{l+1} &: (\pi^{(2)}_1 \circ h)\left(\tilde{\varphi}_l(\omega, R, \tilde{R})\right)(\tilde{\tau}_{l+1}-\tilde{\tau}_l) > 0\\
		&\quad\text{and } (\pi^{(2)}_2 \circ h)\left(\tilde{\varphi}_l(\omega, R, \tilde{R})\right)(\tilde{\tau}_{l+1}-\tilde{\tau}_l) \leq 0\\
		\tilde{\Psi}^Q(\omega, R, \tilde{R})(\tilde{\tau}_{l+1}-) &: \text{otherwise}
		\end{cases}.
		\end{align*}
	\end{itemize}
	\item For $t \geq \tilde{\tau}_{\infty} := \lim_{l\rightarrow \infty}\tilde{\tau}_l$, we set $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})(t)  = (0, 0 , 0, 0).$
\end{itemize}
\end{Def}

The above definition states, that the path of $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})$ is obtained by ``regulating'' the path of $\omega \in D([0,T], \mathbb{R}^4)$ according to the function $G$ and the sequences $(R,\tilde{R})$:  between two hitting times $\tilde{\tau}_l$ and $\tilde{\tau}_{l+1}$, the function $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})$ equals the function $G(\tilde{\varphi}_l(\omega, R, \tilde{R}))(\cdot - \tilde{\tau}_l)$ and each time the cumulative of its $2$nd and $4$th components hits zero (resp. its $1$st and $3$rd components), it jumps to a new position inside $\mathbb{R}^4_+$ taken from the sequence $(R_n)_{n\geq 1}$ (resp. from the sequence $(\tilde{R}_n)_{n\geq 1}$). \par
Similarly as before, we endow the space $D([0,T], \mathbb{R}^4)$ with the Skorokhod topology. Moreover, we endow the set $(\mathbb{R}^4_+)^{\mathbb{N}}$ with the topology induced by cylindrical semi-norms, defined as follows: for a sequence $(R^n)_{n\geq 1} \subset (\mathbb{R}^4_+)^{\mathbb{N}}$,
$$R^n \rightarrow R \in (\mathbb{R}^4_+)^{\mathbb{N}} \Leftrightarrow \Big( \forall k \geq 1,\,  \sup\big\{\|R^n_1-R_1\|, \cdots, \|R^n_k - R_k\|\big\} \rightarrow 0\Big).$$
The space $D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ is then endowed with the corresponding product topology.

\begin{The}[Continuity of $\tilde{\Psi}^{Q}$]\label{CB:res:contPsiQ1}
	Let $(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ be such that
	\begin{enumerate}
		\item[i)] $\omega(0), R_l,$ and $\tilde{R}_l \in (0,\infty)^4$ for all $l\geq 1,$
		\item[ii)] there exists a finite, $\mathbb{N}$-valued $N_T$ such that $\tilde{\tau}_{N_T +1} = T$ and
		\[(h\circ \varphi_{N_T})(\omega, R,\tilde{R})(T-\tilde{\tau}_{N_T}) \in (0,\infty)^2,\]
		\item[iii)] for $i = b,a,$ the projection $\pi_i \omega$ of $\omega$ satisfies the conditions i)-iii) in Lemma \ref{CB:res:contg},
		\item[iv)] for $i = b,a$ and all $1 \leq l \leq N_T$, the projection $\pi_i \varphi_l(\omega, R, \tilde{R})$ of $\varphi_l(\omega, R, \tilde{R})$ introduced in Definition \ref{CB:def:PsiQ1} satisfies the conditions i)-iii) in Lemma \ref{CB:res:contg}, and
		\item[v)] $(0,0) \notin (h \circ \tilde{\Psi}^Q)(\omega, R, \tilde{R})([0,T]).$
		\end{enumerate}
	Then, the function $\tilde{\Psi}^{Q}: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^4_+)$ introduced in Definition \ref{CB:def:PsiQ1} is continuous at $(\omega, R, \tilde{R})$.
\end{The}

The proof of Theorem \ref{CB:res:contPsiQ1} is postponed to Appendix \ref{CB:app:contPsi}. The conditions in the above theorem are not very instructive at first glance. Note that condition i) ensures that $\tilde{\Psi}^Q(\omega, R,\tilde{R})$ starts and is reinitialized at points in the interior of $\mathbb{R}^4_+$. Moreover, condition ii) ensures that the function $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})$ is reinitialized only $N_T<\infty$ often in $[0,T],$ but not at time $T.$ Furthermore, the conditions iii) and iv) ensure that $G$ is continuous at $\omega$ and at the random shift $\varphi_l(\omega, R, \tilde{R})$ of $\omega$ for all $1 \leq l \leq N_T$ (cf. Corollary \ref{CB:res:contG}). Finally, condition v) guarantees that the reinitialization of  $\tilde{\Psi}^{Q}(\omega, R, \tilde{R})$ at $\tilde{\tau}_l$  by either $R_l$ or $\tilde{R}_l$ is continuous in $(\omega, R, \tilde{R}).$ \par
Later, we will apply the map $\tilde{\Psi}^Q$ to a four-dimensional linear Brownian motion $X$ starting in the interior of $\mathbb{R}^4_+$ and random sequences $R^+$ and $R^-$ with almost surely values in $(0,\infty)^4.$ Then, it can be shown that $(X, R^+, R^-)$ lies with probability one in the continuity set of $\tilde{\Psi}^Q$ (cf. the proof of Theorem \ref{CB:res:convQM-1}).\par
In the following, we also work with a slightly different construction of the function $\tilde{\Psi}^Q$ through a sequence of functions $(\tilde{\Psi}^Q_k)_{k\geq 0}.$\par

Before introducing the sequence of functions $(\tilde{\Psi}^Q_k)_{k\geq 0}$, let us first introduce the first hitting time maps $\tilde{\tau},$ $\tilde{\tau}_b,$ and $\tilde{\tau}_a: D([0,T], \mathbb{R}^4)\rightarrow [0,T]$ given by
	\begin{equation}\label{CB:def:tildetauab}
	\begin{split}
	\tilde{\tau}_b(\omega)&:= \inf\{t\geq 0: (\pi^{(2)}_1 \circ h)(\omega)(t) \leq 0\} \wedge T,\\
	\tilde{\tau}_a(\omega)&:=\inf\{t\geq 0: (\pi^{(2)}_2 \circ h)(\omega)(t) \leq 0\} \wedge T,
	\end{split}
	\end{equation}
	and
	\begin{equation}\label{CB:def:tildetau1}
	\tilde{\tau}(\omega) := \tilde{\tau}_b(\omega)\wedge \tilde{\tau}_a(\omega).
	\end{equation}
Note that $\tilde{\tau}(\omega)$ reveals the first hitting time before $T$ of $h(\omega)$ of the axes $\{(0,y): y >0\} \cup \{(x,0): x \geq 0\}.$\newline

We introduce an alternative construction of the function $\tilde{\Psi}^Q$ as follows: let $\tilde{\Psi}^Q_0\equiv G,$
	\begin{equation}\label{CB:def:PsiQ1-1}
	\begin{split}
    &\tilde{\Psi}^{Q}_1: D([0,T], \mathbb{R}^4) \times \mathbb{R}^4_+ \times \mathbb{R}^4_+  \rightarrow D([0,T], \mathbb{R}^4_+)\\
 	(\omega,& R_1, \tilde{R}_1) \mapsto G(\omega)\textup{$\mathds{1}$}_{[0,\tilde{\tau}(\omega))}\\
 	&\hspace{2.2cm} + \textup{$\mathds{1}$}_{[\tilde{\tau}(\omega), T]} \Big\{\textup{$\mathds{1}$}_{\left\{\tilde{\tau}_a(\omega) = \tilde{\tau}(\omega)\right\}} G\Big(R_1 + \omega-\omega(\tilde{\tau}(\omega))\Big) \\
 	&\hspace{4cm} + \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_b(\omega) = \tilde{\tau}(\omega)\right\}} G\Big(\tilde{R}_1 + \omega - \omega(\tilde{\tau}(\omega))\Big)\Big\},
	\end{split}
	\end{equation}
	and for $k \geq 1$ and the short-hand notation $\tilde{\Psi}^{Q}_k := \tilde{\Psi}^{Q}_k(\omega,R,\tilde{R}),$ let
\begin{equation}\label{CB:def:PsiQ1-k}
\begin{split}
	&\tilde{\Psi}^{Q}_{k+1}: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{k+1} \times (\mathbb{R}^4_+)^{k+1} \rightarrow D([0,T], \mathbb{R}^4_+)\\
    &\qquad (\omega, R, \tilde{R}) \mapsto \tilde{\Psi}^{Q}_{k}\textup{$\mathds{1}$}_{\left[0,\tilde{\tau}(\tilde{\Psi}^{Q}_{k})\right)} + \textup{$\mathds{1}$}_{\left[\tilde{\tau}(\tilde{\Psi}^{Q}_{k}), T\right]}\Big\{\\
    &\hspace{4cm}\textup{$\mathds{1}$}_{\left\{\tilde{\tau}_a(\tilde{\Psi}^{Q}_{k}) =\tilde{\tau}(\tilde{\Psi}^{Q}_{k})\right\}}G\Big(R_{k+1} + \omega - \omega\left(\tilde{\tau}(\tilde{\Psi}^{Q}_{k})\right)\Big)\\
    &\hspace{4cm} + \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_b(\tilde{\Psi}^{Q}_{k}) = \tilde{\tau}(\tilde{\Psi}^{Q}_{k})\right\}}G\Big(\tilde{R}_{k+1} + \omega - \omega\left(\tilde{\tau}(\tilde{\Psi}^{Q}_{k})\right)\Big)\Big\}.
\end{split}
\end{equation}
We extend the domain of the sequence of functions $(\tilde{\Psi}^Q_k)_{k\geq 0}$ such that they are defined on the same space $D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ with $\tilde{\Psi}^Q_0(\omega, R, \tilde{R}) = G(\omega)$ and for all $k\geq 1$ the argument of $\tilde{\Psi}^Q_k$ depends only on the first elements $(R_i, \tilde{R}_i)_{i=1,\cdots, k}$ of $(R, \tilde{R}).$

\begin{Rmk}[Connection between $\tilde{\Psi}^Q$ and $(\tilde{\Psi}^Q_k)_{k\geq 0}$]
$ $
	\begin{itemize}
	\item[i)] For the maps $\tilde{\tau}_k: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}\rightarrow [0,T],$ $k\geq 1,$ introduced in Definition \ref{CB:def:PsiQ1}, we have
	\[\tilde{\tau}_k(\omega, R, \tilde{R}) = \tilde{\tau}\left(\tilde{\Psi}^Q_{k-1}(\omega, R, \tilde{R})\right).\]
	In particular, we can interpret the maps $\tilde{\tau}_1,\tilde{\tau}_2,\cdots, \tilde{\tau}_k,\cdots$ as first hitting time maps of the functions $h(\tilde{\Psi}^Q_0), h(\tilde{\Psi}^Q_1),\cdots, h(\tilde{\Psi}^Q_{k-1}), \cdots$.
	\item[ii)] $\tilde{\Psi}^Q_{k-1}(\omega, R, \tilde{R}) \equiv \tilde{\Psi}^Q(\omega, R, \tilde{R})$ on $[0, \tilde{\tau}(\tilde{\Psi}^{Q}_{k-1}(\omega, R, \tilde{R}))),$ but $\tilde{\Psi}^Q_{k-1}(\omega, R, \tilde{R})$ is not reinitialized at time $\tilde{\tau}(\tilde{\Psi}^Q_{k-1}(\omega, R, \tilde{R})).$
	\item[iii)] If there exists a finite, $\mathbb{N}$-valued $N_T := N_T(\omega, R, \tilde{R}) := \inf\{k\geq 0: \tilde{\tau}_{k+1}(\omega, R, \tilde{R}) = T\}$ and $h(\tilde{\Psi}^{Q}_{N_T})(\omega, R, \tilde{R})(T) \in (0,\infty)^2,$ then we can identify $\tilde{\Psi}^Q_{N_T}(\omega, R, \tilde{R}) \equiv \tilde{\Psi}^Q(\omega, R, \tilde{R})$.
	\end{itemize}
\end{Rmk}

	\begin{Rmk}
	Provided that such an $N_T$ in iii) exists, we will use this alternative construction through the sequence of functions $(\tilde{\Psi}^Q_k(\omega, R, \tilde{R}))_{k\geq 0}$ equivalently to $\tilde{\Psi}^Q(\omega, R, \tilde{R})$ without further comment. This will simplify our argumentation at many points.
	\end{Rmk}

Now, the function $\tilde{\Psi}^{Q}$ may be applied to any c\`adl\`ag stochastic process: given a c\`adl\`ag process $X$ with values in $\mathbb{R}^4$ and random sequences $R = (R_n)_{n\geq 1}$ and $\tilde{R} = (\tilde{R}_n)_{n\geq 1}$ with values in $(0,\infty)^4$, the process $\tilde{\Psi}^{Q}(X, R, \tilde{R})$ is a c\`adl\`ag process with values in $\mathbb{R}^4_+$. Finally, we deduce that the queue size process $\tilde{Q}^{(n)}$ may be constructed by this procedure.

\begin{The}\label{CB:res:repQn}
For each $n \in \mathbb{N}$, the queue size process $\tilde{Q}^{(n)}$ is given by
\begin{align}\label{CB:eq:constQS}
\tilde{Q}^{(n)} = \tilde{\Psi}^{Q}\left(\tilde{Q}^{(n)}_0 + X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right),
\end{align}
where
\begin{itemize}
	\setlength\itemsep{0em}
	\item $\tilde{Q}^{(n)}_0 \in (0,\infty)^4$ are the initial queue sizes of the bid/ask queues,
	\item $X^{(n)}$ is the piecewise constant interpolation of the net order flow process,
	\item $\tilde{R}^{+,(n)} \in ((0,\infty)^4)^{\mathbb{N}}$ is the sequence of queue sizes after price increases in $\tilde{S}^{(n)}$,
	\item $\tilde{R}^{-,(n)} \in ((0,\infty)^4)^{\mathbb{N}}$ is the sequence of queue sizes after price decreases in $\tilde{S}^{(n)}.$
\end{itemize}
\end{The}

\begin{proof}
Each component of the net order flow process $X^{(n)}$ represents the sum of incoming order sizes of a specific type $(i,I) \in \{b,a\} \times \{F,G\}$ over time. Then, a comparison of the described dynamics of the individual queue size processes (see the definition in \eqref{CB:def:AD1}-\eqref{CB:def:AD4}) with the construction procedure in Definitions \ref{CB:def:g} and Definition \ref{CB:def:PsiQ1} yields the stated equality \eqref{CB:eq:constQS}.
\end{proof}

\subsection{The number of cross-border trades as a regulated process}\label{CB:sec:regProM}

Let us introduce the piecewise constant interpolation of the discrete-time process $(M^{(n)}_k)_{k=0, \cdots, T_n}$ introduced in \eqref{CB:def:sNoCB} by
\[M^{(n)}(t) = M^{(n)}_k \quad \text{ for } t \in \left[\t_k, \t_{k+1}\right).\]
In this subsection, we want to derive a representation of the number of cross-border trades $M^{(n)}$ as a regulated process of the net order flow process $X^{(n)}$ and the sequences of random variables $\tilde{R}^{+,(n)}$ and $\tilde{R}^{-,(n)}$. To this end, let us introduce the function $\bar{g}: D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2_+)$ which describes the dynamics of the number of cross-border trades triggered by events affecting one side of the shared order book provided that the cumulative queue size process of the corresponding side stays strictly larger than zero (i.e. no price change has been observed).

\begin{Def}\label{CB:def:barg}
Let $\omega \in D([0,T], \mathbb{R}^2)$. We inductively define $\bar{g}(\omega) \in D([0,T], \mathbb{R}^2_+)$ as follows: let $(\hat{\tau}_k)_{k\geq 1} := (\hat{\tau}_k(\omega))_{k\geq 1}$ and $g_k: D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2)$, $k \geq 1$, be as in Definition \ref{CB:def:g}.
	\begin{itemize}
	\item For $t < \hat{\tau}_1$ we set $\bar{g}(\omega)(t) = \bar{g}_1(\omega)(t) := (0,0)$.
	\item Let $k \geq 2$. If $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \notin \mathbb{R}^2_-$, then for $\hat{\tau}_{k-1} \leq t < \hat{\tau}_k,$ we set $\bar{g}(\omega)(t) = \bar{g}_k(\omega)(t)$, where
	\begin{align*}
	\hspace{-0.1cm}\bar{g}_k(\omega)(t) := \bar{g}_{k-1}(\omega)(\hat{\tau}_{k-1} -) &+ \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0\right\}}\hspace{-0.1cm} \left(\sup_{s\leq t}\{-\pi^{(2)}_1 g_{k-1}(\omega)(s), 0\},\, 0\right)\\
	& + \textup{$\mathds{1}$}_{\left\{ \pi^{(2)}_2 g_{k-1}(\omega)(\hat{\tau}_{k-1}) \leq 0 \right\}}\hspace{-0.1cm} \left(0,\, \sup_{s\leq t}\{-\pi^{(2)}_2 g_{k-1}(\omega)(s), 0\}\right).
	\end{align*}
	In contrast, if $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \in \mathbb{R}^2_-$, then for all $t\in [\hat{\tau}_{k-1}, T],$ we set
	\begin{align*}
	\bar{g}(\omega)(t) &:= \bar{g}_{k-1}(\omega)(\hat{\tau}_{k-1}-)\\
	&\qquad + \left(\sup_{s\leq \hat{\tau}_{k-1}}\{-\pi^{(2)}_1 g_{k-1}(\omega)(s), 0\}, \sup_{s\leq \hat{\tau}_{k-1}}\{-\pi^{(2)}_2 g_{k-1}(\omega)(s), 0\}\right).
	\end{align*}
	\item For $t \geq \hat{\tau}_{\infty} := \lim_{k\rightarrow \infty} \hat{\tau}_k$, we set $\bar{g}(\omega)(t) = (0,0).$
	\end{itemize}
\end{Def}

As desired, the function $\bar{g}$ describes the dynamics of the number of cross-border trades $\pi_{i}M^{(n)}$ triggered by events of type $i = b,a$ if we plug in the corresponding components of the net order flow process $\pi_{i}X^{(n)},$ i.e., for each $t \in [0,T]$ such that no price changing event appears during $[0,t]$, it holds
\[\pi_{i} M^{(n)}(s) = \bar{g}\left(\pi_i\tilde{Q}^{(n)}_0 + \pi_iX^{(n)}\right)(s) \quad \text{ for all } s \in [0,t].\]
Moreover, if $g_{k-1}(\omega)(\hat{\tau}_{k-1}) \notin \mathbb{R}^2_-$ but $g_k(\omega)(\hat{\tau}_k) \in \mathbb{R}^2_-,$ our definition of $\bar{g}(\omega)$ on $[\hat{\tau}_k, T]$ ensures that the possibly last cross-border trade has been counted and extends $\bar{g}(\omega)$ from $[0,\hat{\tau}_k)$ to the whole time interval $[0,T].$ Again, we set $\bar{g}(\omega) \equiv 0$ on $[\hat{\tau}_{\infty}, T]$ for technical reasons only. We note, that the continuity set of $\bar{g}$ is characterized analogously as the continuity set of $g$ (cf. Lemma \ref{CB:res:cont-barg}). The following lemma relates the increments of $\bar{g}$ to the reflection maps of the functions $\left(g_k\right)_{k\geq 1}$.

\begin{Lem}[Connection between $g$ and $\bar{g}$]\label{CB:res:conn-gbarg}
Let $\omega \in D([0,T], \mathbb{R}^2)$, $(\hat{\tau}_k)_{k\geq 1}$ be the functions introduced in Definition \ref{CB:def:g}, and $\ell_t(x) := \sup_{s\leq t}\{-x(s), 0\}$ be the reflection at zero of $x \in D([0,T], \mathbb{R}).$ Then, for all $k\geq 1$ and $t\in [0,\tau) \cap [\hat{\tau}_k, \hat{\tau}_{k+1})$ with $\tau := \tau(\omega) := \inf\{t \geq 0: h_1(\omega)(t) \leq 0\} \wedge T,$ we have
\[\bar{g}(\omega)(t) - \bar{g}(\omega)(\hat{\tau}_k-) = \left(\ell_t\left(\pi^{(2)}_1 g_k(\omega)\right), \ell_t\left(\pi^{(2)}_2 g_k(\omega)\right)\right).\]
Moreover, if $\tau = \hat{\tau}_{k_0}$, then for all $t \geq \tau,$ we have
$$\bar{g}(\omega)(t) - \bar{g}(\omega)(\tau-) = \left(\ell_{\tau}\left(\pi^{(2)}_1 g_{k_0}(\omega)\right), \ell_{\tau}\left(\pi^{(2)}_2 g_{k_0}(\omega)\right)\right).$$
\end{Lem}

The proof is given in Appendix \ref{CB:app:auxRes}. Let us introduce another function $\bar{G}$ that maps c\`adl\`ag processes with values in $\mathbb{R}^4$ to the space of c\`adl\`ag processes with values of $\mathbb{R}^4_+$. For $\omega \in D([0,T], \mathbb{R}^4),$ let
\begin{align}\label{CB:def:barG}
& \bar{G}: D([0,T], \mathbb{R}^4) \rightarrow D([0,T], \mathbb{R}^4_+),\\
\bar{G}\left(\omega\right) =& \left(\pi^{(2)}_1 \bar{g}(\pi_{b}\omega), \pi^{(2)}_1\bar{g}(\pi_{a}\omega), \pi^{(2)}_2\bar{g}(\pi_{b}\omega), \pi^{(2)}_2\bar{g}(\pi_{a}\omega)\right).
\end{align}

We observe, that the function $\bar{G}$ describes the dynamics of the number of cross-border trades $M^{(n)}$ if we plug in the net order flow process $X^{(n)},$ i.e., for each $t \in [0,T]$ such that no price changing event appears during $[0,t]$, we have
\[M^{(n)}(s) = \bar{G}\left(\tilde{Q}^{(n)}_0 + X^{(n)}\right)(s) \quad \text{ for all } s \in [0,t].\]

Next, we introduce a function $\tilde{\Psi}^M$ that will be used to describe the evolution of the number of cross-border trades $M^{(n)}$ over time. Note that also $M^{(n)}$ is affected by the reinitialization of the queue size process after each price change.

\begin{Def}\label{CB:def:PsiM1}
Let $\omega \in D([0,T], \mathbb{R}^4)$ and $R = (R_n)_{n \geq 1}, \, \tilde{R} = (\tilde{R}_n)_{n\geq 1} \in ((0,\infty)^4)^{\mathbb{N}}.$ Let $(\tilde{\tau}_l)_{l\geq 1} := (\tilde{\tau}_l(\omega, R, \tilde{R}))_{l\geq 1}$ and $(\tilde{\varphi}_l(\omega,R, \tilde{R}))_{l\geq 1}$ be as in Definition \ref{CB:def:PsiQ1}. We define $\tilde{\Psi}^{M}(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4_+)$ as follows:
\begin{itemize}[left=0.5cm]
	\item For $t \leq \tilde{\tau}_1$, let $\tilde{\Psi}^{M}(\omega, R, \tilde{R})(t) = \bar{G}(\omega)(t).$
	\item For $l \geq 1$ and $0 < t \leq \tilde{\tau}_{l+1} - \tilde{\tau}_l$, we set
	\[\tilde{\Psi}^{M}(\omega, R, \tilde{R})(\tilde{\tau}_l + t) = \tilde{\Psi}^{M}(\omega, R, \tilde{R})(\tilde{\tau}_l) + \bar{G}\Big(\tilde{\varphi}_l(\omega, R, \tilde{R})\Big)(t).\]
	\item Moreover, for $t > \tilde{\tau}_{\infty}:= \lim_{l\rightarrow \infty}\tilde{\tau}_l,$ we set $\tilde{\Psi}^M(\omega, R, \tilde{R})(t) = (0,0,0,0).$
\end{itemize}
\end{Def}

Note that $R, \tilde{R} \in ((0,\infty)^4)^{\mathbb{N}}$ ensure that $\bar{G}(\tilde{\varphi}_l(\omega, R,\tilde{R}))(0) = 0$, $\tilde{\Psi}^M(\omega, R, \tilde{R})(\tilde{\tau}_l) = \tilde{\Psi}^M(\omega, R, \tilde{R})(\tilde{\tau}_l+)$, for all $l\geq 1,$ and hence indeed $\tilde{\Psi}^M(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4_+).$\par
Moreover, observe that for each $l\geq 1$ and $\tilde{\tau}_0 := 0$, the increment $\tilde{\Psi}^{M}(\omega, R, \tilde{R})(\cdot + \tilde{\tau}_{l-1}) - \tilde{\Psi}^M(\omega, R, \tilde{R})(\tilde{\tau}_{l-1})$ equals $\bar{G}(\tilde{\varphi}_l(\omega, R, \tilde{R}))$ on $[0, \tilde{\tau}_{l}-\tilde{\tau}_{l-1}]$. \par

In contrast to $\tilde{\Psi}^{Q}$, the function $\tilde{\Psi}^{M}$ is not reinitialized at $\tilde{\tau}_l$, $l \geq 1,$ but still depends on the reinitialization of $\Psi^{Q}(\omega,R,\tilde{R})$ at $\tilde{\tau}_l,$ $l \geq 1$. Thanks to the characterization of the continuity set of $\tilde{\Psi}^Q$ in Theorem \ref{CB:res:contPsiQ1}, we immediately obtain an understanding of the continuity points of $\tilde{\Psi}^M.$

\begin{The}[Continuity of $\tilde{\Psi}^M$]\label{CB:res:contPsiM1}
	Let $(\omega, R,\tilde{R}) \in D([0,T], \mathbb{R}^4) \times ((0,\infty)^4)^{\mathbb{N}} \times ((0,\infty)^4)^{\mathbb{N}}$ satisfy the conditions i)-v) in Theorem \ref{CB:res:contPsiQ1}. Then the map
	$$\tilde{\Psi}^{M}: D([0,T], \mathbb{R}^4) \times ((0,\infty)^4)^{\mathbb{N}} \times ((0,\infty)^4)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^4_+)$$
	introduced in Definition \ref{CB:def:PsiM1} is continuous at $(\omega, R, \tilde{R})$.
\end{The}

\begin{proof}
Since $(\omega, R, \tilde{R})$ satisfies the conditions of Theorem \ref{CB:res:contPsiQ1}, we know that $\tilde{\Psi}^Q$ is continuous at $(\omega, R, \tilde{R}).$ Moreover, since condition ii) holds, the map $N_T: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}\rightarrow \mathbb{N}_0$ is continuous at $(\omega, R, \tilde{R})$ and $N_T(\omega, R, \tilde{R}) < \infty.$ Moreover, condition iii), iv), and Lemma \ref{CB:res:cont-barG} imply that $\bar{G}$ is continuous at $\omega$ and at $\tilde{\varphi}_l(\omega, R, \tilde{R})$ for all $1\leq l \leq N_T.$ Hence, $\tilde{\Psi}^M$ is the sum of finitely many functions that are continuous at $(\omega, R, \tilde{R})$ and hence must be itself continuous at $(\omega, R, \tilde{R}).$
\end{proof}

Finally, we can construct $M^{(n)}$ with the procedure introduced in Definition \ref{CB:def:PsiM1}.

\begin{The}\label{CB:res:repScalNumber}
For each $n \in \mathbb{N}$, the process $M^{(n)}$ is given by
\begin{align}\label{CB:eq:constM}
M^{(n)} = \tilde{\Psi}^{M}\left(\tilde{Q}^{(n)}_0 + X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right)
\end{align}
where
\begin{itemize}
\setlength\itemsep{0em}
	\item $\tilde{Q}^{(n)}_0 \in (0,\infty)^4$ are the initial queue sizes of the bid/ask queues,
	\item $X^{(n)}$ is the piecewise constant interpolation of the net order flow process,
	\item $\tilde{R}^{+,(n)} \in ((0,\infty)^4)^{\mathbb{N}}$ is the sequence of queue sizes after a price increase in $\tilde{S}^{(n)}$,
	\item $\tilde{R}^{-,(n)} \in ((0,\infty)^4)^{\mathbb{N}}$ is the sequence of queue sizes after a price decrease in $\tilde{S}^{(n)}$.
\end{itemize}
\end{The}

\begin{proof}
Each component of the net order flow process $X^{(n)}$ represents the sum of incoming order sizes of a specific type $(i,I) \in \{b,a\} \times \{F,G\}$ over time. Then, a comparison of the described dynamics of the number of cross-border trades (see the definition in \eqref{CB:def:sNoCB}) with the construction procedure in Definition \ref{CB:def:barg} and \ref{CB:def:PsiM1} yields the stated equality \eqref{CB:eq:constM}.
\end{proof}

\subsection{The price process as a regulated process}

Similarly as for $\tilde{Q}^{(n)}$ and $M^{(n)}$, we introduce in this subsection a new representation of the bid price process $\tilde{B}^{(n)}$ in terms of transformations of the net order flow process $X^{(n)}$ and the sequences of random variables $\tilde{R}^{+,(n)}$ and $\tilde{R}^{-,(n)}$ corresponding to the order sizes after price changes. Therefore, consider the sequence of functions
\[\tilde{\Psi}^Q_k: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^4_+), \quad k\geq 0,\]
introduced in equations \eqref{CB:def:PsiQ1-1} and \eqref{CB:def:PsiQ1-k}. Note that for each $(\omega, R, \tilde{R})$ satisfying the conditions in Theorem \ref{CB:res:contPsiQ1}, there exists a finite, $\mathbb{N}$-valued $N_T := N_T(\omega, R, \tilde{R})$ with
\[\tilde{\Psi}^Q(\omega, R,\tilde{R}) \equiv \tilde{\Psi}^Q_{N_T}(\omega, R, \tilde{R}).\]

\begin{Def}\label{CB:def:PsiB1}
	Let $\omega \in D([0,T], \mathbb{R}^4)$ and $R, \tilde{R}$ be sequences with values in $\mathbb{R}^4_+.$ For $k\geq 1,$ let $\tilde{\tau}_k := \tilde{\tau}_k(\omega, R, \tilde{R})$ be as in Definition \ref{CB:def:PsiQ1} and $\tilde{\Psi}^Q_{k-1} := \tilde{\Psi}^Q_{k-1}(\omega, R, \tilde{R})$ be as in equation \eqref{CB:def:PsiQ1-1} and \eqref{CB:def:PsiQ1-k}. Then, we define $N_b(\omega, R, \tilde{R}),$ $N_a(\omega, R, \tilde{R}) \in D([0,T], \mathbb{N}_0)$, and $\tilde{\Psi}^B(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^2)$ by
	\[N_a(\omega, R, \tilde{R}) := \sum_{\tilde{\tau}_k \leq \cdot} \textup{$\mathds{1}$}_{\left\{\left(\pi^{(2)}_2 \circ h \circ \tilde{\Psi}^Q_{k-1}\right)(\tilde{\tau}_k) \leq 0\right\}},\hspace{0.2cm} N_b(\omega, R, \tilde{R}) := \sum_{\tilde{\tau}_k \leq \cdot} \textup{$\mathds{1}$}_{\left\{\left(\pi^{(2)}_1 \circ h \circ \tilde{\Psi}^Q_{k-1}\right)(\tilde{\tau}_k) \leq 0\right\}},\]
	and
	\[\tilde{\Psi}^B(\omega, R, \tilde{R}) := \delta \left(N_a(\omega, R, \tilde{R}) - N_b(\omega, R, \tilde{R})\right) (1,1).\]
\end{Def}

It can be easily deduced from the described bid price dynamics in equations \eqref{CB:def:AD1}-\eqref{CB:def:AD4} that
\begin{equation}\label{CB:res:repPrices}
\tilde{B}^{(n)}  = \tilde{B}^{(n)}_0 + \tilde{\Psi}^B\left(\tilde{Q}^{(n)}_0 + X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right),
\end{equation}
where $\tilde{B}^{(n)}_0 = (\tilde{B}^{F,(n)}_0, \tilde{B}^{F,(n)}_0) \in (\delta\mathbb{Z})^2$ by the assumptions on the initial states. The following theorem characterizes the continuity set of the map $\tilde{\Psi}^B.$

\begin{The}[Continuity of $\tilde{\Psi}^B$]\label{CB:res:contPsiB1} Let $(\omega, R, \tilde{R})\in D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ satisfy the conditions i)-v) in Theorem \ref{CB:res:contPsiQ1}. Moreover, assume that
\begin{equation}\label{CB:cond:noSimJumps}
\ensuremath\mathrm{Disc}\left(N_a(\omega, R, \tilde{R})\right) \cap \ensuremath\mathrm{Disc}\left(N_b(\omega, R, \tilde{R})\right) = \text{\O},
\end{equation}
where $\ensuremath\mathrm{Disc}(N_i(\omega, R, \tilde{R})) := \{t\in [0,T]: N_i(\omega, R, \tilde{R})(t-) \neq N_i(\omega, R, \tilde{R})(t)\}$ for $i = b,a.$ Then, the map $\tilde{\Psi}^B$ is continuous at $(\omega, R, \tilde{R}).$
\end{The}

\begin{proof}
	Since the conditions i)-v) of Theorem \ref{CB:res:contPsiQ1} hold for $(\omega, R, \tilde{R})$, we know by Corollary \ref{CB:res:contNba}, that the maps $N_b, N_a: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{N}_0)$ are continuous at $(\omega, R, \tilde{R}).$ Since also the condition in \eqref{CB:cond:noSimJumps} is satisfied, we can apply Theorem 4.1 in Whitt \cite{W80} and conclude that also the map $\tilde{\Psi}^B: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)\times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T],\mathbb{R}^2)$ is continuous at $(\omega, R, \tilde{R}).$ This finishes the proof.
\end{proof}

\subsection{A heavy traffic approximation of the active dynamics}\label{CB:sec:HTAppr}

In this subsection, we derive heavy traffic approximations of the queue size process $\tilde{Q}^{(n)}$, the number of cross-border trades $M^{(n)}$, and the bid price process $\tilde{B}^{(n)}$. As a consequence, we obtain a joint limit result of the active market dynamics $\tilde{S}^{(n)}.$ The proofs of these approximations are based on the functional central limit theorem for $X^{(n)}$ (cf. Proposition \ref{CB:res:fCLT}) combined with the continuous mapping theorem. \par

In the following, let us introduce the sequence of stopping times corresponding to the prices changes in $\tilde{S}^{(n)}$ by $\tilde{\tau}^{(n)}_0 := 0,$
\begin{align}\label{CB:def:tau1}
   \tilde{\tau}^{(n)}_k := \tilde{\tau}\left(\tilde{\Psi}^{Q}_{k-1}\left(\tilde{Q}^{(n)}_0 + X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right)\right), \quad k\geq 1,
\end{align}
where the first hitting time map $\tilde{\tau}$ is defined in \eqref{CB:def:tildetau1}.\par
Next, for $(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4) \times ((0,\infty)^4)^{\mathbb{N}} \times ((0,\infty)^4)^{\mathbb{N}}$ and $(b_0, q_0) \in \mathbb{R}^2 \times (0,\infty)^4$, let us introduce the function
\begin{align*}
\tilde{\Psi}^{(B,Q,M)}&((b_0, q_0); \omega, R, \tilde{R})\\
&:= \left(b_0 + \tilde{\Psi}^B(q_0 + \omega, R, \tilde{R}), \tilde{\Psi}^Q(q_0 + \omega, R, \tilde{R}), \tilde{\Psi}^M(q_0+\omega, R, \tilde{R})\right)
\end{align*}
and observe that by Theorem \ref{CB:res:repQn}, Theorem \ref{CB:res:repScalNumber}, and equation \eqref{CB:res:repPrices}
\[(\tilde{B}^{(n)}, \tilde{Q}^{(n)}, M^{(n)}) = \tilde{\Psi}^{(B,Q,M)}\left((\tilde{B}^{(n)}_0, \tilde{Q}^{(n)}_0); X^{(n)}, R^{+,(n)}, R^{-,(n)}\right).\]

\begin{The}[Joint limit theorem of $\tilde{B}^{(n)},$ $\tilde{Q}^{(n)},$ and $M^{(n)}$]\label{CB:res:convQM-1}
	Let Assumptions \ref{CB:ass:prob}, \ref{CB:ass:scaling}, and \ref{CB:ass:R} be satisfied. Assume that $\tilde{Q}^{(n)}_0 \rightarrow \tilde{Q}_0 \in (0,\infty)^4$ and $\tilde{B}^{(n)}_0 := (\tilde{B}^{F,(n)}_0, \tilde{B}^{F,(n)}_0)\rightarrow (\tilde{B}^F_0,\tilde{B}^F_0)=: \tilde{B}_0 \in (\delta \mathbb{Z})^2.$ Then, we have
\begin{align*}
		\left(\tilde{B}^{(n)}, \tilde{Q}^{(n)}, M^{(n)}\right) =\, &\tilde{\Psi}^{(B,Q,M)}\left((\tilde{B}^{(n)}_0, \tilde{Q}^{(n)}_0), X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right)\\
		&\quad \Rightarrow \tilde{\Psi}^{(B,Q,M)}\left((\tilde{B}_0, \tilde{Q}_0); X, \tilde{R}^{+}, \tilde{R}^{-}\right) =: (\tilde{B}, \tilde{Q}, M)
\end{align*}
in the Skorokhod topology on the space $D([0,T], \mathbb{R}^2 \times \mathbb{R}^4_+ \times \mathbb{R}^4_+).$
\end{The}

\begin{proof}
Proposition \ref{CB:res:fCLT} combined with the assumption on the initial state shows, that the net order flow process $Y^{(n)}:=\tilde{Q}^{(n)}_0 + X^{(n)}$ satisfies a functional central limit theorem, i.e.,
\begin{equation} \label{CB:clt}
Y^{(n)}\Rightarrow Y := \left(\tilde{Q}_0 + \Sigma^{1/2} B(t) + t \mu\right)_{t\geq 0}
\end{equation}
in the Skorokhod topology on $D([0,T], \mathbb{R}^4)$, where the drift $\mu$ and the covariance matrix $\Sigma$ are defined in Proposition \ref{CB:res:fCLT}. With as slight abuse of notion, by the Skorokhod representation theorem and Assumption \ref{CB:ass:R} we may assume the existence of independent sequences of iid random variables $(\epsilon^+_k)_{k\geq 1}$, $(\epsilon^-_k)_{k\geq 1}$ on $(\Omega, \mathcal{F}, \ensuremath\mathbb{P})$ where $\epsilon^+_1 \sim f^+,$ $\epsilon^-_1 \sim f^-$, and
\[\ensuremath\mathbb{P}\left[ Y^{(n)}\rightarrow Y, \text{ and for all } k \geq 1,\, \epsilon^{+,(n)}_k \rightarrow \epsilon^+_k,\, \epsilon^{-,(n)}_k \rightarrow \epsilon^-_k\right] = 1.\]
Applying Theorem \ref{CB:res:repQn}, Theorem \ref{CB:res:repScalNumber}, and equation \eqref{CB:res:repPrices}, we can represent the bid price process, the queue size process, and the number of cross-border trades via the function $\tilde{\Psi}^{(B,Q,M)}$. In the following, we will show that $(\tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}) \rightarrow (\tilde{R}^+, \tilde{R}^-)$ $\ensuremath\mathbb{P}$-almost surely for appropriately constructed sequences $\tilde{R}^+$ and $\tilde{R}^-$, and $(Y, \tilde{R}^+, \tilde{R}^-)$ lies, with probability one, in the continuity set of $\tilde{\Psi}^{I}$, $I = B, Q, M$. Then, the assumption on the initial states, equation \eqref{CB:clt}, and the continuous mapping theorem yield the stated result. \par
Let us first show that $(\tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}) \rightarrow (\tilde{R}^+, \tilde{R}^-)$ $\ensuremath\mathbb{P}$-almost surely.  Since $Y$ is a four-dimensional linear Brownian motion, its paths almost surely fulfill the assumptions in Corollary \ref{CB:res:contG}.
Hence, $G$ is continuous in $Y$ almost surely. Proposition \ref{CB:res:fCLT} and the continuous mapping theorem (cf. Billingsley \cite{B99}, Theorem 2.7) imply
\[G(Y^{(n)}) \rightarrow G(Y) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
in the Skorokhod topology.
Next, we construct the limit process $\tilde{Q}$ of $\tilde{Q}^{(n)}$ by induction:
let $\tilde{\tau}^*_1 = \tilde{\tau}(G(Y))$ be the first hitting time of $(h \circ G)(Y)$ of the axes $\{(0,y): y > 0\} \cup \{(x,0): x > 0\}$ and let us set $\tilde{Q}(t) = G(Y)(t)$ for $t< \tilde{\tau}^*_1.$ Since $(h \circ G)(Y) = h(Y)$ on $[0, \tilde{\tau}^*_1)$ (cf. Lemma \ref{CB:res:sumCompg}), $\tilde{\tau}^*_1$ equals the first hitting time of $h(Y)$ of the axes and $h(Y)$ crosses the axes at $\tilde{\tau}^*_1$ with probability one. Since $Y$ is a four-dimensional linear Brownian motion, $h(Y)$ is again a planar Brownian motion and has therefore almost surely sample paths in $C_0'([0,T], \mathbb{R}^2\setminus\{(0,0)\}).$ Now, the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}) and the continuous mapping theorem imply
\[\left(\tilde{\tau}^{(n)}_1, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_1-)\right) \rightarrow \left(\tilde{\tau}^*_1, \tilde{Q}(\tilde{\tau}^*_1-)\right) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
Let us set
\[\tilde{Q}(\tilde{\tau}^*_1) := \Phi\big(G(Y)(\tilde{\tau}^*_1-), \epsilon^+_1\big) \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_{a,1} = \tilde{\tau}^*_1\right\}} + \Phi\big(G(Y)(\tilde{\tau}^*_1-), \epsilon^-_1\big) \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_{b,1} =\tilde{\tau}^*_1\right\}},\]
where, for the first hitting time maps $\tilde{\tau}_b,$ $\tilde{\tau}_a$ introduced in equation \eqref{CB:def:tildetauab}, we set
\[\tilde{\tau}_{a,1} := \tilde{\tau}_a(G(Y)) \quad \text{and} \quad \tilde{\tau}_{b,1} := \tilde{\tau}_b(G(Y)).\]
Again, since we can relate $\tilde{\tau}^*_1$ (and also $\tilde{\tau}_{b,1}, \tilde{\tau}_{a,1}$) to the first hitting time of $h(Y)$ of the axes, and $h(Y)$ has, with probability one, sample paths in $C_0'([0,T], \mathbb{R}^2\setminus \{(0,0)\})$, we can apply Lemma \ref{CB:res:contH} to conclude that $G(Y)$ lies, with probability one, in the continuity set of $H_i: \omega \mapsto \textup{$\mathds{1}$}_{\{\tilde{\tau}(\omega) = \tilde{\tau}_i(\omega)\}},$ $i = b,a.$ So, using Assumption \ref{CB:ass:R} (i.e. $\|\Phi^{(n)} - \Phi\|_{\infty} \rightarrow 0$ and the continuity of $\Phi(\cdot, \cdot)$), the continuity of the last value map, and the continuous mapping theorem, we conclude
\[\tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_1)\rightarrow \tilde{Q}(\tilde{\tau}^*_1) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
Let us now show the induction step: assume that we have defined $\tilde{Q}$ on $[0, \tilde{\tau}^*_k]$ and have shown that
\begin{align*}
\Big(\tilde{\tau}^{(n)}_1, \cdots, \tilde{\tau}^{(n)}_{k}, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_1), \cdots, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_{k})\Big)\rightarrow \left(\tilde{\tau}^*_1, \cdots, \tilde{\tau}^*_{k}, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k})\right) \quad \ensuremath\mathbb{P}\text{-a.s.}
\end{align*}
Next, we define the stopping time $\tilde{\tau}^*_{k+1} :=\tilde{\tau}(\tilde{\Psi}^{Q}_{k}(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k})))$, i.e., $\tilde{\tau}^*_{k+1}$ denotes the first hitting time of $(h \circ \tilde{\Psi}^{Q}_{k})(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))$ of the axes $\{(0,y): y > 0\} \cup \{(x,0): x > 0\}.$ Now, we want to extend the definition of $\tilde{Q}$ to $[0, \tilde{\tau}^*_{k+1}]$ by setting
\begin{align*}
\tilde{Q}(t) &:= \tilde{\Psi}^{Q}_{k}(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))(t), \quad \text{for } t < \tilde{\tau}^*_{k+1},\\
\tilde{Q}(\tilde{\tau}^*_{k+1}) &:= \Phi\left(\tilde{Q}(\tilde{\tau}^*_{k+1}-), \epsilon^+_{k}\right) \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_{a,k+1} = \tilde{\tau}^*_{k+1}\right\}}+ \Phi\left(\tilde{Q}(\tilde{\tau}^*_{k+1}-), \epsilon^-_{k+1}\right) \textup{$\mathds{1}$}_{\left\{\tilde{\tau}_{b,k+1} = \tilde{\tau}^*_{k+1}\right\}},
\end{align*}
where
\[\tilde{\tau}_{a,k+1} := \tilde{\tau}_a\left(\tilde{\Psi}^{Q}_{k}(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))\right), \quad \tilde{\tau}_{b,k+1} := \tilde{\tau}_b\left(\tilde{\Psi}^{Q}_{k}(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))\right).\]
Let us denote $\tilde{B}_k := \tilde{Q}(\tilde{\tau}^*_k) + Y(\cdot + \tilde{\tau}^*_k) - Y(\tilde{\tau}^*_k).$ Then, we observe that $(h \circ G)(\tilde{B}_k) = h(\tilde{B}_k)$ on $[0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k)$. Therefore, $\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k$ equals the first hitting time of $h(\tilde{B}_k)$ of the axes. Since $Y$ is a four-dimensional linear Brownian motion, we conclude that $h(\tilde{B}_k)$ is a planar Brownian motion and has therefore, almost surely, sample paths in $C_0'([0,T-\tilde{\tau}^*_k], \mathbb{R}^2 \setminus \{(0,0)\}).$ Further, applying Assumption  \ref{CB:ass:R}, we have that $\tilde{Q}(\tilde{\tau}^*_1), \dots, \tilde{Q}(\tilde{\tau}^*_k) \in (0,\infty)^4$ with probability one. Now, by the definition of $\tilde{\Psi}^{Q}_k$, we conclude that $(0,0) \notin (h \circ \tilde{\Psi}^{Q}_{k})(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))([0, \tilde{\tau}^*_{k+1}])$ almost surely. Hence, by Lemma \ref{CB:res:contPsik}, $(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_{k}))$ lies with probability one in the continuity set of $\tilde{\Psi}^{Q}_{k}$.
So, by the continuous mapping theorem, we have
\[\tilde{\Psi}^{Q}_k(Y^{(n)}, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_1), \cdots, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_k)) \rightarrow \tilde{\Psi}^{Q}_k(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_k)) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
in the Skorokhod topology. Again, combining the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}) and the continuous mapping theorem, we conclude that
\[(\tilde{\tau}^{(n)}_{k+1}, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_{k+1}-)) \rightarrow (\tilde{\tau}^*_{k+1}, \tilde{Q}(\tilde{\tau}^*_{k+1}-)) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
Again, since we can relate $\tilde{\tau}^*_{k+1}$ (and also $\tilde{\tau}_{b,k+1}, \tilde{\tau}_{a,k+1}$) to the first hitting time of $h(\tilde{B}_k)$ of the axes and $h(\tilde{B}_k)$ has, with probability one, sample paths in $C_0'([0,T-\tilde{\tau}^*_k], \mathbb{R}^2\setminus\{(0,0)\})$, we can apply Lemma \ref{CB:res:contH} to conclude that $\tilde{\Psi}^{Q}_k(Y, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_k))$ lies, with probability one, in the continuity set of $H_i: \omega \mapsto \textup{$\mathds{1}$}_{\{\tilde{\tau}(\omega) = \tilde{\tau}_i(\omega)\}}$, $i = b,a.$ So, using again Assumption \ref{CB:ass:R}, the continuity of the last value map, and the continuous mapping theorem, we conclude that $\tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_{k+1}) \rightarrow \tilde{Q}(\tilde{\tau}^*_{k+1})$ $\ensuremath\mathbb{P}$-almost surely. So finally, we have shown for all $k \geq 1,$ that
\begin{align*}
\Big(\tilde{\tau}_1^{(n)}, \cdots, \tilde{\tau}_k^{(n)}, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_1), \cdots, \tilde{Q}^{(n)}(\tilde{\tau}^{(n)}_k)\Big)\rightarrow \left(\tilde{\tau}^*_1, \cdots, \tilde{\tau}^*_k, \tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_k)\right)\quad \ensuremath\mathbb{P}\text{-a.s.}
\end{align*}
Now, we construct the sequences $\tilde{R}^+, \tilde{R}^-$ by setting $\tilde{R}^+_k := \Phi(\tilde{Q}(\tilde{\tau}^*_k-), \epsilon^+_k)$ and $\tilde{R}^-_k := \Phi(\tilde{Q}(\tilde{\tau}^*_k-), \epsilon^-_k),$ for $k \geq 1.$ Then, we have
$$\tilde{Q}(\tilde{\tau}^*_k) = \tilde{R}^+_k \text{ if } \tilde{\tau}_{a,k} =  \tilde{\tau}^*_k, \quad \tilde{Q}(\tilde{\tau}^*_k) = \tilde{R}^-_k \text{ if } \tilde{\tau}_{b,k} = \tilde{\tau}^*_k,$$
and $(\tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}) \rightarrow (\tilde{R}^+, \tilde{R}^-)$ $\ensuremath\mathbb{P}$-almost surely. \par
By Assumption \ref{CB:ass:R} we have $\tilde{Q}(\tilde{\tau}^*_1), \cdots, \tilde{Q}(\tilde{\tau}^*_k) \in (0,\infty)^4$ with probability one. Applying Lemma \ref{CB:res:sumCompg} and Theorem \ref{CB:res:fCLT}, we conclude that $h(\tilde{B}_k),$ $k\geq 0,$ are independent planar Brownian motions starting in the interior of the positive orthant. Note that the distribution of the first exit time of the planar Brownian motion is well studied (cf. e.g. \cite{CL21, I85, M10, Z01} or the conditional distribution in \eqref{CB:def:condDist-Duration}). Hence, for any $t  > 0$ and $y \in (0,\infty)^4$, $k \geq 0$, we conclude that
\begin{equation}\label{CB:eq:hittingPBM}
\ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \,|\, \tilde{B}_k(0) = y\right] =: \delta(t,y) \in (0,1)
\end{equation}
and further note that this probability is decreasing in $y$ (componentwise). Next, for each $\varepsilon > 0,$ let us denote by $U(\varepsilon):= \{x\in \mathbb{R}^4_+: \pi_j x \leq \varepsilon\quad  \forall\,  j =1,2,3,4\}$ the $\varepsilon$-ball with respect to the $\sup$-norm and $\epsilon_k := \epsilon_k^+\textup{$\mathds{1}$}_{\{\tilde{\tau}_{a,k} = \tilde{\tau}^*_{k}\}} + \epsilon^-_k\textup{$\mathds{1}$}_{\{\tilde{\tau}_{b,k}=\tilde{\tau}^*_k\}}.$ Thanks to the independence of $\tilde{B}_k,$ $k\geq 0,$ and Assumption \ref{CB:ass:R} i), there exists an $\varepsilon> 0$ such that for all $m \geq 0,$
\begin{equation*}
\begin{split}
	\ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \,\, \forall \, k\leq m\right] &= \prod_{k\leq m} \int_{\mathbb{R}^4_+} \ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \, |\, \tilde{B}_k(0) = \tilde{y}_k\right] \ensuremath\mathbb{P}[\tilde{B}_k(0) \in d\tilde{y}_k]\\
	&=\prod_{k\leq m} \int_{\mathbb{R}^4_+ \setminus U(\varepsilon)} \ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \,|\, \tilde{B}_k(0) = \tilde{y}_k\right] \ensuremath\mathbb{P}[\tilde{B}_k(0) \in d\tilde{y}_k]\\
	&\qquad  + \prod_{k\leq m} \int_{U(\varepsilon)} \ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \,|\, \tilde{B}_k(0) = \tilde{y}_k\right] \ensuremath\mathbb{P}[\tilde{B}_k(0) \in d\tilde{y}_k]\\
	&\leq \prod_{k\leq m} \ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_k) \leq t \, | \, \tilde{B}_k(0) = (\varepsilon,\varepsilon,\varepsilon,\varepsilon)\right] + \prod_{k\leq m} \ensuremath\mathbb{P}\left[\tilde{B}_k(0) \in U(\varepsilon)\right]\\
	&\leq \prod_{k\leq m} \ensuremath\mathbb{P}\left[\tilde{\tau}(\tilde{B}_1) \leq t \,|\, \tilde{B}_1(0) = (\varepsilon,\varepsilon,\varepsilon,\varepsilon)\right] + \prod_{k\leq m} \ensuremath\mathbb{P}\left[\alpha \epsilon_k \in U(\varepsilon)\right]\\
	&\leq (\delta_1(t, \varepsilon))^m + \prod_{k\leq m}\max\left\{\ensuremath\mathbb{P}(\alpha \epsilon^+_1 \in U(\varepsilon)),\ensuremath\mathbb{P}(\alpha\epsilon^-_1 \in U(\varepsilon))\right\}\\
	&\leq (\delta_1(t,\varepsilon))^m + \max\{\delta_2^+(\varepsilon), \delta_2^-(\varepsilon)\}^m \overset{m\rightarrow \infty}{\longrightarrow} 0,
\end{split}
\end{equation*}
where $\delta_1(t,\varepsilon),$ $\delta_2^+(\varepsilon),$ and $\delta_2^-(\varepsilon) \in (0,1)$ thanks to equation \eqref{CB:eq:hittingPBM} and Assumption \ref{CB:ass:R} i). Hence, with probability one, there exists a finite, $\mathbb{N}$-valued random variable $N_T$ such that $\tilde{\tau}^*_{N_T+1} = T$ and
\[\tilde{Q} := \tilde{\Psi}^{Q}_{N_T}(Y, \tilde{R}^+, \tilde{R}^-) = \tilde{\Psi}^{Q}(Y, \tilde{R}^+, \tilde{R}^-) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
Let us now show that $(Y, \tilde{R}^+, \tilde{R}^-)$ lies with probability one in the continuity set of $\tilde{\Psi}^{Q}$. Therefore, we verify the conditions i)-v) of Theorem \ref{CB:res:contPsiQ1}. First, $Y(0) = \tilde{Q}_0 \in (0,\infty)^4$ and by Assumption \ref{CB:ass:R} the elements of $R$ and $\tilde{R}$ have values in $(0,\infty)^4$ with probability one. Hence, condition i) holds with probability one. As argued above, we can relate the stopping times $\tilde{\tau}^*_k$, $k\geq 1,$ to the first hitting times of $h(\tilde{B}_{k-1})$, $k\geq 1,$ of the axes and as we already have shown, with probability one, there exists a finite $\mathbb{N}$-valued random variable $N_T$ such that $\tilde{\tau}^*_{k+1} = T$ and either $\tilde{\tau}^*_k = \tilde{\tau}_{a,k}$ or $\tilde{\tau}^*_k = \tilde{\tau}_{b,k}$ for all $k \leq N_T.$ Since $h(\tilde{B}_{N_T})$ is again a planar Brownian motion, $h(\tilde{B}_{N_T})(T-\tilde{\tau}^*_{N_T})\in (0,\infty)^2$ with probability one. Hence, also condition ii) and v) are satisfied. Moreover, the processes $\tilde{B}_k,$ $k\geq 0,$ are independent, four-dimensional linear Brownian motions and therefore also the conditions in iii) and iv) are satisfied with probability one for all $k \leq N_T$. An application of Theorem \ref{CB:res:contPsiQ1} and Theorem \ref{CB:res:contPsiM1} yields that $(Y,\tilde{R}^+,\tilde{R}^-)$ lies with probability one in the continuity set of $\tilde{\Psi}^Q$ and $\tilde{\Psi}^M.$ Again since $h(\tilde{B}_k)$, $k\geq 0$, are independent planar Brownian motions, we conclude that also condition \eqref{CB:cond:noSimJumps} in Theorem \ref{CB:res:contPsiB1} is satisfied implying that $(Y, \tilde{R}^+, \tilde{R}^-)$ lies with probability one in the continuity set of $\tilde{\Psi}^B.$ We can therefore apply the continuous mapping theorem and the assumption on the initial state to conclude that
\[\tilde{\Psi}^{(B,Q,M)}((\tilde{B}^{(n)}_0, \tilde{Q}^{(n)}_0); X^{(n)}, \tilde{R}^{+,(n)} ,\tilde{R}^{-,(n)}) \rightarrow \tilde{\Psi}^{(B,Q,M)}((\tilde{B}_0, \tilde{Q}_0); X,\tilde{R}^+, \tilde{R}^-) \quad \ensuremath\mathbb{P}\text{-a.s.}\]
in the Skorokhod topology. This finishes the proof.
\end{proof}

In the following, we denote by $(\tilde{\tau}^*_k)_{k\geq 1}$ the stopping times corresponding to the times of price changes in the limit $\tilde{S}$ of $\tilde{S}^{(n)}$, i.e., by Theorem \ref{CB:res:convQM-1} we have $\tilde{\tau}^*_0 := 0,$
\begin{align}\label{CB:def:tau1-limit}
    \tilde{\tau}^*_1 := \tilde{\tau}\left(G\left(\tilde{Q}_0 + X\right)\right), \hspace{0.2cm} \tilde{\tau}^*_k := \tilde{\tau}\left(\tilde{\Psi}^{Q}_{k-1}\left(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^-\right)\right) \hspace{0.2cm} \text{ for all } k \geq 2,
\end{align}
where the random sequences $\tilde{R}^+, \tilde{R}^-$ describe the queue sizes after price changes in $\tilde{S}$ with
\begin{align} \label{CB:def:R1-limit}
\tilde{R}^+_k := \Phi\left(\tilde{Q}(\tilde{\tau}^*_k-), \epsilon^+_k\right), \quad \tilde{R}^-_k := \Phi\left(\tilde{Q}(\tilde{\tau}^*_k-), \epsilon^-_k\right)
\end{align}
for independent sequences of iid random variables $(\epsilon^+_k)_{k\geq 1}$ and $(\epsilon^-_k)_{k\geq 1}$ with $\epsilon^+_1 \sim f^+$ and $\epsilon^-_1 \sim f^-$, and the distributions $f^+, f^-$ in Assumption \ref{CB:ass:R}.
Analogously to the proof of Theorem \ref{CB:res:convQM-1}, we define the stopping times $(\tilde{\tau}_{b,k}, \tilde{\tau}_{a,k})_{k\geq 1}.$ \par
Now, our construction procedure based on the function $\tilde{\Psi}^{(B,Q,M)}: (\mathbb{R}^2 \times (0,\infty)^4) \times D([0,T], \mathbb{R}^4) \times ((0,\infty)^4)^{\mathbb{N}} \times ((0,\infty)^4)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^2 \times \mathbb{R}^4_+ \times \mathbb{R}^4_+) $ allows us to identify the distribution of the limit $(\tilde{B}, \tilde{Q},M)$ in further detail. First, let us study the distribution of the process $\tilde{Q}.$

\begin{The}[Limit distribution of the queue size process in active regimes]\label{CB:res:idQ-1}
	Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied.
	\begin{enumerate}
	\item[i)] On each interval $[\tilde{\tau}^*_k, \tilde{\tau}^*_{k+1}),$ $k \geq 0$, the limit $\tilde{Q}$ is distributed as a continuous semimartingale. In more detail,
	\begin{align}\label{CB:def:limitQ}
	\tilde{Q}\left(\cdot + \tilde{\tau}^*_k\right) \simeq \tilde{A}_k + \tilde{B}_k\, \text{ on } \left[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k\right),
	\end{align}
	where
	\begin{equation}\label{CB:def:tildeB}
	\tilde{B}_k := \tilde{Q}(\tilde{\tau}^*_k) + X(\cdot + \tilde{\tau}^*_k) - X(\tilde{\tau}^*_k)
	\end{equation}
	 is a four-dimensional linear Brownian motion with mean $\mu$ and covariance matrix $\Sigma$ (cf. Theorem \ref{CB:res:fCLT}) and
	 \begin{equation}\label{CB:def:tildeA}
	 \tilde{A}_k := (\tilde{A}^{b,F}_k, \tilde{A}_k^{a,F}, \tilde{A}_k^{b,G}, \tilde{A}_k^{a,G})
	 \end{equation}
	  is a four-dimensional continuous process of finite variation. Moreover, for $i = b,a$ and $t\in [0,\tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k)$, the components of $\tilde{A}_k$ are given by
	\begin{align*}
	   \tilde{A}^{i,F}_{k}(t) &:= \sum_{\hat{\tau}^i_{k,l} \leq t}\Bigg\{ \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_{l}(\pi_i\tilde{B}_k)(\hat{\tau}^i_{k,l}) \leq 0\right\}} \ell_{t}\left(\pi^{(2)}_1 g_{l}(\pi_i\tilde{B}_k)\right)\\
	   &\hspace{4cm} - \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_2 g_{l}(\pi_i\tilde{B}_k)(\hat{\tau}^i_{k,l}) \leq 0\right\}} \ell_{t}\left(\pi^{(2)}_2 g_{l}(\pi_i\tilde{B}_k)\right)\Bigg\},\\
	    \tilde{A}^{i,G}_{k}(t) &:= \sum_{\hat{\tau}^i_{k,l} \leq t}\Bigg\{ \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_2 g_{l}(\pi_i\tilde{B}_k)(\hat{\tau}^i_{k,l}) \leq 0\right\}} \ell_{t}\left(\pi^{(2)}_2 g_{l}(\pi_i\tilde{B}_k)\right)\\
	    &\hspace{4cm}- \textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_{l}(\pi_i\tilde{B}_k)(\hat{\tau}^i_{k,l}) \leq 0\right\}} \ell_t\left(\pi^{(2)}_1 g_{l}(\pi_i\tilde{B}_k)\right)\Bigg\}.
	\end{align*}
	For $k \geq 0$ and $i = b,a,$ the sequence of stopping times $(\hat{\tau}^i_{k,l})_{l\geq 1} := (\hat{\tau}_l(\pi_i \tilde{B}_k))_{l\geq 1}$ and the sequence of functions $g_l : D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2_+)$, $l\geq 1,$ are given in Definition \ref{CB:def:g}.
	\item[ii)]
	At each time $t = \tilde{\tau}^*_k$, $k\geq 1$, the process $\tilde{Q}$ is reinitialized by the random variable $\tilde{R}^+_k$ or $\tilde{R}^-_k$, i.e.,
\[\tilde{Q}(\tilde{\tau}^*_k) = \tilde{R}^{+}_k \text{ if } \tilde{\tau}_{a,k} = \tilde{\tau}^*_k \quad \text{ or } \quad \tilde{Q}(\tilde{\tau}^*_k) = \tilde{R}^-_k \text{ if } \tilde{\tau}_{b,k} = \tilde{\tau}^*_k.\]
	\item[iii)] Finally, the process $\tilde{Q}$ has only finitely many discontinuities and is therefore a semimartingale.
	\end{enumerate}
\end{The}

\begin{proof}
The relation $\tilde{Q} = \tilde{\Psi}^{Q}(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^{-})$ gives us a unique construction of our process $\tilde{Q}$, where $\tilde{Q}_0$ is its initial state, $X$ is introduced in Proposition \ref{CB:res:fCLT} and defines a four-dimensional linear Brownian motion with drift $\mu$ and covariance matrix $\Sigma,$ and $\tilde{R}^+, \tilde{R}^-$ are the sequences of random variables introduced in \eqref{CB:def:R1-limit} with values in $((0,\infty)^4)^{\mathbb{N}}$. Now, for all $k \geq 0,$ $\tilde{B}_k$ introduced in \eqref{CB:def:tildeB} is a four-dimensional linear Brownian motions starting at $\tilde{Q}(\tilde{\tau}^*_k)$ with drift $\mu$ and covariance matrix $\Sigma.$ By the definition of $\tilde{\Psi}^{Q}$, for $k \geq 0,$ we have
$$\tilde{Q}\left(\cdot + \tilde{\tau}^*_k\right)  = \tilde{\Psi}^{Q}\left(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^-\right)\left(\cdot + \tilde{\tau}^*_k\right) \simeq G(\tilde{B}_k) \quad \text{ on } \left[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k\right).$$
We note that $G$ is inductively defined through the sequence of functions $(g_l)_{l\geq 1}$ (cf. Definition \ref{CB:def:g} and equation \eqref{CB:def:G}). In what follows, we fix $i = b$ and $k\geq 0.$ Moreover, let $\hat{\tau}_l := \hat{\tau}^b_{k,l}$ for $l\geq 1$ and $\hat{\tau}_0 := 0.$\par
We prove the statement in i) by induction, i.e., setting $g_0(x_1, x_2) := (x_1, x_2),$ we show for $j\geq 1$ with $g_{j-1}(\pi_{b}\tilde{B}_k)(\hat{\tau}_{j-1}) \notin \mathbb{R}^2_-$ that $g_j(\pi_{b} \tilde{B}_k)$ is a continuous semimartingale. First, let $j = 1.$ By Definition \ref{CB:def:g} we have $g_1(\pi_{b}\tilde{B}_k) := \pi_{b}\tilde{B}_k.$ By construction, $\pi_{b}\tilde{B}_k$ is a planar Brownian motion, hence a continuous semimartingale. Next, assume that we have already shown for $j = 1, \cdots, l$ for which $g_{j-1}(\pi_{b}\tilde{B}_k)(\hat{\tau}_{j-1}) \notin \mathbb{R}^2_-$ that $g_j(\pi_{b}\tilde{B}_k)$ is a continuous semimartingale. Now, let us consider $j = l+1$ with $g_l(\pi_{b}\tilde{B}_k)(\hat{\tau}_l) \notin \mathbb{R}^2_-.$ By construction of the $(g_l)_{l\geq 1}$ in Definition \ref{CB:def:g}, we have
\begin{align}\label{CB:eq:gl1}
	g_{l+1}\left(\pi_{b}\tilde{B}_k\right) = Y^{(1),b}_{k,l}\textup{$\mathds{1}$}_{\left\{\pi^{(2)}_1 g_l\left(\pi_{b} \tilde{B}_k\right)(\hat{\tau}_l) \leq 0\right\}}+ Y^{(2),b}_{k,l}\textup{$\mathds{1}$}_{\left\{\pi^{(2)}_2 g_l\left(\pi_{b}\tilde{B}_k\right)(\hat{\tau}_l) \leq 0\right\}},
\end{align}
where
\begin{align}\label{CB:def:cRS1}
\begin{aligned}
    Y^{(1),b}_{k,l}(t)&:= g_l\left(\pi_{b}\tilde{B}_k\right)(t) + \ell_{t}\left(\pi^{(2)}_1 g_l\left(\pi_{b}\tilde{B}_k\right)\right) (1,-1),\\
    Y^{(2),b}_{k,l}(t)&:=g_l\left(\pi_{b}\tilde{B}_k\right)(t) + \ell_{t}\left(\pi^{(2)}_2 g_l\left(\pi_{b}\tilde{B}_k\right)\right) (-1, 1),
\end{aligned}
\end{align}
and $\ell_t(x):=\sup_{s\leq t}\{-x(s), 0\}$ defines the reflection at zero of $x\in D([0,T],\mathbb{R}).$ Now, the first component of $Y^{(1),b}_{k,l}$ and the second component of $Y^{(2),b}_{k,l}$ define the at zero reflected processes of a continuous semimartingale (note that $g_l(\pi_{b}\tilde{B}_k)$ is a continuous semimartingale by the induction hypothesis). In contrast, the second component of $Y^{(1),b}_{k,l}$ and the first component of $Y^{(2),b}_{k,l}$ are differences of a continuous semimartingale and the reflection at zero of another continuous semimartingale. Now, for any continuous semimartingale $X,$ we have that $\ell_t(X)$ is a monotone, non-decreasing, continuous process (cf. e.g. Lemma 2.1 in \cite{Z17}). Hence, $\ell_t(X)$ is a process of finite variation. We conclude that the processes $Y^{(1),b}_{k,l}$ and $Y^{(2),b}_{k,l}$ are continuous semimartingales. Therefore, also $g_{l+1}(\pi_{b}\tilde{B}_k)$ is a continuous semimartingale. This finishes our induction. \par
Since $\tilde{B}_k,$ $k \geq 0,$ are Brownian motions, for each $k \geq 0$, there exists a finite random variable $\kappa_{k} \in \mathbb{N}$ such that $g(\pi_{b}\tilde{B}_k) = g_{\kappa_{k}}(\pi_{b}\tilde{B}_k)$ on $[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k).$ Arguing analogously for $g(\pi_{a}\tilde{B}_k)$, we conclude that the process $\tilde{Q}$ is distributed between consecutive stopping times $\tilde{\tau}^*_k$ and $\tilde{\tau}^*_{k+1}$ according to $G(\tilde{B}_k)(\cdot -\tilde{\tau}^*_k)$ and is hence a continuous semimartingale on $[\tilde{\tau}^*_k, \tilde{\tau}^*_{k+1}).$\par
Finally, studying the construction of $\tilde{\Psi}^Q$ via the functions $G$ and $g_l$, $l\geq 1,$ (cf. equation \eqref{CB:def:G} and \eqref{CB:eq:gl1}), the finite-dimensional distributions of $\tilde{Q}$ satisfy \eqref{CB:def:limitQ}. This proves part i).\par

In order to prove part ii) and iii), we note that ii) and the fact that $\tilde{\Psi}^Q(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^-)$ has almost surely only finitely many discontinuities are already shown in the proof of Theorem \ref{CB:res:convQM-1}. Together with part i) we conclude that $\tilde{Q} = \tilde{\Psi}^Q(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^-)$ is a semimartingale which finally yields the statement iii).
\end{proof}

Before identifying the limit distribution of $M^{(n)}$, we state an intermediate result which links the reflection map applied to the continuous processes $\pi^{(2)}_1g_l(\pi_i \tilde{B}_k)$ and $\pi^{(2)}_2 g_l(\pi_i\tilde{B}_k)$ to the local times of its corresponding at zero reflected processes $\pi^{(2)}_1 Y^{(1),i}_{k,l}$ and $\pi^{(2)}_2 Y^{(2),i}_{k,l}$ (cf. \eqref{CB:def:cRS1} for their definitions), for $i = b,a,$ $k\geq 0,$ and $l\geq 1$. Let us first introduce the concept of a local time of a continuous semimartingale $X$.

\begin{Def}[Local time of a continuous semimartingale]
For an $\mathbb{R}$-valued continuous semimartingale $(X_t)_{t\geq 0}$ with quadratic variation process $(\langle X \rangle_t)_{t\geq 0}$, there exits a measurable process $(L^a_t)_{t\geq 0, a\in \mathbb{R}}$ such that almost surely for all bounded $\varphi: \mathbb{R} \rightarrow \mathbb{R}$
$$\int_0^t \varphi(X_s) d \langle X\rangle_s = \int_{\mathbb{R}} \varphi(a) L^a_t(X) da, \quad \text{ for } t \geq 0.$$
\end{Def}

The process $(L^a_t)_{t\geq 0, a \in \mathbb{R}}$ is called the local time of the process $X$ and has a modification such that almost surely $(t,a) \mapsto L^a_t$ is continuous in $t$ and c\`adl\`ag in $a$.  Without loss of generality, we will always work with this continuous modification without further comments. Moreover, for all $a \in \mathbb{R}$ the process $(L^a_t)_{t\geq 0}$ is monotone non-decreasing, hence of finite variation. For more details, see e.g. \cite{YY13, Z17}. In the following, we write $(L_t)_{t\geq 0} := (L^0_t)_{t\geq 0}.$

\begin{Lem}[Link between reflection maps and local times]\label{CB:res:LinkRMLT}
Let $\tilde{B}_k$, $k\geq 0$, be the Brownian motions introduced in \eqref{CB:def:tildeB} and $g_l : D([0,T], \mathbb{R}^2) \rightarrow D([0,T], \mathbb{R}^2_+),$ $\hat{\tau}_l: D([0,T], \mathbb{R}^2) \rightarrow [0,T],$ $l\geq 1,$ be as in Definition \ref{CB:def:g}. Moreover, for $i = b,a,$ let
\begin{equation}\label{CB:def:y1122}
Y^{(1,1), i}_{k,l}:= \pi_1^{(2)} Y^{(1),i}_{k,l} \quad \text{ and }\quad Y^{(2,2),i}_{k,l}:= \pi_2^{(2)} Y^{(2),i}_{k,l}
\end{equation}
 be the at zero reflected processes of $\pi^{(2)}_1 g_l(\pi_i \tilde{B}_k)$ respectively $\pi_2^{(2)} g_l(\pi_i \tilde{B}_k)$ introduced in \eqref{CB:def:cRS1}. Denote by $\hat{\tau}^i_{k,l} := \hat{\tau}_l(\pi_i \tilde{B}_k).$ Then, for all $t\in [0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k),$ we have
\begin{itemize}
	\item[i)] for $l = 1,$ $\ell_t(\pi_1^{(2)}g_1(\pi_i \tilde{B}_k)) = \frac{1}{2}L_t(Y^{(1,1),i}_{k,1})$ and $\ell_t(\pi_2^{(2)} g_1(\pi_i \tilde{B}_k)) = \frac{1}{2}L_t(Y^{(2,2),i}_{k,1}).$
	\item[ii)] For $l = 2$ and $j = 1,2,$ it holds
	\[\ell_t\left(\pi^{(2)}_j g_2(\pi_i\tilde{B}_k)\right) = \begin{cases}
	0 &: \pi^{(2)}_j g_1(\pi_i \tilde{B}_k)(\hat{\tau}^i_{k,1}) \leq 0\\
	\frac{1}{2}L_t(Y^{(j,j),i}_{k,2}) &: \text{otherwise}
	\end{cases}.\]
	\item[iii)] For $l \geq 3$ with $g_{l-1}(\pi_i \tilde{B}_k)(\hat{\tau}^i_{k,l-1}) \notin \mathbb{R}^2_-$ and $j = 1,2,$ we have
	\[\ell_t\left(\pi^{(2)}_j g_l(\pi_i \tilde{B}_k)\right)  = \frac{1}{2}L_t(Y^{(j,j),i}_{k,l}) - \frac{1}{2}L_t(Y^{(j,j),i}_{k,l-1})\]
	which equals zero if $\pi_j^{(2)}g_{l-1}(\pi_i \tilde{B}_k)(\hat{\tau}^i_{k,l-1}) \leq 0$ and equals $\frac{1}{2}L_t(Y^{(j,j),i}_{k,l}) - \frac{1}{2}L_t(Y^{(j,j),i}_{k,l-2})$ otherwise.
\end{itemize}
\end{Lem}

The proof is presented in Appendix \ref{CB:app:auxRes}. Next, with the help of the previous lemma, we can identify the limit $M$ of the number of cross-border trades $M^{(n)}$ in more detail.

\begin{The}[Limit distribution of the number of cross-border trades]\label{CB:res:idM-1}
Under the assumptions of Theorem \ref{CB:res:convQM-1}, the limit process $M := (M^{b,F}, M^{a,F},$ $M^{b,G}, M^{a,G})$ of the number of cross-border trades $M^{(n)}$ is a continuous process of finite variation. \par
In more detail, for $i = b,a,$ $k\geq 0,$ let $\tilde{B}_k$ be the Brownian motion introduced in \eqref{CB:def:tildeB}, $\tilde{\tau}^*_k$ be as in \eqref{CB:def:tau1-limit}, and $\hat{\tau}^i_{k,l} := \hat{\tau}_l(\pi_i \tilde{B}_k),$ $l\geq 1,$ be as in Definition \ref{CB:def:g}. Moreover, for $\kappa^i_k:= \min\{j\in \mathbb{N}: \hat{\tau}^i_{k,j} \geq \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k\}$, which is almost-surely finite, we have
\[\pi_i M(\cdot + \tilde{\tau}^*_k) \equiv \pi_i M(\tilde{\tau}^*_k) \quad \text{ on } [0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [0, \hat{\tau}^i_{k,1})\]
and on $[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k) \cap [\hat{\tau}^i_{k,l} , \hat{\tau}^i_{k,l+1}),$ $1 \leq l < \kappa^i_{k}$, we have
\begin{align*}
M^{i,F}(\cdot + \tilde{\tau}^*_k) &\simeq M^{i,F}(\tilde{\tau}^*_k) + \frac{1}{2}\left(L_{\cdot}(Y^{(1,1),i}_{k,l}) - U^{(1,1),i}_k\right),\\
M^{i,G}(\cdot + \tilde{\tau}^*_k) &\simeq M^{i,G}(\tilde{\tau}^*_k) + \frac{1}{2}\left(L_{\cdot}(Y^{(2,2),i}_{k,l}) - U^{(2,2),i}_k\right),
\end{align*}
where $Y^{(1,1),i}_{k,l}$ and $Y^{(2,2),i}_{k,l}$ were introduced in \eqref{CB:def:y1122}. For $j = 1,2$, the remainder process $U^{(j,j),i}_k\equiv 0$ on $[0,\tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k) \cap [0, \hat{\tau}^i_{k,2})$ and for $t \in [0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k) \cap [\hat{\tau}^i_{k,l}, \hat{\tau}^i_{k,l+1}),$ $2 \leq l < \kappa_k^i,$
\[U^{(j,j),i}_k(t) = L_t(Y^{(j,j),i}_{k,l-1}) - L_{\hat{\tau}^i_{k,l}-}(Y^{(j,j),i}_{k,l-1}) + \sum_{m=2}^{l-1}\left(L_{\hat{\tau}_{k,m+1}-}(Y^{(j,j),i}_{k,m-1}) - L_{\hat{\tau}^i_{k,m}-}(Y^{(j,j),i}_{k,m-1})\right).\]
\end{The}

\begin{Rmk}
	Observe that $U^{(1,1),i}_k$ and $U^{(2,2),i}_k$, $i = b,a,$ are monotonically non-decreasing processes on $[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k)$. In particular, for $l\geq 2,$ they only increase on the interval $[0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [\hat{\tau}^i_{k,l}, \hat{\tau}^i_{k,l+1})$ if and only if $t \mapsto L_{t}(Y^{(1,1),i}_{k,l-1})$ respectively $t \mapsto L_t(Y^{(2,2),i}_{k,l-1})$ increases on this interval as well.
\end{Rmk}

\begin{proof}
By the definition of $\tilde{\Psi}^{M}$, for each $k \geq 0,$ we have
\[M\left(\cdot + \tilde{\tau}^*_k\right) - M\left(\tilde{\tau}^*_k\right) \simeq \bar{G}(\tilde{B}_k) \quad \text{ on } \left[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k\right],\]
where $(\tilde{\tau}^*_k)_{k\geq 0}$ are the stopping times introduced in \eqref{CB:def:tau1-limit} and $\bar{G}(\tilde{B}_k)(0) \equiv 0.$ Because of the diffusive laws of the independent Brownian motions $\tilde{B}_k,$ $k\geq 0,$ there exist finite $\mathbb{N}$-valued $\kappa_k^b$ and $\kappa_k^a$ such that $(\hat{\tau}^b_{k, \kappa^b_k} \wedge \hat{\tau}^a_{k,\kappa^a_k}) = \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k.$ Without loss of generality, let us concentrate on the cross-border trades caused by bid orders and for $k\geq 0$ and $l\geq 1$, let us introduce the short-hand notations
\[\hat{\tau}_{k,l} := \hat{\tau}_l(\pi_{b}\tilde{B}_k), \quad g_k := g(\pi_b\tilde{B}_k),\quad\bar{g}_k := \bar{g}(\pi_{b}\tilde{B}_k), \quad \text{and} \quad g_{k,l} := g_l(\pi_{b}\tilde{B}_k).\]
By the definition of the function $\bar{g}$ (cf. Definition \ref{CB:def:barg}), for $k\geq 0,$ we have $\bar{g} \equiv 0$ on $[0,\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [0, \hat{\tau}_{k,1}).$ Hence, also $\pi_b M(\cdot + \tilde{\tau}^*_{k}) \equiv \pi_b M(\tilde{\tau}^*_k)$ on $[0,\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [0, \hat{\tau}_{k,1})$ for all $k \geq 0.$ Applying Lemma \ref{CB:res:conn-gbarg}, for $k\geq 0$, $1 \leq l< \kappa^b_k$, and $i=1,2$, we conclude for all $t \in [0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [\hat{\tau}_{k,l}, \hat{\tau}_{k,l+1})$,
\begin{align*}
    \pi^{(2)}_i\bar{g}_k(t)  & =  \sum_{j = 2}^l\left\{ \pi^{(2)}_i\bar{g}_k(\hat{\tau}_{k,j}-) -  \pi^{(2)}_i\bar{g}_k(\hat{\tau}_{k,j-1}-)\right\} + \pi^{(2)}_i\bar{g}_k(t) -  \pi^{(2)}_i\bar{g}_k(\hat{\tau}_{k,l}-)\\
    & = \sum_{j=2}^l \ell_{\hat{\tau}_{k,j}-}\left(\pi^{(2)}_ig_{k,j-1}\right) + \ell_t\left(\pi^{(2)}_i g_{k,l}\right).
\end{align*}
Moreover, for $k \geq 0$ and $i = 1,2,$ applying Lemma \ref{CB:res:LinkRMLT}, we are able to link the reflection map applied to the processes $(\pi^{(2)}_i g_{k,j})_{j\geq 1}$ to the local times of the at zero reflected processes $(Y^{(i,i)}_{k,j})_{j\geq 1}:= (Y^{(i,i),b}_{k,j})_{j\geq 1}$ introduced in \eqref{CB:def:y1122}. Hence, for $k\geq 0$, $1 \leq l < \kappa^b_k,$ and $i =1,2,$ we can write for $t \in [0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [\hat{\tau}_{k,l},\hat{\tau}_{k,l+1}),$
\begin{align*}
   \pi^{(2)}_i\bar{g}_k(t) = \frac{1}{2}L_t\left(Y^{(i,i)}_{k,l}\right) - \frac{1}{2}U^{(i,i)}_{k}(t),
\end{align*}
where $U^{(i,i)}_k(t) = 0$ for $t\in [0,\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap [0, \hat{\tau}_{k,2})$ and for $t \in [0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) \cap \left[\hat{\tau}_{k,l}, \hat{\tau}_{k,l+1}\right),$ $2 \leq l < \kappa^b_k,$
$$U^{(i,i)}_{k}(t) := L_t\left(Y^{(i,i)}_{k,l-1}\right) - L_{\hat{\tau}_{k,l}-}\left(Y^{(i,i)}_{k,l-1}\right) + \sum_{j=2}^{l-1} \left(L_{\hat{\tau}_{k,j+1}-}\left(Y^{(i,i)}_{k,j-1}\right) - L_{\hat{\tau}_{k,j}-}\left(Y^{(i,i)}_{k,j-1}\right)\right).$$
Since $(Y^{(i,i)}_{k,l})$, $1 \leq l <\kappa^b_k,$ $k\geq 0,$ and $i = 1,2,$ are continuous semimartingales (cf. Theorem \ref{CB:res:idQ-1}), the process $U^{(i,i)}_k$ is, with probability one, the sum of finitely many local times on $[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k).$ Moreover, observe that $U^{(i,i)}_k$, $k\geq 0$ and $i = 1,2,$ is a monotonically non-decreasing continuous process which increases on the interval $[0,\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k)\cap [\hat{\tau}_{k,l}, \hat{\tau}_{k,l+1})$, $l\geq 2,$ if and only if the $t\mapsto L_t(Y^{(i,i)}_{k,l-1})$ increases on the same interval. Hence, $U^{(i,i)}_k$ is a continuous process of finite variation on each $[0,\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k)$ and therefore, also $\pi^{(2)}_i \bar{g}_k$ restricted to $[0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_{k})$ is a continuous process of finite variation. Moreover, for $k\geq 0$ and $i = 1,2,$ applying again Lemma \ref{CB:res:conn-gbarg} for $t = \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k$ and the continuity of the reflection map, we conclude that
\begin{equation*}
\pi_i^{(2)}\bar{g}_k(\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k) = \pi^{(2)}_i \bar{g}_k(\tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k-).
\end{equation*}
Hence, for $k\geq 0$ and $i =1,2,$ we conclude that $\pi_i^{(2)} \bar{g}_k$ is a continuous process of finite variation on $[0, \tilde{\tau}^*_{k+1} - \tilde{\tau}^*_k].$ By construction of $\bar{G}(\tilde{B}_k)$, we conclude that $\bar{G}(\tilde{B}_k)$, $k\geq 0,$ is a continuous process of finite variation on $[0, \tilde{\tau}^*_{k+1}-\tilde{\tau}^*_k]$ for all $k\geq 0.$ Arguing again as in the proof of Theorem \ref{CB:res:convQM-1}, there exists a finite, $\mathbb{N}$-valued $N_T$ such that $\tilde{\tau}^*_{N_T+1} = T$ with probability one. Hence, $M$ is almost surely the sum of $N_T+1$ continuous processes of finite variation. Hence, $M$ is itself a continuous process of finite variation.
\end{proof}

Finally, by the continuity of each component of the limit $M$, we conclude a heavy traffic result for the non-restricted capacity process. Therefore, for $\omega \in D([0,T], \mathbb{R}^4)$, let us introduce
$$h_2: D([0,T], \mathbb{R}^4) \rightarrow D([0,T], \mathbb{R}), \quad h_2(\omega) = \pi_2\omega + \pi_3\omega - \pi_1\omega - \pi_4 \omega.$$

\begin{Cor}[Limit theorem for the capacity process in active regimes]\label{CB:res:convC}
Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied and assume that $\tilde{C}^{(n)}_0 \rightarrow \tilde{C}_0 \in \mathbb{R}$. Then, the capacity process $\tilde{C}^{(n)} = \tilde{C}^{(n)}_0 + h_2(M^{(n)})$ converges weakly in the Skorokhod topology on the space $D([0,T], \mathbb{R})$ to the continuous, finite-variation process $\tilde{C} =\tilde{C}_0 + h_2(M)$.
\end{Cor}

\begin{proof}
By Theorem \ref{CB:res:convQM-1} we have that $M^{(n)} \Rightarrow M$ in the Skorokhod topology. Moreover, by Theorem \ref{CB:res:idM-1} we conclude that each component of $M$ is a continuous process of finite variation. Therefore, using continuous mapping, we are able to conclude the weak convergence in the Skorokhod topology also for the sum, i.e.,
\begin{align*}
    \tilde{C}^{(n)} = \tilde{C}^{(n)}_0 + h_2(M^{(n)}) \Rightarrow \tilde{C}_0 + h_2(M) =: \tilde{C}
\end{align*}
in the Skorohkod topology and in particular the limit $\tilde{C}$ is again a continuous process of finite variation.
\end{proof}

\begin{Rmk}[Queue size and capacity process live on different time scales]
    By construction, both the queue size process $\tilde{Q}^{(n)}$ and the capacity process $\tilde{C}^{(n)}$ change by incoming orders of size $\pm \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}.$ However, the above limit results yield under the chosen scaling of $(\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)})^2 \approx \Delta t^{(n)}$ that $\tilde{Q}^{(n)}$ is approximated in the limit by a semimartingale with a non-trivial martingale part whereas the capacity process is approximated by a continuous process of finite variation. Hence, $\tilde{Q}^{(n)}$ lives on a finer time scale than the capacity process. This can already be deduced from the microscopic construction, as the queue size process changes by each incoming market and limit order but the capacity process only changes by market orders leading to cross-border trades.
\end{Rmk}

Next, we derive a limit theorem for the cumulative queue size process $h(\tilde{Q}^{(n)}).$

\begin{The}[Limit theorem for the cumulative queue size process]\label{CB:res:convcumQ}
Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied. Then, $h(\tilde{Q}^{(n)})$ converges weakly in the Skorokhod topology on the space $D([0,T], \mathbb{R}^2_+)$ to the $\mathbb{R}^2_+$-valued process $h(\tilde{Q}).$
\end{The}

	The proof resembles the idea of the proof of Theorem \ref{CB:res:convQM-1} and Theorem 2 in \cite{CL21}. Therefore, we only present a proof sketch.

\begin{proof}
First note that times of reinitialization of $\tilde{Q}^{(n)}$ and $h(\tilde{Q}^{(n)})$ coincide and are equal to $\tilde{\tau}^{(n)}_1, \tilde{\tau}^{(n)}_2,\cdots$ for all $n\in \mathbb{N}.$ Moreover, the reinitialization of $h(\tilde{Q}^{(n)})$ at $\tilde{\tau}^{(n)}_k$ equals $h(R^{+,(n)}_k)$ or $h(R^{-,(n)}_k)$ for $k\geq 1.$ Then, by Theorem \ref{CB:res:convQM-1}, we conclude that also for all $k\geq 1,$
	\begin{align*}
	&\left(\tilde{\tau}^{(n)}_1, \cdots, \tilde{\tau}^{(n)}_k, h(R^{+,(n)}_1), \cdots, h(R^{+,(n)}_k), h(R^{-,(n)}_1), \cdots, h(R^{-,(n)}_k)\right) \overset{n\rightarrow \infty}{\Rightarrow}\\
	&\hspace{5cm} \left(\tilde{\tau}^*_1, \cdots, \tilde{\tau}^*_k, h(R^+_1), \cdots, h(R^+_k), h(R^-_1), \cdots, h(R^-_k)\right).
	\end{align*}
	Let us introduce the sequences $h(R^{+,(n)}) := (h(R^{+,(n)}_k))_{k\geq 1},$ $h(R^{-,(n)}) := (h(R^{-,(n)}_k))_{k\geq 1},$ $h(R^+):=(h(R^+_k))_{k\geq 1},$ and $h(R^-) := (h(R^-_k))_{k\geq 1}.$ Now, similarly as for $\tilde{Q}^{(n)}$ and $\tilde{Q}$, we can construct a function
\[\hat{\Psi}^Q: D([0,T], \mathbb{R}^2) \times (\mathbb{R}^2_+)^{\mathbb{N}} \times (\mathbb{R}^2_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^2_+)\]
such that
\[h(\tilde{Q}^{(n)}) = \hat{\Psi}^Q\left(h(\tilde{Q}^{(n)}_0 + X^{(n)}), h(R^{+,(n)}), h(R^{-,(n)})\right)\]
and $(h(\tilde{Q}_0 + X), h(R^+), h(R^-))$ lies, with probability one, in the continuity set of $\hat{\Psi}^Q.$ Then, applying Corollary \ref{CB:res:fCLT-cNOF} and the continuous mapping theorem, we conclude that
\[h(\tilde{Q}^{(n)}) = \hat{\Psi}^Q\left(h(\tilde{Q}^{(n)}_0 + X^{(n)}), h(R^{+,(n)}), h(R^{-,(n)})\right) \Rightarrow \hat{\Psi}^Q\left(h(\tilde{Q}_0 + X), h(R^+), h(R^-)\right)\]
in the Skorokhod topology. In particular, one can show that $\hat{\Psi}^Q(h(\tilde{Q}_0 + X), h(R^+), h(R^-)) = h(\tilde{Q}).$ For details, we refer to Theorem 2 in Cont and de Larrard \cite{CL21}.
\end{proof}

The next result gives us a characterization of the limit distribution of the cumulative queue size process $h(\tilde{Q}^{(n)})$. For that, we denote $F^+:= h(f^+)$ and $F^- := h(f^-)$, where $f^+, f^-$ are the distributions introduced in Assumption \ref{CB:ass:R}. Moreover, for $\Phi \in C^2(\mathbb{R}^4_+ \times \mathbb{R}^4_+, (0,\infty)^4)$ introduced in Assumption \ref{CB:ass:R} ii), we need the following additional assumption that guarantees the existence of a function $\hat{\Phi} \in C^2(\mathbb{R}^2_+ \times \mathbb{R}^2_+, (0,\infty)^2)$ such that $h(\Phi(x,y)) = \hat{\Phi}(h(x), h(y))$ for all $x,y \in \mathbb{R}^4_+.$

\begin{Ass}\label{CB:ass:addPhi}
	Let $\Phi \in C^2(\mathbb{R}^4_+ \times \mathbb{R}^4_+, (0,\infty)^4)$ be as in Assumption \ref{CB:ass:R} ii) and denote $\Phi_i = \pi_i \Phi \in C^2(\mathbb{R}^4_+ \times \mathbb{R}^4_+, (0,\infty))$ for $i = 1,2,3,4.$ There exist $\tilde{\Phi}_i \in C^2(\mathbb{R}^2_+ \times \mathbb{R}^2_+, (0,\infty))$ for $i =1,2, 3,4,$ and $\alpha, \tilde{\alpha} \in C^2(\mathbb{R}^4_+ \times \mathbb{R}^4_+, \mathbb{R})$ satisfying for all $x,y \in \mathbb{R}^4_+$
	\[\min_{i =1,3}|\tilde{\Phi}_i(h(x),h(y))| > |\alpha(x,y)|, \quad \min_{i=2,4}|\tilde{\Phi}_i(h(x),h(y))| > |\tilde{\alpha}(x,y)|,\]
	and
	\begin{align*}
			\Phi_1(x,y) &= \tilde{\Phi}_1(h(x),h(y)) + \alpha(x,y), & \quad \Phi_3(x,y)&=\tilde{\Phi}_3(h(x), h(y)) - \alpha(x,y),\\
			\Phi_2(x,y) &= \tilde{\Phi}_2(h(x),h(y)) + \tilde{\alpha}(x,y), &\quad \Phi_4(x,y) &= \tilde{\Phi}_4(h(x), h(y)) - \tilde{\alpha}(x,y).
	\end{align*}
\end{Ass}

\begin{Prop}\label{CB:res:idcumQ}
Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied. The limit process $h(\tilde{Q})$ of the cumulative queue size process satisfies the following: let $(\tilde{\tau}^*_k)_{k\geq 0}$ be the sequence of stopping times introduced in \eqref{CB:def:tau1-limit}.
\begin{enumerate}
    \item[i)] For all $k \geq 0,$ inbetween two stopping times $\tilde{\tau}^*_k$ and $\tilde{\tau}^*_{k+1},$ the distribution of $h(\tilde{Q})$ follows that of a planar Brownian motion starting at $h(\tilde{Q})(\tilde{\tau}^*_k)$ with drift $\hat{\mu}$ and covariance matric $\hat{\Sigma}$ being defined in Corollary \ref{CB:res:fCLT-cNOF}.
    \item[ii)] In addition, let Assumption \ref{CB:ass:addPhi} be satisfied. Then,  $h(\tilde{Q})$ is a Markov process with values almost surely in $\mathbb{R}^2_+ \setminus \{(0,0)\}$ with infinitesimal generator $\tilde{\mathcal{G}}$ given by
    \begin{align}\label{CB:eq:defGen}
    \begin{aligned}
        \tilde{\mathcal{G}}g(x,y) &= \hat{\mu}^b \frac{\partial g}{\partial x} + \hat{\mu}^a \frac{\partial g}{\partial y} + \frac{\hat{\sigma}_{1,1}^2}{2}\frac{\partial^2 g}{\partial x^2} + \frac{\hat{\sigma}_{2,2}^2}{2}\frac{\partial^2 g}{\partial y^2} + \frac{\hat{\sigma}_{1,2} \partial^2 g}{\partial x \partial y},\\
        \tilde{\mathcal{G}}g(x,0) &= \int_{\mathbb{R}^2_+} \tilde{\mathcal{G}}g\left(\hat{\Phi}((x,0),(u,v))\right)F^+(du, dv),\\
        \tilde{\mathcal{G}}g(0,y) &= \int_{\mathbb{R}^2_+} \tilde{\mathcal{G}}g\left(\hat{\Phi}((0,y) ,(u,v))\right)F^-(du, dv),
    \end{aligned}
    \end{align}
    for $x > 0$, $y > 0$, whose domain is the set $\ensuremath\mathrm{dom}(\tilde{\mathcal{G}})$ of functions $C^2((0,\infty)^2, \mathbb{R}) \cap C^0(\mathbb{R}^2_+, \mathbb{R})$ verifying the boundary conditions
    \begin{align*}
        \forall \,x> 0, \quad &g(x,0) = \int_{\mathbb{R}^2_+}g\left(\hat{\Phi}((x,0), (u,v))\right)F^+(du, dv),\\
        \forall \,y >0, \quad &g(0,y) = \int_{\mathbb{R}^2_+}g\left(\hat{\Phi}((0,y), (u,v))\right)F^-(du, dv).
    \end{align*}
\end{enumerate}
\end{Prop}

The proof of the above theorem resembles the ideas of the proof of Lemma 5 in \cite{CL21}, but for completeness sake, we present the proof in Appendix \ref{CB:app:B-cumQlimit}. \par
The above proposition gives us a detailed description of the limit distribution of the price dynamics in the active regimes. In particular, the limit approximation of the bid price process is entirely characterized by hitting times of the axes of the two-dimensional process $h(\tilde{Q}).$

\begin{Cor}[Limit distribution of the prices process in active regimes]
Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied. Then, the limit $\tilde{B} = (\tilde{B}^{F}, \tilde{B}^F)$ satisfies
\[\tilde{B}^F(t)  = \tilde{B}^F_0 + \delta \sum_{0\leq s \leq t}\left(\textup{$\mathds{1}$}_{\left\{\left(\pi^{(2)}_2 \circ h\right)(\tilde{Q})(s-) = 0\right\}} - \textup{$\mathds{1}$}_{\left\{ \left(\pi^{(2)}_1 \circ h\right)(\tilde{Q})(s-) = 0\right\}}\right)\]
and is a piecewise constant càdlàg process which
\begin{itemize}
	\item increases by one tick every time the process $h(\tilde{Q})$ hits the horizontal axis $\{y = 0\}$,
	\item decreases by one tick every time the process $h(\tilde{Q})$ this the vertical axis $\{x = 0\}.$
\end{itemize}
\end{Cor}

\begin{proof}
	This follows directly by the definition of the function $\tilde{\Psi}^B.$
\end{proof}

We end this subsection by presenting the joint limit theorem for the active dynamics $\tilde{S}^{(n)}.$ First, we rewrite the active dynamics by
\[\tilde{S}^{(n)} = \left(\tilde{B}^{(n)}, \tilde{Q}^{(n)}, \tilde{C}^{(n)}\right) = \tilde{\Psi}\left(\tilde{S}^{(n)}_0; X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right),\]
where
$$ \tilde{\Psi}: E \times D([0,T], \mathbb{R}^4) \times ((0,\infty)^4)^{\mathbb{N}} \times ((0,\infty)^4)^{\mathbb{N}} \rightarrow D([0,T], E)$$
is given for $s_0 := (b_0, q_0, c_0) \in E$ by
\begin{align*}
    \tilde{\Psi}&(s_0; \omega, R, \tilde{R})\\
    &:= \left(b_0 + \tilde{\Psi}^{B}(q_0 + \omega, R, \tilde{R}),\, \tilde{\Psi}^{Q}(q_0 + \omega, R, \tilde{R}),\, c_0 + \left(h_2\circ \tilde{\Psi}^{M}\right)(q_0 + \omega, R, \tilde{R})\right).
\end{align*}

Then, applying the previous convergence results, we derive the joint convergence of the active dynamics $\tilde{S}^{(n)}$ in the Skorokhod topology on $D([0,T], E).$

\begin{The}[Limit theorem for the active dynamics]\label{CB:res:convS1}
Let the assumptions of Theorem \ref{CB:res:convQM-1} be satisfied and let $\tilde{S}^{(n)}_0 \in (\delta \mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{Z}$ with $\tilde{B}^{F,(n)}_0 = \tilde{B}^{G,(n)}_0$ for all $n\in \mathbb{N}$ such that $\tilde{S}^{(n)}_0 \rightarrow \tilde{S}_0 \in (\delta \mathbb{Z})^2 \times (0,\infty)^4 \times \mathbb{R}$. Then we have
$$\tilde{S}^{(n)} = \tilde{\Psi}\left(\tilde{S}^{(n)}_0; X^{(n)}, \tilde{R}^{+,(n)}, \tilde{R}^{-,(n)}\right) \Rightarrow \tilde{\Psi}\left(\tilde{S}_0; X, \tilde{R}^+, \tilde{R}^-\right) =: \tilde{S}$$
in the Skorokhod topology on the space $D([0,T], E).$
\end{The}

\begin{proof}
	Thanks to Theorem \ref{CB:res:convQM-1}, we conclude the joint convergence of $(\tilde{B}^{(n)}, \tilde{Q}^{(n)}, M^{(n)})$ $\Rightarrow (\tilde{B}, \tilde{Q}, M)$ in the Skorokhod topology on the space $D([0,T], \mathbb{R}^2 \times \mathbb{R}^4_+ \times \mathbb{R}^4_+).$ Corollary \ref{CB:res:convC} yields the weak convergence of the capacity process in the Skorokhod topology and since its limit approximation\[\tilde{C} := \tilde{C}_0 + (h_2 \circ \tilde{\Psi}^M)(\tilde{Q}_0 + X, \tilde{R}^+, \tilde{R}^-)\]
	has almost surely continuous sample paths, also $(\tilde{B}^{(n)}, \tilde{Q}^{(n)}, \tilde{C}^{(n)}) \Rightarrow (\tilde{B}, \tilde{Q}, \tilde{C})$ in the Skorokhod topology on the space $D([0,T], E).$ This finishes the proof.
\end{proof}

\subsection{Probabilistic results of the limit dynamics}\label{CB:sec:ProbLimit}

We finish this section by presenting important properties of the process $\tilde{S}.$ They become important when deriving a limit theorem for the cross-border market model $S^{(n)}$.

\begin{Lem}\label{CB:res:cumQatSwitching}
    Let us introduce $\sigma := \inf\{t \geq 0: \tilde{C}(t) \leq -\kappa_- \text{ or } \tilde{C}(t) \geq \kappa_+\} \wedge T,$ the first time at which the process $\tilde{C}$ hits the boundary of $[-\kappa_-, \kappa_+].$ Then, with probability one, the limit $h(\tilde{Q})$ of the cumulative queue size process satisfies
    \begin{equation}\label{CB:eq:signeqtau}
    \ensuremath\mathbb{P}\left[h(\tilde{Q})(\sigma) \in (0,\infty)^2\right] = 1.
    \end{equation}
    In words, with probability one, the hitting time $\sigma$ do not coincide with the random time of a price change in $\tilde{S}.$
\end{Lem}

Note that thanks to the above result, with probability one, a regime switch from an active to an inactive regime in $\tilde{S}$ already occurs if $\tilde{C}$ hits the boundary of $[-\kappa_-, \kappa_+].$ Recall that $\tilde{Q} := (\tilde{Q}^{b,F}, \tilde{Q}^{a,F}, \tilde{Q}^{b,G}, \tilde{Q}^{a,G}).$ The following result show that, with probability one, no combination of bid and ask queue is simultaneous zero.

\begin{Lem}\label{CB:res:locLOBnotzero}
   	Let $I, J \in \{F,G\}.$ With probability one, we have
    \[\ensuremath\mathbb{P}\left[\exists \, t \in [0,T] : \left(\tilde{Q}^{b,I}, \tilde{Q}^{a,J}\right)(t) = (0,0)\right] = 0.\]
    In particular, if $I = J$, we conclude, with probability one, that no national limit order book in $\tilde{S}$ is empty.
\end{Lem}

The proofs are omitted at this point and are presented in Appendix \ref{CB:app:B-hfl}.

\section{A brief overview of the inactive dynamics}\label{CB:sec:anaID}

In this section, we introduce a heavy traffic approximation for the inactive dynamics. Note that the subsequent analysis and approximation of the inactive dynamics is, with one exception, a straight-forward extension of the reduced-form representation of a LOB introduced in Cont and de Larrard \cite{CL21} to two non-interacting LOBs. In the following, we only present the ideas and results and refer to Section \ref{CB:sec:anaAD} and \cite{CL21} for the technical details of the proofs. \par

\begin{Rmk}
Note that in Cont and de Larrard \cite{CL21} the weak convergence of prices is established in the slightly weaker and not so standard Skorokhod $M_1$-topology (cf. e.g. Whitt \cite{W02} for an introduction to the Skorokhod $M_1$-topology). However, by replicating our strategy for the analysis of the active dynamics in Section \ref{CB:sec:anaAD}, we are able to prove weak convergence of prices in the stronger Skorokhod $J_1$-topology.
\end{Rmk}

Let us recall that the inactive dynamics $\dbtilde{S}^{(n)} = (\dbtilde{S}^{(n)}(t))_{t\in [0,T]}$ is given by
\[\dbtilde{S}^{(n)}(t) = \dbtilde{S}^{(n)}_k \quad \text{ for } t \in \left[\t_k, \t_{k+1}\right),\]
where its discrete-time dynamics is defined in the equations \eqref{CB:def:ID1}-\eqref{CB:def:ID4} in Section \ref{CB:sec:defID}. Let $\dbtilde{S}^{(n)}_0 := (\dbtilde{B}^{(n)}_0, \dbtilde{Q}^{(n)}_0, \dbtilde{C}^{(n)}_0) \in (\delta\mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{Z}$ be the initial state of $\dbtilde{S}^{(n)}.$ Again, the limit results for the queue size process $\dbtilde{Q}^{(n)}$ and the price process $\dbtilde{B}^{(n)}$ are based on a limit result for the net order flow process $X^{(n)}$ (cf. Proposition \ref{CB:res:fCLT}) and Assumption \ref{CB:ass:R} guaranteeing the convergence of the sequences $\dbtilde{R}^{+,(n)}$, $\dbtilde{R}^{-,(n)}.$ \par
First, let us introduce a function $\dbtilde{\Psi}^{Q}: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^4_+)$ that can be used to describe the dynamics of the queue size process $\dbtilde{Q}^{(n)}$ over time.

\begin{Def}\label{CB:def:PsiQ2}
    For $\omega \in D([0,T], \mathbb{R}^4)$ and $R =  (R_n)_{n\geq 1},$ $\tilde{R} = (\tilde{R}_n)_{n\geq 1} \in (\mathbb{R}^4_+)^{\mathbb{N}}$ the function $\dbtilde{\Psi}^{Q}(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4_+)$ is defined as follows:
    \begin{itemize}
        \item For $t < \dbtilde{\tau}_1$, we have that $\dbtilde{\Psi}^{Q}(\omega,R, \tilde{R})(t) = \dbtilde{\varphi}_0(\omega, R, \tilde{R})(t) := \omega(t),$ where
        \[\dbtilde{\tau}_1 := \dbtilde{\tau}_1(\omega) := \inf\{t \geq 0: \exists \, i=1,\cdots, 4 \text{ such that } \pi_i \omega(t) \leq 0\} \wedge T\]
        is the first hitting time of $\omega$ of the axes.
        \item We set
        \begin{align*}
        		\pi_F \dbtilde{\Psi}^Q(\omega, R,\tilde{R})(\dbtilde{\tau}_1) = \begin{cases}
        		\pi_F \tilde{R}_1 &: \pi_1 \omega(\dbtilde{\tau}_1) \leq 0\\
        		\pi_F R_1 &: \pi_1 \omega(\dbtilde{\tau}_1) > 0 \text{ and } \pi_2 \omega(\dbtilde{\tau}_1) \leq 0\\
        		\pi_F \dbtilde{\Psi}^Q(\omega, R, \tilde{R})(\dbtilde{\tau}_1-)&: \text{otherwise}
        		\end{cases}
        \end{align*}
        and
        \begin{align*}
        \pi_G \dbtilde{\Psi}^Q(\omega, R, \tilde{R})(\dbtilde{\tau}_1) = \begin{cases}
        \pi_G \tilde{R}_1 &: \pi_3\omega(\dbtilde{\tau}_1) \leq 0\\
        \pi_G R_1 &: \pi_3 \omega(\dbtilde{\tau}_1) > 0 \text{ and } \pi_4 \omega(\dbtilde{\tau}_1) \leq 0\\
        \pi_G \dbtilde{\Psi}^Q(\omega, R, \tilde{R})(\dbtilde{\tau}_1 - ) &: \text{otherwise}
        \end{cases}.
        \end{align*}
        \item Let $l \geq 1$.
        \begin{itemize}
            \item[$\circ$] For $0 \leq t < \dbtilde{\tau}_{l+1} - \dbtilde{\tau}_{l}$, we set $\dbtilde{\Psi}^{Q}(\omega, R, \tilde{R})(t + \dbtilde{\tau}_{l}) = \dbtilde{\varphi}_{l}(\omega,R, \tilde{R})(t),$ where
            \[\dbtilde{\varphi}_{l}(\omega, R, \tilde{R}) := \dbtilde{\Psi}^{Q}(\omega,R, \tilde{R})(\dbtilde{\tau}_{l}) + \omega(\cdot + \dbtilde{\tau}_{l}) - \omega(\dbtilde{\tau}_{l})\]
            and
            \begin{align*}
            \dbtilde{\tau}_{l+1} &:= \dbtilde{\tau}_{l+1}(\omega,R, \tilde{R})\\
            &:=\inf\Big\{t\geq \dbtilde{\tau}_{l}: \exists \, i = 1,\cdots, 4 \text{ with } \pi_i \dbtilde{\varphi}_{l}(\omega,R,\tilde{R})(t-\dbtilde{\tau}_{l}) \leq 0\Big\} \wedge T.
            \end{align*}
            \item[$\circ$] We set
            \begin{align*}
            \pi_F \dbtilde{\Psi}^Q&(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1})\\
            & = \begin{cases}
            	\pi_F \tilde{R}_{l+1} &: \pi_1 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) \leq 0\\
            	\pi_F R_{l+1} &: \pi_1 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) > 0\\
            	&\quad \text{and } \pi_2 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) \leq 0\\
            	\pi_F \dbtilde{\Psi}^Q(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-) &: \text{otherwise}
            \end{cases}
            \end{align*}
            and
            \begin{align*}
            \pi_G \dbtilde{\Psi}^Q&(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1})\\
            &= \begin{cases}
            	\pi_G \tilde{R}_{l+1} &: \pi_3 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) \leq 0\\
            	\pi_G R_{l+1} &: \pi_3 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) > 0\\
            	& \quad \text{and } \pi_4 \dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-\dbtilde{\tau}_{l}) \leq 0\\
            	\pi_G \dbtilde{\Psi}^Q(\omega, R, \tilde{R})(\dbtilde{\tau}_{l+1}-) &: \text{otherwise}
            \end{cases}.
            \end{align*}
        \end{itemize}
        \item For $t \geq \dbtilde{\tau}_{\infty} := \lim_{l\rightarrow \infty}\dbtilde{\tau}_l,$ we set $\dbtilde{\Psi}^{Q}(\omega, R, \tilde{R})(t) = (0,0,0,0).$
    \end{itemize}
\end{Def}

The function $\dbtilde{\Psi}^{Q}$ is obtained by regulating the path of $\omega$ with help of the sequences $R, \tilde{R}:$ between two consecutive times $\dbtilde{\tau}_l$ and $\dbtilde{\tau}_{l+1}$ the function $\dbtilde{\Psi}^{Q}(\omega,R,\tilde{R})$ equals
$\dbtilde{\varphi}_{l}(\omega, R, \tilde{R})(\cdot - \dbtilde{\tau}_l).$
The first time $\dbtilde{\varphi}_{l}(\omega,R,\tilde{R})$ hits the axes, the components corresponding to the origin of the incoming order are reinitialized according to $R_{l+1}$ or $\tilde{R}_{l+1}$, whereas the components corresponding to the foreign order book stay unchanged.

Next, let us introduce the continuity set of $\dbtilde{\Psi}^{Q}$. Therefore, let us introduce the functions space $C_0'([0,T], \mathbb{R}^2 \setminus\{(0,0)\} \times \mathbb{R}^2 \setminus \{(0,0)\})$ that contains all continuous functions $\omega \in C([0,T], \mathbb{R}^4)$ whose components cross the axes each time they touch them and whose projections $\pi_F\omega,$ $\pi_G\omega$ avoid the origin $(0,0)$.

\begin{The}[Continuity of $\dbtilde{\Psi}^{Q}$]\label{CB:res:contPsi2}
    Let $(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ be such that
    \begin{itemize}
    \item[i)] $\omega(0),$ $R_l$, and $\tilde{R}_l \in (0,\infty)^4$ for all $l\geq 1,$
    	\item[ii)] there exists a finite, $\mathbb{N}$-valued $N_T$ such that $\dbtilde{\tau}_{N_T+1} = T$, and
    	\item[iii)] $\omega$ and $\dbtilde{\varphi}_l(\omega, R, \tilde{R}) \in C'_0([0,T], \mathbb{R}^2\setminus\{(0,0)\} \times \mathbb{R}^2\setminus\{(0,0)\})$ for all $1\leq l \leq N_T,$ and
    	\item[iv)] for $I = F,G$,
    	\[\pi_I \dbtilde{\varphi}_{N_T}(\omega, R, \tilde{R})(T-\dbtilde{\tau}_{N_T}) \in (0,\infty)^2.\]
    	\end{itemize}
    	Then, the function $\dbtilde{\Psi}^Q: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^4_+)$ introduced in Definition \ref{CB:def:PsiQ2} is continuous at $(\omega, R, \tilde{R}).$
\end{The}

Using the procedure introduced by the function $\dbtilde{\Psi}^Q$, we found a new representation of the queue size process $\dbtilde{Q}^{(n)},$ i.e., for all $n\in \mathbb{N}$, we have
    $$\dbtilde{Q}^{(n)} = \dbtilde{\Psi}^{Q}\left(\dbtilde{Q}^{(n)}_0 + X^{(n)}, \dbtilde{R}^{+,(n)}, \dbtilde{R}^{-,(n)}\right),$$
    where $\dbtilde{Q}^{(n)}_0$ are the initial queue sizes, $X^{(n)}$ is introduced in \eqref{CB:def:netorderflow}, and $\dbtilde{R}^{+,(n)}$ respectively $\dbtilde{R}^{-,(n)}$ describe the queue sizes after price increases/decreases in $\dbtilde{S}^{(n)}$. \par
    Next, let us turn to the price dynamics in the inactive dynamics $\dbtilde{S}^{(n)}$. Therefore, let us introduce a function $\dbtilde{\Psi}^{B}: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^2)$ that can be used to describe the price dynamics $\dbtilde{B}^{(n)}$ over time.

\begin{Def}\label{CB:def:PsiB2}
    Let $\omega \in D([0,T], \mathbb{R}^4)$ and let $R, \tilde{R}$ be sequences with values in $\mathbb{R}^4_+.$ Let the functions $(\dbtilde{\varphi}_k(\omega,R,\tilde{R}))_{k\geq 0}$ and $(\dbtilde{\tau}_k(\omega, R, \tilde{R}))_{k\geq 1}$ be as in Definition \ref{CB:def:PsiQ2}. Moreover, let us denote $\dbtilde{\varphi}_k := \dbtilde{\varphi}_k(\omega, R, \tilde{R})$ and $\dbtilde{\tau}_k := \dbtilde{\tau}_k(\omega, R, \tilde{R})$ with $\dbtilde{\tau}_0 := 0.$ Then, we define
    \begin{align*}
    N_i(\omega, R, \tilde{R})(t) := \sum_{\dbtilde{\tau}_k \leq t} \textup{$\mathds{1}$}_{\left\{\pi_i \dbtilde{\varphi}_{k-1}(\dbtilde{\tau}_k-\dbtilde{\tau}_{k-1})\leq 0\right\}} \quad \text{for } i = 1,2,3,4,
    \end{align*}
    and
    \begin{align*}
        \dbtilde{\Psi}^{B}&(\omega, R, \tilde{R})(t)\\
        &:= \delta \left(N_2(\omega,R, \tilde{R})(t) - N_1(\omega,R, \tilde{R})(t), \, N_4(\omega, R, \tilde{R})(t) - N_3(\omega, R, \tilde{R})(t)\right).
    \end{align*}
\end{Def}

Next, let us introduce the continuity set of $\dbtilde{\Psi}^B.$

\begin{The}
	Let $(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ satisfy the conditions i)-iv) in Theorem \ref{CB:res:contPsi2}. Moreover, for each $1\leq i, j \leq 4$ with $i \neq j,$ assume that
	\begin{equation}
	\ensuremath\mathrm{Disc}\left(N_i(\omega, R, \tilde{R})\right) \cap \ensuremath\mathrm{Disc}\left(N_j(\omega, R, \tilde{R})\right) = \text{\O},
	\end{equation}
	where $\ensuremath\mathrm{Disc}(N_i(\omega, R, \tilde{R})) := \{t\in [0,T]: N_i(\omega, R, \tilde{R})(t-) \neq N_i(\omega, R, \tilde{R})(t)\}$ for $i = 1,2,3,4.$ Then, the map $\dbtilde{\Psi}^B: D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], \mathbb{R}^2)$ is continuous at $(\omega, R, \tilde{R}).$
\end{The}

Again, with the procedure introduced by the function $\dbtilde{\Psi}^B,$ we can rewrite the bid price process $\dbtilde{B}^{(n)}$, i.e., for all $n\in \mathbb{N}$, we have
\[\dbtilde{B}^{(n)} = \dbtilde{B}^{(n)}_0 + \dbtilde{\Psi}^B\left(\dbtilde{Q}^{(n)}_0 + X^{(n)}, \dbtilde{R}^{+,(n)}, \dbtilde{R}^{-,(n)}\right).\]

Next, let us introduce the function
    $$\dbtilde{\Psi}: E \times D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}} \rightarrow D([0,T], E),$$
where for $s_0 := (b_0, q_0, c_0) \in E$ and $(\omega, R, \tilde{R}) \in D([0,T], \mathbb{R}^4) \times (\mathbb{R}^4_+)^{\mathbb{N}} \times (\mathbb{R}^4_+)^{\mathbb{N}}$ we have
\begin{align*}
    \dbtilde{\Psi}(s_0; \omega, R, \tilde{R}):= \left(b_0 + \dbtilde{\Psi}^{B}(q_0 + \omega, R, \tilde{R}), \dbtilde{\Psi}^{Q}(q_0 + \omega, R, \tilde{R}), c_0\right).
    \end{align*}

Replicating the arguments of Theorem \ref{CB:res:convQM-1}, we are able to derive a joint limit theorem for the inactive dynamics $\dbtilde{S}^{(n)} := (\dbtilde{B}^{(n)}, \dbtilde{Q}^{(n)}, \dbtilde{C}^{(n)}).$

\begin{The}[Limit theorem for the inactive dynamics]\label{CB:res:convS2}
    Let Assumption \ref{CB:ass:prob}, \ref{CB:ass:scaling}, and \ref{CB:ass:R} be satisfied. Assume that $\dbtilde{S}^{(n)}_0 \in (\delta \mathbb{Z})^2 \times (\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{N})^4 \times \Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} \mathbb{Z}$ with $\dbtilde{S}^{(n)}_0 \rightarrow \dbtilde{S}_0 \in (\delta \mathbb{Z})^2 \times (0,\infty)^4 \times \mathbb{R} \subset E.$ Then, we have
    \begin{align*}
    \dbtilde{S}^{(n)} =  \dbtilde{\Psi}\left(S^{(n)}_0; X^{(n)}, \dbtilde{R}^{+,(n)}, \dbtilde{R}^{-,(n)}\right) \Rightarrow \dbtilde{\Psi}\left(\dbtilde{S}_0; X, \dbtilde{R}^+, \dbtilde{R}^-\right) =: \dbtilde{S}
    \end{align*}
    in the Skorokhod topology on the space $D([0,T], E).$
\end{The}

Next, we want to describe the distribution of the limit processes $\dbtilde{B}$ and  $\dbtilde{Q}.$ Therefore, we need to introduce the sequence of stopping times at which we observe price changes in the high-frequency limit. Let $\dbtilde{\tau}: D([0,T], \mathbb{R}^4) \rightarrow [0,T]$ be given by
\[\dbtilde{\tau}(\omega):= \inf\{t\geq 0: \exists\, i = 1,2,3,4 \text{ with } \pi_i \omega(t) \leq 0\}\wedge T.\]
Then, we set $\dbtilde{\tau}^*_0 := 0,$ $\dbtilde{\tau}_1:=\dbtilde{\tau}(\dbtilde{Q}_0 + X),$ and
\begin{align}\label{CB:def:tau2-limit}
    \dbtilde{\tau}^*_k := \dbtilde{\tau}^*_{k-1} + \dbtilde{\tau}\left(\dbtilde{\varphi}_{k-1}\left(\dbtilde{Q}_0 + X, \dbtilde{R}^{+}, \dbtilde{R}^-\right)\right), \quad \text{ for } k\geq 2,
\end{align}
where the random sequences $\dbtilde{R}^+, \dbtilde{R}^-$ describe the queue sizes after price changes in $\dbtilde{S}$ with
\[\dbtilde{R}^+_k := \Phi\left(\dbtilde{Q}(\dbtilde{\tau}^*_k-), \epsilon^+_k\right)  \quad \text{ and } \quad \dbtilde{R}^-_k := \Phi\left(\dbtilde{Q}(\dbtilde{\tau}^*_k-), \epsilon^-_k\right)\]
for independent sequences of iid random variables $(\epsilon^+_k)_{k\geq 1}$ and $(\epsilon^-_k)_{k\geq 1}$ with $\epsilon^+_1 \sim f^+$ and $\epsilon^-_1 \sim f^-,$ and the distributions $f^+, f^-$ in Assumption \ref{CB:ass:R}. In the next proposition, we identify the limit process $\dbtilde{Q}$ as a Markov process similarly to Theorem 2 and Lemma 5 in \cite{CL21}.

\begin{Prop}[Limit distribution of the queue size process in inactive regimes]\label{CB:res:idQ-2}
    Let the assumptions of Theorem \ref{CB:res:convS2} be satisfied and let $(\dbtilde{\tau}^*_k)_{k\geq 0}$ be the sequence of stopping times introduced in \eqref{CB:def:tau2-limit}. The limit $\dbtilde{Q}$ is a four-dimensional Markov process with values almost surely in $\mathbb{R}^4_+ \setminus \{(x_1, x_2, x_3, x_4): \exists\, i, j, \, i \neq j \text{ with } x_i = x_j = 0\}$ satisfying the following:
    \begin{enumerate}
        \item[i)] Inbetween two consecutive stopping times $\dbtilde{\tau}^*_k$ and $\dbtilde{\tau}^*_{k+1}$ the process $\dbtilde{Q}$ follows a four-dimensional linear Brownian motion starting at $\dbtilde{Q}(\dbtilde{\tau}^*_k)$ with drift $\mu$ and covariance matrix $\Sigma$ (see Proposition \ref{CB:res:fCLT} for the definitions of $\mu$ and $\Sigma$).
        \item[ii)] The infinitesimal generator $\dbtilde{\mathcal{G}}$ of $\dbtilde{Q}$ is given by
        \begin{align}\label{CB:eq:defGen-Q2a}
        \begin{aligned}
            \dbtilde{\mathcal{G}}g(x_1,x_2, y_1, y_2)&=\mu^{b,F}\frac{\partial g}{\partial x_1} + \mu^{a,F}\frac{\partial g}{\partial x_2} + \mu^{b,G}\frac{\partial g}{\partial y_1} + \mu^{a,G}\frac{\partial g}{\partial y_2}\\
            &\hspace{-2.5cm} \quad + \frac{(\sigma^{b,F})^2}{2}\frac{\partial^2 g}{\partial x_1^2} + \frac{(\sigma^{a,F})^2}{2}\frac{\partial^2 g}{\partial x_2^2} + \frac{(\sigma^{b,G})^2}{2}\frac{\partial^2 g}{\partial y_1^2} + \frac{(\sigma^{a,G})^2}{2}\frac{\partial^2 g}{\partial y_2^2}\\
            &\hspace{-2.5cm}\quad  + \sigma^{(b,F),(a,F)}\frac{\partial^2 g}{\partial x_1 \partial x_2} + \sigma^{(b,F),(b,G)}\frac{\partial^2 g}{\partial x_1 \partial y_1} + \sigma^{(b,F), (a,G)}\frac{\partial^2 g}{\partial x_1 \partial y_2}\\
            &\hspace{-2.5cm}\quad  + \sigma^{(a,F),(b,G)}\frac{\partial^2 g}{\partial x_2 \partial y_1} + \sigma^{(a,F), (a,G)}\frac{\partial^2 g}{\partial x_2 \partial y_2} + \sigma^{(b,G), (a,G)}\frac{\partial^2 g}{\partial y_1 \partial y_2}
        \end{aligned}
        \end{align}
        and
        \begin{align}\label{CB:eq:defGen-Q2b}
        \begin{aligned}
            \dbtilde{\mathcal{G}}g(x_1,0, y_1, y_2) &= \int_{\mathbb{R}^4_+}\dbtilde{\mathcal{G}}g(\pi_{F}\Phi((x_1,0, y_1, y_2),u), y_1, y_2)f^+(du),\\
            \dbtilde{\mathcal{G}}g(0, x_2 ,y_1, y_2) &= \int_{\mathbb{R}^4_+}\dbtilde{\mathcal{G}}g(\pi_{F}\Phi((0, x_2, y_1, y_2),u), y_1, y_2)f^-(du),\\
            \dbtilde{\mathcal{G}}g(x_1, x_2, y_1, 0) &= \int_{\mathbb{R}^4_+}\dbtilde{\mathcal{G}}g( x_1, x_2, \pi_{G}\Phi((x_1, x_2, y_1, 0),u))f^+(du),\\
            \dbtilde{\mathcal{G}}g(x_1, x_2, 0, y_2) &= \int_{\mathbb{R}^4_+}\dbtilde{\mathcal{G}}g(x_1, x_2, \pi_{G}\Phi((x_1, x_2, 0, y_2),u))f^-(du),
        \end{aligned}
        \end{align}
        for $x_1,x_2, y_1, y_2 > 0$ and $u \in \mathbb{R}^4_+$, whose domain is the set $\ensuremath\mathrm{dom}(\dbtilde{\mathcal{G}})$ of functions $C^2((0,\infty)^4, \mathbb{R}) \cap C^0(\mathbb{R}^4_+, \mathbb{R})$ verifying  for all $x_1, x_2, y_1, y_2 > 0$ and $u \in \mathbb{R}^4_+$ the boundary conditions
        \begin{align*}
            &g(x_1,0, y_1, y_2) = \int_{\mathbb{R}^4_+}g(\pi_{F}\Phi((x_1,0, y_1, y_2),u), y_1, y_2)f^+(du),\\
            &g(0,x_2, y_1, y_2) = \int_{\mathbb{R}^4_+}g(\pi_{F}\Phi((0,x_2, y_1, y_2),u),y_1, y_2)f^-(du),\\
            &g(x_1, x_2, y_1, 0) = \int_{\mathbb{R}^4_+} g(x_1, x_2, \pi_{G}\Phi((x_1, x_2, y_1, 0), u))f^+(du),\\
            &g(x_1, x_2, 0, y_2) = \int_{\mathbb{R}^4_+} g(x_1, x_2, \pi_{G}\Phi((x_1, x_2, 0, y_2), u))f^-(du) .
        \end{align*}
    \end{enumerate}
\end{Prop}

The next result follows directly from the definition of the function $\dbtilde{\Psi}^B$ introduced in Definition \ref{CB:def:PsiB2} and identifies the distribution of the limit process $\dbtilde{B}.$

\begin{The}[Limit distribution of the price process in inactive regimes]
    Let the assumptions of Theorem \ref{CB:res:convS2} be satisfied. Then, the limit $\dbtilde{B} = (\dbtilde{B}^{F}, \dbtilde{B}^G)$ satisfies
    \begin{align*}
        \dbtilde{B}(t) = \dbtilde{B}_0 & + \delta \sum_{0 \leq s \leq t}\Bigg(\textup{$\mathds{1}$}_{\left\{\pi_2\dbtilde{Q}(s-) = 0 \right\}} - \textup{$\mathds{1}$}_{\left\{\pi_1\dbtilde{Q}(s-) = 0\right\}} , \textup{$\mathds{1}$}_{\left\{\pi_4\dbtilde{Q}(s-) = 0\right\}} - \textup{$\mathds{1}$}_{\left\{\pi_3\dbtilde{Q}(s-) = 0\right\}} \Bigg)
    \end{align*}
    which is a piecewise constant c\`adl\`ag process whose components do almost surely not jump simultaneously.
\end{The}

\section{Analysis of the cross-border market dynamics}\label{CB:sec:mainResult}

In this section, we finally derive the weak convergence of the sequence of the cross-border market models $S^{(n)}$ being introduced in Definition \ref{CB:def:Sn}. For that, we need the convergence results corresponding to the active dynamics $\tilde{S}^{(n)}$ and inactive dynamics $\dbtilde{S}^{(n)}$ derived in Theorem \ref{CB:res:convS1} and Theorem \ref{CB:res:convS2}, respectively. Before stating our main theorem, let us introduce the limits $(\rho_k)_{k\geq 0}$ and $(\sigma_k)_{k\geq 1}$ of the discrete-time stopping times $(\rho^{(n)}_k)_{k\geq 0}$ and $(\sigma^{(n)}_k)_{k\geq 1}.$ Therefore, let $S := (B, Q, C)$ denote the limit process of $S^{(n)}$ (defined below). Again, let $\rho_0 := 0,$ and for $k\geq 1$, we set
\[\sigma_k := \sigma^{Im}_k \wedge \sigma^{Ex}_k,\]
where
\begin{equation*}
	\sigma^{Im}_k := \inf\left\{t \geq \rho_{k-1}: C(t) = \kappa_+\right\} \wedge T, \quad \sigma^{Ex}_k := \inf\left\{t \geq \rho_{k-1}: C(t) = -\kappa_-\right\} \wedge T,
\end{equation*}
and
\[\rho_k := \rho^{Im}_k \textup{$\mathds{1}$}_{\{C(\sigma_k) = -\kappa_-\}} + \rho^{Ex}_k \textup{$\mathds{1}$}_{\{C(\sigma_k) = \kappa_+\}},\]
where
\begin{equation*}
\begin{split}
	\rho^{Im}_k &:= \inf\left\{t \geq \sigma_k: \left|B^F(t) - B^G(t)\right| = 0 \text{ and } \exists \, (i,I) \in \mathcal{I}^{Im} \text{ with } Q^{i,I}(t-) \leq 0\right\} \wedge T,\\
	\rho^{Ex}_k &:= \inf\left\{t\geq \sigma_k: \left|B^F(t) - B^G(t)\right| = 0 \text{ and } \exists \, (i,I) \in \mathcal{I}^{Ex} \text{ with } Q^{i,I}(t-) \leq 0\right\} \wedge T.
\end{split}
\end{equation*}
We observe that the limits $\sigma_k,$ $k\geq 1,$ have a much easier representation as its discrete-time versions $\sigma^{(n)}_k,$ $k\geq 1,$ as we directly switch in the high-frequency limit from an active to an inactive regime if all capacities in one direction are occupied (cf. Lemma \ref{CB:res:cumQatSwitching}). In contrast, $\rho_k$, $k \geq 0,$ are the canonical high-frequency versions of $\rho^{(n)}_k,$ $k\geq 0.$ \par
Similarly to the discrete-time setting, we denote $(\tau_k)_{k\geq 1}$ the sequence of stopping times at which we observe a price change in $S$, $l(t) \in \mathbb{N}_0$ the number of price changes of $S$ in $[0,t],$ $t\in [0,T],$ and $R^+_k:= \Phi(Q(\tau_k-), \epsilon^+_k),$ $R^-_k:=\Phi(Q(\tau_k-), \epsilon^-_k),$ $k \geq 1,$ the queue sizes after price changes. Finally, we denote for all $s \in [0,T]$
\begin{align*}
\tilde{S}^s &:= \tilde{\Psi}\left(S(s); X(\cdot + s) - X(s), (R^+_{l(s) + j})_{j\geq 1}, (R^-_{l(s) + j})_{j\geq 1}\right),\\
\dbtilde{S}^s &:= \dbtilde{\Psi}\left(S(s); X(\cdot +s) - X(s), (R^+_{l(s)+j})_{j\geq 1}, (R^-_{l(s)+j})_{j\geq 1}\right),
\end{align*}
the active  and inactive dynamics starting in $S(s) \in E,$ respectively.

\begin{The}[Main result]\label{CB:res:mainTheorem}
    Let Assumption \ref{CB:ass:initialCon}--\ref{CB:ass:R} be satisfied. Then, the microscopic dynamics $S^{(n)} = (S^{(n)}(t))_{t\in [0,T]}$ of the cross-border market model converges weakly in the Skorokhod topology on $D([0,T], E)$ to a continuous-time regime switching process $S$, whose dynamics is described as follows: for all $k \geq 0,$ we have
    \begin{itemize}
    		\item $S\simeq \tilde{S}^{\rho_k}$ on $[\rho_k, \sigma_{k+1})$, i.e., on the interval $[\rho_k, \sigma_{k+1})$ the price dynamics $B$ follows a two-dimensional pure jump process jumping at hitting times of the cumulative queue size process $h(Q)$ of the axes. In particular, we have that $B^F \equiv B^G$. The queue size dynamics $Q$ follows a four-dimensional, non-negative semimartingale with a non-trivial martingale part. The cumulative queue size process $h(Q)$ is a two-dimensional process with values in $\mathbb{R}_+^2$ behaving like a planar Brownian motion between its jump times. The dynamics of the capacity process $C$ follows a continuous process of finite variation with values in $[-\kappa_-,  \kappa_+]$.
    		\item $S \simeq \dbtilde{S}^{\sigma_{k+1}}$ on $[\sigma_{k+1}, \rho_{k+1})$, i.e., on the interval $[\sigma_{k+1}, \rho_{k+1})$ the price dynamics $B$ follows a two-dimensional pure jump process whose components jump at hitting times of the corresponding components of the queue size process of the axes. In particular, $B^F$ and $B^G$ follow different one-dimensional pure jump processes which do almost surely not jump simultaneously. The queue size dynamics $Q$ follows a four-dimensional Markov process with values in $\mathbb{R}_+^4$ behaving like a four-dimensional linear Brownian motion between its jump times. The dynamics of the capacity process $C$ stays constant and equal to either $-\kappa_-$ or $\kappa_+.$
    \end{itemize}
\end{The}

We inductively construct a candidate for the limit process $S = (S(t))_{t\in [0,T]}$ and show that $S^{(n)} \Rightarrow S$ in the Skorokhod topology by applying Theorem \ref{CB:res:convS1} and Theorem \ref{CB:res:convS2} together with the continuity of the first hitting time and last value map (see Theorem 13.6.4 in \cite{W02} and Lemma \ref{CB:res:cont-firstExit-lastValue}).

\begin{proof}
    We apply Skorokhod representation in order to be able to refer to arguments of almost-sure convergence instead of weak convergence. Hence, with a slight abuse of notation, applying Proposition \ref{CB:res:fCLT} and Assumption \ref{CB:ass:R} we may assume
    $$\ensuremath\mathbb{P}\left[Y^{(n)} \rightarrow Y, \quad \forall\, k \geq 1, \, \epsilon^{+,(n)}_{k} \rightarrow \epsilon^+_{k},\, \epsilon^{-,(n)}_{k} \rightarrow \epsilon^-_{k}\right] = 1,$$
    where $Y^{(n)}:= Q^{(n)}_0 + X^{(n)},$ $Y := Q_0 + X,$ $\epsilon^+_k \sim f^+,$ and $\epsilon^-_k \sim f^-$ for the distributions $f^+,f^-$ in Assumption \ref{CB:ass:R}. \par
Recall, that $(\tau^{(n)}_k)_{k\geq 1}$ denotes the sequence of stopping times at which we observe a  price change in $S^{(n)}.$ Moreover, $R^{+,(n)}_k := \Phi^{(n)}(Q^{(n)}(\tau^{(n)}_k-), \epsilon^{+,(n)}_k)$ and $R^{-,(n)}_k := \Phi^{(n)}(Q^{(n)}(\tau^{(n)}_k-), \epsilon^{-,(n)}_k)$, $k \geq 1,$ denote the queue sizes after a price increase respectively decrease in $S^{(n)}$, and $l^{(n)}(t) \in \mathbb{N}_0$ denotes the number of price changes in $S^{(n)}$ in $[0,t]$, $t\in [0,T]$. Then, we define for $s\in [0,T]$
    \begin{align*}
    \tilde{S}^{(n), s} &:= \tilde{\Psi}\left(S^{(n)}(s); X^{(n)}(\cdot + s) - X^{(n)}(s), \left(R^{+,(n)}_{l^{(n)}(s)+j}\right)_{j\geq 1}, \left(R^{-,(n)}_{l^{(n)}(s)+j}\right)_{j\geq 1}\right),\\
	\dbtilde{S}^{(n),s} &:= \dbtilde{\Psi}\left(S^{(n)}(s); X^{(n)}(\cdot + s) - X^{(n)}(s), \left(R^{+,(n)}_{l^{(n)}(s)+j}\right)_{j\geq 1}, \left(R^{-,(n)}_{l^{(n)}(s)+j}\right)_{j\geq 1}\right)
	\end{align*}
	the microscopic active and inactive dynamics starting in $S^{(n)}(s) \in E.$
    Similarly, we introduce their components $\tilde{B}^{(n),s}, \tilde{Q}^{(n),s}$ $\tilde{C}^{(n),s}$, $\dbtilde{B}^{(n),s},$ and $\dbtilde{Q}^{(n),s}.$ In the following, we will prove the stated convergence result by induction.\newline

     \textit{Induction start:} Let us introduce the functions $\sigma^{Im}, \sigma^{Ex}, \sigma: D([0,T], \mathbb{R}) \rightarrow [0,T]$ by
    \begin{align*}
        \sigma^{Im}(\omega) := \inf\left\{t \geq 0: \omega(t) \geq \kappa_+ \right\} \wedge T,\quad \sigma^{Ex}(\omega) := \inf\left\{t \geq 0: \omega(t) \leq -\kappa_- \right\} \wedge T,
    \end{align*}
    and $\sigma(\omega) := \sigma^{Im}(\omega) \wedge \sigma^{Ex}(\omega).$ Note that for the stopping time $\sigma^{(n)}_1$ introduced in Definition \ref{CB:def:Sn}, there exists a function $\tilde{\sigma}: D([0,T], E) \rightarrow [0,T]$ satisfying $\sigma^{(n)}_1 = \tilde{\sigma}(\tilde{S}^{(n), 0}) \geq \sigma(\tilde{C}^{(n),0}) =: \sigma^{(n)}_{1,1}.$ Then, we set $S(t) = \tilde{S}^{0}(t)$ for $0 \leq t < \sigma_1,$ where $\sigma_1 := \sigma(\tilde{C}^{0}).$  By Assumption \ref{CB:ass:initialCon} and Theorem \ref{CB:res:convS1}, we conclude that
    \begin{equation}\label{CB:eq:convS0}
    \tilde{S}^{(n),0} \rightarrow \tilde{S}^{0}\quad \ensuremath\mathbb{P}\text{-a.s.}
    \end{equation}
    in the Skorokhod topology. In particular, we have that $\tilde{C}^{(n),0}\rightarrow \tilde{C}^{0}$ $\ensuremath\mathbb{P}$-almost surely in the Skorokhod topology. By Theorem \ref{CB:res:idM-1} and Theorem \ref{CB:res:convC}, the paths of $\tilde{C}^{0}$ take with probability one their values in $C_{\kappa_+}'([0,T],\mathbb{R})\cap C_{-\kappa_-}'([0,T],\mathbb{R}).$ Hence, we can apply the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}) to conclude that
    $$(\sigma^{(n)}_{1,1}, S^{(n)}(\sigma^{(n)}_{1,1}-)) \rightarrow (\sigma_1, S(\sigma_1-)) \quad \ensuremath\mathbb{P}\text{-a.s.}$$
    Now, by Lemma \ref{CB:res:cumQatSwitching}, we have that
    $$\ensuremath\mathbb{P}\left[h(\tilde{Q}^0)(\sigma_1) \notin (0,\infty)^2\right] = 0,$$
	i.e., with probability one, no price changing event occurs at $\sigma_1$ in $\tilde{S}^0$. Hence, $\sigma_1 = \tilde{\sigma}(\tilde{S}^0).$ Now, together with \eqref{CB:eq:convS0}, we conclude that also $(\sigma^{(n)}_1, S^{(n)}(\sigma^{(n)}_1-)) \rightarrow (\sigma_1, S(\sigma_1-))$ $\ensuremath\mathbb{P}$-almost surely. \par
    Next, let us introduce the function $Z: \mathbb{R}^2 \rightarrow \{-1, 0, 1\}$ by
    \begin{equation}\label{CB:def:Z}
    Z(x) := \begin{cases}
    0 &: x \in (0,\infty)^2\\
    -1 &: x \in \mathbb{R}_- \times (0,\infty)\\
    1 &: x\in (0,\infty) \times \mathbb{R}_-
    \end{cases}
    \end{equation}
    and set $\tilde{Z}^{I}_1 := Z(\pi_I \tilde{Q}^0(\sigma_1))$ for $I = F,G.$ Then, we define $S(\sigma_1) = (B(\sigma_1), Q(\sigma_1), C(\sigma_1))$ as follows: let $C(\sigma_1) = C(\sigma_1-),$
    \begin{align*}
        \pi_IQ(\sigma_1) = \pi_IQ(\sigma_1-)\textup{$\mathds{1}$}_{\{\tilde{Z}^I_1 = 0\}} + \pi_I R^+_{l(\sigma_1-) +1} \textup{$\mathds{1}$}_{\{\tilde{Z}^I_1 = 1\}} + \pi_IR^{-}_{l(\sigma_1-)+1}\textup{$\mathds{1}$}_{\{\tilde{Z}^I_1 = -1\}},
    \end{align*}
    and
    $$B^{I}(\sigma_1) = B^{I}(\sigma_1-) + \delta\left(\textup{$\mathds{1}$}_{\{\tilde{Z}^I_1 = 1\}} - \textup{$\mathds{1}$}_{\{\tilde{Z}^I_1 = -1\}}\right).$$
       Let us show that $\tilde{Z}^{I,(n)}_1(\sigma^{(n)}_1) \rightarrow \tilde{Z}^I_1$ $\ensuremath\mathbb{P}$-almost surely for $I = F,G$, where the random process $\tilde{Z}^{I,(n)}_1$, $I = F,G$, is introduced in \eqref{CB:def:Zn}. First, we define $\tilde{Z}^{I,(n),*}_1 := Z(\pi_I \tilde{Q}^{(n),0}(\sigma^{(n)}_1))$ for $I = F,G.$ Since almost surely no price changing event occurs at $\sigma_1$ and no national limit order book is empty in $\tilde{S}^0$ (cf. Lemma \ref{CB:res:cumQatSwitching} and Lemma \ref{CB:res:locLOBnotzero}), we conclude that
    \begin{equation}\label{CB:eq:2approxZn}
    \left|\tilde{Z}^{I,(n)}_1(\sigma^{(n)}_1) - \tilde{Z}^{I,(n), *}_1\right| \rightarrow 0 \quad \ensuremath\mathbb{P}\text{-a.s.}
    \end{equation}
    By Lemma \ref{CB:res:cumQatSwitching}, Lemma \ref{CB:res:locLOBnotzero}, and the construction of $\tilde{C}^0,$ we conclude with probability one, that exactly one component of $\tilde{Q}^0(\sigma_1)$ is in  $\mathbb{R}_-$ while the remaining components are in $(0,\infty).$ Because of \eqref{CB:eq:convS0}, we conclude for all $n$ large enough, that the same component of $\tilde{Q}^{(n),0}(\sigma^{(n)}_1)$ is in $\mathbb{R}_-$ while the others are in $(0,\infty).$ Hence, for $I =F,G$, $\pi_I\tilde{Q}^{(n),0}(\sigma^{(n)}_1)$ and $\pi_I\tilde{Q}^0(\sigma_1)$ are both either in $(0,\infty)^2$, $\mathbb{R}_- \times (0,\infty),$ or $(0,\infty)  \times \mathbb{R}_-$ with probability one for all $n$ large enough. We conclude that $Z(\pi_I \tilde{Q}^{(n),0}(\sigma_1^{(n)})) \rightarrow Z(\pi_I \tilde{Q}^0(\sigma_1))$ $\ensuremath\mathbb{P}$-almost surely. Thus, together with \eqref{CB:eq:2approxZn}, $\tilde{Z}^{I,(n)}_1(\sigma^{(n)}_1) \rightarrow \tilde{Z}^I_1$ $\ensuremath\mathbb{P}$-almost surely. Applying \eqref{CB:eq:convS0}, Lemma \ref{CB:res:contNT}, and the fact that almost surely no price changing event occurs at $\sigma_1$ in $\tilde{S}^0$ (cf. Lemma \ref{CB:res:cumQatSwitching}), we further conclude that $l^{(n)}(\sigma^{(n)}_1-) \rightarrow l(\sigma_1-)$ $\ensuremath\mathbb{P}$-almost surely. Hence, together with the continuity of the last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}), Assumption \ref{CB:ass:R}, and equation \eqref{CB:eq:convS0}, we get
    \begin{align}\label{CB:eq:convSwitchState1}
        S^{(n)}(\sigma^{(n)}_1)\rightarrow S(\sigma_1) \quad \mathbb{P}\text{-a.s.}
    \end{align}

    In order to extend the definition of $S$ beyond $[0, \sigma_1]$, we introduce a function $\bar{\rho}: D([0,T], \mathbb{R}^2) \rightarrow [0,T]$ by
    $$\bar{\rho}(\omega) := \inf\left\{t \geq 0: \left|\pi^{(2)}_1\omega(t) - \pi^{(2)}_2\omega(t)\right| < \frac{\delta}{2}\right\}\wedge T.$$
    Furthermore, we define the functions $\bar{\tau}^{Im}$, $\bar{\tau}^{Ex}$, $\bar{\tau}: D([0,T], \mathbb{R}^4) \rightarrow [0,T]$ by
    \begin{align*}
        \bar{\tau}^{Im}(\omega) &:= \inf\{t \geq 0: \exists \, i \in \{2,3\} \text{ with } \pi_i\omega(t) \leq 0\} \wedge T,\\
        \bar{\tau}^{Ex}(\omega) &:= \inf\{t \geq 0: \exists \, i \in \{1,4\} \text{ with } \pi_i\omega(t) \leq 0\} \wedge T,
    \end{align*}
    and $\bar{\tau}(\omega) := \bar{\tau}^{Im}(\omega) \,\wedge \,\bar{\tau}^{Ex}(\omega).$ Then, for $l,j \geq 1$ and $\bar{\tau}^{(n)}_{l,0} := \sigma^{(n)}_l$, we define
    \begin{align}\label{CB:def:barSTn}
        \begin{aligned}
            \bar{\rho}^{(n)}_{l,j} &:= \bar{\tau}^{(n)}_{l,j-1} + \bar{\rho}\left(\dbtilde{B}^{(n), \bar{\tau}^{(n)}_{l,j-1}}\right), \text{ and }\\
            \bar{\tau}^{(n)}_{l,j} &:= \bar{\rho}^{(n)}_{l,j} + \bar{\tau}\left(Q^{(n)}(\bar{\rho}^{(n)}_{l,j})+ X^{(n)}(\cdot + \bar{\rho}^{(n)}_{l,j}) - X^{(n)}(\bar{\rho}^{(n)}_{l,j})\right).
        \end{aligned}
    \end{align}
    Now, we define for $t\in [0, \bar{\tau}_{1,1} - \sigma_1),$
    $$S(t+\sigma_1) := \dbtilde{S}^{\sigma_1}(t),$$
    where $\bar{\tau}_{1,1} := \bar{\rho}_{1,1} + \bar{\tau}\left(\dbtilde{Q}^{\sigma_1}(\bar{\rho}_{1,1}) + X(\cdot + \bar{\rho}_{1,1}) -X(\bar{\rho}_{1,1})\right)$ and $\bar{\rho}_{1,1}:= \sigma_1 + \bar{\rho}(\dbtilde{B}^{\sigma_1}).$  By Theorem \ref{CB:res:convS2} together with \eqref{CB:eq:convSwitchState1}, we conclude that
    $\dbtilde{S}^{(n), \sigma_1} \rightarrow \dbtilde{S}^{\sigma_1}$ $\ensuremath\mathbb{P}$-almost surely in the Skorokhod topology. In particular, $\dbtilde{B}^{(n),\sigma_1} \rightarrow \dbtilde{B}^{\sigma_1}$ $\ensuremath\mathbb{P}$-almost surely in the Skorkhod topology. Note, that $\bar{\rho}^{(n)}_{1,1} \leq \rho^{(n)}_1$ and by Assumption \ref{CB:ass:R}, we have that $\bar{\rho}_{1,1}$ does not indicate the start of the next inactive regime. Hence, together with the continuity of the first exit time and last value map (cf. Theorem 13.6.4 in Whitt \cite{W02}), we conclude that
    $$\left(\bar{\rho}^{(n)}_{1,1}, S^{(n)}(\bar{\rho}^{(n)}_{1,1}-), S^{(n)}(\bar{\rho}^{(n)}_{1,1})\right) \rightarrow \left(\bar{\rho}_{1,1}, S(\bar{\rho}_{1,1}-), S(\bar{\rho}_{1,1})\right) \quad \ensuremath\mathbb{P}\text{-a.s.}$$
    In particular, $Q(\bar{\rho}_{1,1}) = \dbtilde{Q}^{\sigma_1}(\bar{\rho}_{1,1}).$ By Proposition \ref{CB:res:fCLT}, with probability one, the paths of $Q(\bar{\rho}_{1,1}) + X(\cdot + \bar{\rho}_{1,1}) - X(\bar{\rho}_{1,1})$ are in $C_0'([0,T-\bar{\rho}_{1,1}], \mathbb{R}^4).$ Hence, applying the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}), we conclude that
    $$\left(\bar{\tau}^{(n)}_{1,1}, S^{(n)}(\bar{\tau}^{(n)}_{1,1}-)\right) \rightarrow \left(\bar{\tau}_{1,1}, S(\bar{\tau}_{1,1}-)\right)\quad \ensuremath\mathbb{P}\text{-a.s.}$$
    Now, depending on the first component of the queue size process being depleted, we either switch to the next active regime or stay in the current inactive regime, i.e., we set
    \begin{align*}
        S(\bar{\tau}_{1,1}) &:= S(\bar{\tau}_{1,1}-)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_1) = -\kappa_-\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,1} = \bar{\tau}^{Im}_{1,1}\right\}} + \textup{$\mathds{1}$}_{\left\{C(\sigma_1) = \kappa_+\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,1} = \bar{\tau}^{Ex}_{1,1}\right\}}\right\}\\
        &\qquad + \dbtilde{S}^{\sigma_1}(\bar{\tau}_{1,1} - \sigma_1)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_1) = -\kappa_-\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,1} = \bar{\tau}^{Ex}_{1,1}\right\}} + \textup{$\mathds{1}$}_{\left\{C(\sigma_1) = \kappa_+\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,1} = \bar{\tau}^{Im}_{1,1}\right\}}\right\}.
    \end{align*}
The above assignment is well-defined, since the queue size process of $\dbtilde{S}^{\sigma_1}$ follows between two consecutive price changes the distribution of a four-dimensional linear Brownian motion (cf. Theorem \ref{CB:res:convS2}). Hence, there are almost surely no two queues hitting zero at $\bar{\tau}_{1,1}$ and therefore, the cumulative best bid and ask queues are almost surely strictly greater than zero. Therefore, we do not observe a price change when transmitting from the current inactive to the next active regime. By Lemma \ref{CB:res:contH}, we conclude the continuity of the maps $H^{Im}: \omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Im}(\omega) = \bar{\tau}(\omega)\right\}}$ and $H^{Ex}: \omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Ex}(\omega) = \bar{\tau}(\omega)\right\}}$ and therefore,
    $$S^{(n)}(\bar{\tau}^{(n)}_{1,1}) \rightarrow S(\bar{\tau}_{1,1}) \quad \ensuremath\mathbb{P}\text{-a.s.}$$
Now, let us assume that $S(\bar{\tau}_{1,1}) = \dbtilde{S}^{\sigma_1}(\bar{\tau}_{1,1} - \sigma_1),$ i.e., we stay in the current inactive regime. Moreover, assume for $j \geq 2$ and $0 \leq t \leq \bar{\tau}_{1,j-1} - \sigma_1$, we have inductively constructed $S(t+ \sigma_1)=\dbtilde{S}^{\sigma_1}(t)$ provided we did not switch the regime in $[0, \bar{\tau}_{1,j-1}- \sigma_1).$ Then, we extend the definition of $S$ to $[\bar{\tau}_{1,j-1}, \bar{\tau}_{1,j})$ by setting $S(t + \sigma_1) = \dbtilde{S}^{\sigma_1}(t)$ for $0 \leq t < \bar{\tau}_{1,j} - \sigma_1,$ where $\bar{\tau}_{1,j} := \bar{\rho}_{1,j} + \bar{\tau}(\dbtilde{Q}^{\sigma_1}(\bar{\rho}_{1,j}) + X(\cdot + \bar{\rho}_{1,j}) - X(\bar{\rho}_{1,j}))$ and $\bar{\rho}_{1,j} := \bar{\tau}_{1,j-1} + \bar{\rho}(\dbtilde{B}^{\bar{\tau}_{1,j-1}}).$
    Then, we can first argue as for $\bar{\rho}_{1,1}$ and therefore conclude that
    $$\left(\bar{\rho}^{(n)}_{1,j}, S^{(n)}(\bar{\rho}^{(n)}_{1,j}-), S^{(n)}(\bar{\rho}^{(n)}_{1,j})\right) \rightarrow \left(\bar{\rho}_{1,j}, S(\bar{\rho}_{1,j}-), S(\bar{\rho}_{1,j})\right) \quad \ensuremath\mathbb{P}\text{-a.s.}$$
   In particular, $Q(\bar{\rho}_{1,j}) = \dbtilde{Q}^{\sigma}(\bar{\rho}_{1,j}).$ Furthermore, by Proposition \ref{CB:res:fCLT} with probability one the path of $Q(\bar{\rho}_{1,j}) + X(\cdot + \bar{\rho}_{1,j}) - X(\bar{\rho}_{1,j})$ is in $C_0'([0,T-\bar{\rho}_{1,j}], \mathbb{R}^4).$ Hence, applying the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}), we conclude that $(\bar{\tau}^{(n)}_{1,j}, S^{(n)}(\bar{\tau}^{(n)}_{1,j}-)) \rightarrow (\bar{\tau}_{1,j}, S(\bar{\tau}_{1,j}-))$ $\ensuremath\mathbb{P}$-almost surely. Again, depending on which component of the queue size process hits zero first, we either switch to the next active regime or stay in the current inactive regime, i.e., we define
    \begin{align*}
        S(\bar{\tau}_{1,j}) &:= S(\bar{\tau}_{1,j}-)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_1) = -\kappa_-\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,j} = \bar{\tau}^{Im}_{1,j}\right\}} + \textup{$\mathds{1}$}_{\left\{C(\sigma_1) = \kappa_+\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,j} = \bar{\tau}^{Ex}_{1,j}\right\}}\right\}\\
        &\qquad + \dbtilde{S}^{\sigma_1}(\bar{\tau}_{1,j}-\sigma_1)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_1) = -\kappa_-\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,j} = \bar{\tau}^{Ex}_{1,j}\right\}} + \textup{$\mathds{1}$}_{\left\{C(\sigma_1) = \kappa_+\right\}} \textup{$\mathds{1}$}_{\left\{\bar{\tau}_{1,j} = \bar{\tau}^{Im}_{1,j}\right\}}\right\}.
    \end{align*}
    Again, the above assignment is well-defined. By Lemma \ref{CB:res:contH}, we conclude the  continuity of the map $H^{Im}: \omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Im}(\omega) = \bar{\tau}(\omega)\right\}}$ and $H^{Ex}: \omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Ex}(\omega) = \bar{\tau}(\omega)\right\}}$. Hence, $S^{(n)}(\bar{\tau}^{(n)}_{1,j}) \rightarrow S(\bar{\tau}_{1,j})$ $\ensuremath\mathbb{P}$-almost surely. In contrast, let us assume that there exists a $j \geq 1$ such that $S(\bar{\tau}_{1,j}) = S(\bar{\tau}_{1,j}-).$ By construction, we have that $\rho^{(n)}_1 = \bar{\tau}^{(n)}_{1,j} \rightarrow \bar{\tau}_{1,j} = \rho_1$ $\ensuremath\mathbb{P}$-almost surely. In particular, we start with the next active regime. Note that $C(\rho_1) = \tilde{C}^{\rho_1}(\rho_1)$ and since the paths of $\tilde{C}^{\rho_1}$ are elements in $C([0,T-\rho_1], \mathbb{R})$ and a single component of $Q$ hits zero at $\rho_1$, we almost surely do not switch back to the inactive regime at $\rho_1.$ \newline

    \textit{Induction hypothesis:} Assume that for all $k \geq 2,$ we have already shown that
    \begin{align}\label{CB:eq:indAss}
    \begin{aligned}
        &\Big(\sigma^{(n)}_1, \rho^{(n)}_1, \cdots, \sigma^{(n)}_{k-1}, \rho^{(n)}_{k-1}, S^{(n)}(\sigma^{(n)}_1), S^{(n)}(\rho^{(n)}_1), \cdots, S^{(n)}(\sigma^{(n)}_{k-1}), S^{(n)}(\rho^{(n)}_{k-1})\Big)\\
        &\hspace{2cm} \rightarrow \Big(\sigma_1, \rho_1,\cdots, \sigma_{k-1}, \rho_{k-1}, S(\sigma_1), S(\rho_1),\cdots, S(\sigma_{k-1}), S(\rho_{k-1})\Big)
    \end{aligned}
    \end{align}
    $\ensuremath\mathbb{P}$-almost surely and that we have constructed our candidate $S$ on the whole interval $[0, \rho_{k-1}].$  \newline

    \textit{Induction step:} We aim to extend the definition of $S$ to the interval $[\rho_{k-1}, \rho_k].$ Again, we have $\sigma^{(n)}_k = \tilde{\sigma}(\tilde{S}^{(n), \rho^{(n)}_{k-1}}) \geq \sigma(\tilde{C}^{(n), \rho^{(n)}_{k-1}}) =: \sigma^{(n)}_{k,1}.$ Then, for $0 \leq t < \sigma_k - \rho_{k-1}$, we set
    $$S(t+ \rho_{k-1}) = \tilde{S}^{\rho_{k-1}}(t),$$
    where $\sigma_k := \sigma(\tilde{C}^{\rho_{k-1}}).$ Again, by Theorem \ref{CB:res:convS1} and since $S^{(n)}(\rho^{(n)}_{k-1}) \rightarrow S(\rho_{k-1})$ $\ensuremath\mathbb{P}$-almost surely by \eqref{CB:eq:indAss}, we have that
    $$\tilde{S}^{(n), \rho^{(n)}_{k-1}} \rightarrow \tilde{S}^{\rho_{k-1}} \quad \ensuremath\mathbb{P}\text{-a.s.}$$
    in the Skorokhod topology on the space $D([0,T], E).$  Again, by Theorem \ref{CB:res:convC}, the path of $\tilde{C}^{\rho_{k-1}}$ takes with probability one its values in $C_{\kappa_+}'([0,T-\rho_{k-1}], \mathbb{R})\cap C_{-\kappa_-}'([0,T-\rho_{k-1}],\mathbb{R})$. Hence, we can apply the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}), to conclude that $(\sigma^{(n)}_{k,1}, S^{(n)}(\sigma^{(n)}_{k,1}-)) \rightarrow (\sigma_k, S(\sigma_k-))$ $\ensuremath\mathbb{P}$-almost surely. Now, again by Lemma \ref{CB:res:cumQatSwitching} with probability one, we have that $h(\tilde{Q}^{\rho_{k-1}})(\sigma_k) \in (0,\infty)^2.$ Hence, $\sigma_k = \tilde{\sigma}(\tilde{S}^{\rho_{k-1}})$ and therefore, $(\sigma^{(n)}_k, S^{(n)}(\sigma^{(n)}_k-)) \rightarrow (\sigma_k, S(\sigma_k-))$ $\ensuremath\mathbb{P}$-almost surely. Now, we define $S(\sigma_k) = (B(\sigma_k), Q(\sigma_k), C(\sigma_k))$ as follows: let $\tilde{Z}^I_k := Z(\pi_I \tilde{Q}^{\rho_{k-1}}(\sigma_k)),$ where the function $Z$ is introduced in \eqref{CB:def:Z}. Moreover, we set $C(\sigma_k) = C(\sigma_k-)$, for $I = F,G,$
     \begin{align*}
        \pi_IQ(\sigma_k) = \pi_IQ(\sigma_k-) \textup{$\mathds{1}$}_{\{\tilde{Z}^I_k = 0\}} + \pi_IR^+_{l(\sigma_k-) + 1} \textup{$\mathds{1}$}_{\{\tilde{Z}^I_k = 1\}} + \pi_IR^-_{l(\sigma_k-) +1} \textup{$\mathds{1}$}_{\{\tilde{Z}^I_k = -1\}},
    \end{align*}
    and
    $$B^I(\sigma_k) = B^{I}(\sigma_k-) + \delta \left(\textup{$\mathds{1}$}_{\{\tilde{Z}^I_k = 1\}} - \textup{$\mathds{1}$}_{\{\tilde{Z}^I_k = -1\}}\right).$$
    Now, we can argue similarly as for $S(\sigma_1)$ and obtain that
    \begin{align}\label{CB:eq:convSwitchState}
    \begin{aligned}
        &\left(l^{(n)}(\sigma^{(n)}_k -), \tilde{Z}_k^F(\sigma^{(n)}_k), \tilde{Z}_k^G(\sigma^{(n)}_k), S^{(n)}(\sigma^{(n)}_k)\right)\\
        &\hspace{4.5cm} \rightarrow \left(l(\sigma_k-), \tilde{Z}^F_k, \tilde{Z}^G_k, S(\sigma_k)\right)\quad \ensuremath\mathbb{P}\text{-a.s.}
    \end{aligned}
    \end{align}

    Now, we extend the definition of $S$ such that for all $0 \leq t < \bar{\tau}_{k,1} - \sigma_k,$
    $$S(t+\sigma_k) = \dbtilde{S}^{\sigma_k}(t),$$
    where $\bar{\tau}_{k,1} := \bar{\rho}_{k,1} + \bar{\tau}(\dbtilde{Q}^{\sigma_k}(\bar{\rho}_{k,1}) + X(\cdot + \bar{\rho}_{k,1}) - X(\bar{\rho}_{k,1}))$ and $\bar{\rho}_{k,1} := \sigma_k +  \bar{\rho}(\dbtilde{B}^{\sigma_k}).$ By Theorem \ref{CB:res:convS2} and \eqref{CB:eq:convSwitchState}, we conclude that $\dbtilde{S}^{(n), \sigma^{(n)}_k} \rightarrow \dbtilde{S}^{\sigma_k}$ $\ensuremath\mathbb{P}$-almost surely in the Skorokhod  topology. Together with the continuity of the first hitting time and last value map (cf. Theorem 13.6.4 in Whitt \cite{W02}), we conclude that $(\bar{\rho}^{(n)}_{k,1}, S^{(n)}(\bar{\rho}^{(n)}_{k,1}-), S^{(n)}(\bar{\rho}^{(n)}_{k,1})) \rightarrow (\bar{\rho}_{k,1}, S(\bar{\rho}_{k,1}-), S(\bar{\rho}_{k,1}))$ $\ensuremath\mathbb{P}$-almost surely. In particular, $\dbtilde{Q}^{\sigma_k}(\bar{\rho}_{k,1}) = Q(\bar{\rho}_{k,1})$. By Proposition \ref{CB:res:fCLT}, the process $Q(\bar{\rho}_{k,1}) + X(\cdot + \bar{\rho}_{k,1}) - X(\bar{\rho}_{k,1})$ takes almost surely its path in $C_0'([0,T-\bar{\rho}_{k,1}], \mathbb{R}^4).$ Hence, we can apply the continuity of the first hitting time and last value map (cf. Lemma \ref{CB:res:cont-firstExit-lastValue}) to conclude that
     $(\bar{\tau}^{(n)}_{k,1}, S^{(n)}(\bar{\tau}^{(n)}_{k,1}-)) \rightarrow (\bar{\tau}_{k,1}, S(\bar{\tau}_{k,1}-))$ $\ensuremath\mathbb{P}$-almost surely. Depending on which component of the queue size process has been depleted first, we either switch to the next active regime or stay in the current inactive regime, i.e., we set
    \begin{align*}
        S(\bar{\tau}_{k,1}) &:= S(\bar{\tau}_{k,1}-)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_k) = -\kappa_-\right\}}\textup{$\mathds{1}$}_{\left\{\bar{\tau}_{k,1} = \bar{\tau}^{Im}_{k,1}\right\}} + \textup{$\mathds{1}$}_{\{C(\sigma_k) = \kappa^+\}}\textup{$\mathds{1}$}_{\{\bar{\tau}_{k,1} = \bar{\tau}^{Ex}_{k,1}\}}\right\}\\
        &\qquad + \dbtilde{S}^{\sigma_k}(\bar{\tau}_{k,1} - \sigma_k)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_k) = -\kappa_-\right\}}\textup{$\mathds{1}$}_{\left\{\bar{\tau}_{k,1} = \bar{\tau}^{Ex}_{k,1}\right\}} + \textup{$\mathds{1}$}_{\{C(\sigma_k) = \kappa_+\}}\textup{$\mathds{1}$}_{\{\bar{\tau}_{k,1} = \bar{\tau}^{Im}_{k,1}\}}\right\}.
    \end{align*}
    Again, the above assignment is well-defined. By Lemma \ref{CB:res:contH}, we conclude the continuity of the map $H^{Im}: \omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Im}(\omega) = \bar{\tau}(\omega)\right\}}$ and $H^{Ex}:\omega \mapsto \textup{$\mathds{1}$}_{\left\{\bar{\tau}^{Ex}(\omega) = \bar{\tau}(\omega)\right\}}.$ Hence, $S^{(n)}(\bar{\tau}^{(n)}_{k,1}) \rightarrow S(\bar{\tau}_{k,1})$ $\ensuremath\mathbb{P}$-almost surely.
    Now, assume that $S(\bar{\tau}_{k,1}) = \dbtilde{S}^{\sigma_k}(\bar{\tau}_{k,1} - \sigma_k).$ Further, assume for $j \geq 2$ and $0 \leq t \leq \bar{\tau}_{k,j-1} - \sigma_k,$ that we have constructed $S(t + \sigma_k) = \dbtilde{S}^{\sigma_k}(t)$ provided we did not switch the regime in $[0, \bar{\tau}_{k, j-1} - \sigma_k).$ Again, we extend the definition of $S$ to $[\bar{\tau}_{k,j-1}, \bar{\tau}_{k,j}]$ by setting $S(t + \sigma_k) = \dbtilde{S}^{\sigma_k}(t)$ for $0 \leq t  < \bar{\tau}_{k,j} - \sigma_k,$ where $\bar{\tau}_{k,j} := \bar{\rho}_{k,j} + \bar{\tau}(\dbtilde{Q}^{\sigma_k}(\bar{\rho}_{k,j}) + X(\cdot + \bar{\rho}_{k,j}) - X(\bar{\rho}_{k,j}))$ and $\bar{\rho}_{k,j} := \bar{\tau}_{k,j-1} + \bar{\rho}(\dbtilde{B}^{\bar{\tau}_{k,j-1}}).$ Then, we can argue as for $\bar{\rho}_{k,1}$ to conclude that $(\bar{\rho}^{(n)}_{k,j}, S^{(n)}(\bar{\rho}^{(n)}_{k,j}-), S^{(n)}(\bar{\rho}^{(n)}_{k,j}))\rightarrow (\bar{\rho}_{k,j}, S(\bar{\rho}_{k,j}-), S(\bar{\rho}_{k,j}))$ $\ensuremath\mathbb{P}$-almost surely. In particular $\dbtilde{Q}^{\sigma_k}(\bar{\rho}_{k,j}) = Q(\bar{\rho}_{k,j}).$ Moreover, again by Proposition \ref{CB:res:fCLT}, the process $Q(\bar{\rho}_{k,j}) + X(\cdot + \bar{\rho}_{k,j}) - X(\bar{\rho}_{k,j})$ has almost surely paths in $C_0'([0,T-\bar{\rho}_{k,j}], \mathbb{R}^4).$ Hence, we can apply the continuity of the first hitting time and last value map (cf. Theorem \ref{CB:res:cont-firstExit-lastValue}) to conclude that $(\bar{\tau}^{(n)}_{k,j}, S^{(n)}(\bar{\tau}^{(n)}_{k,j}-)) \rightarrow (\bar{\tau}_{k,j}, S(\bar{\tau}_{k,j}-))$ $\ensuremath\mathbb{P}$-almost surely. Again, depending on which component of the queue size process has been depleted first, we either switch to the next active regime or stay in the current inactive regime, i.e., we define
     \begin{align*}
        S(\bar{\tau}_{k,j}) &:= S(\bar{\tau}_{k,j}-)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_k) = -\kappa_-\right\}}\textup{$\mathds{1}$}_{\left\{\bar{\tau}_{k,j} = \bar{\tau}^{Im}_{k,j}\right\}} + \textup{$\mathds{1}$}_{\{C(\sigma_k) = \kappa^+\}}\textup{$\mathds{1}$}_{\{\bar{\tau}_{k,j} = \bar{\tau}^{Ex}_{k,j}\}}\right\}\\
        &\qquad + \dbtilde{S}^{\sigma_k}(\bar{\tau}_{k,j} - \sigma_k)\left\{\textup{$\mathds{1}$}_{\left\{C(\sigma_k) = -\kappa_-\right\}}\textup{$\mathds{1}$}_{\left\{\bar{\tau}_{k,j} = \bar{\tau}^{Ex}_{k,j}\right\}} + \textup{$\mathds{1}$}_{\{C(\sigma_k) = \kappa^+\}}\textup{$\mathds{1}$}_{\{\bar{\tau}_{k,j} = \bar{\tau}^{Im}_{k,j}\}}\right\}.
    \end{align*}
    Arguing as above, we can conclude that $S^{(n)}(\bar{\tau}^{(n)}_{k,j}) \rightarrow S(\bar{\tau}_{k,j})$ $\ensuremath\mathbb{P}$-almost surely. In contrast, let us assume that there exists a $j \geq 1$ such that $S(\bar{\tau}_{k,j}) = S(\bar{\tau}_{k,j}-).$ Then $\bar{\tau}_{k,j} = \rho_k$ and we conclude that
    $$\rho^{(n)}_k = \bar{\tau}^{(n)}_{k,j} \rightarrow  \rho_k \quad \ensuremath\mathbb{P}\text{-a.s.}$$
    i.e., we switch to the next active regime. Note again that $C(\rho_k) = \tilde{C}^{\rho_k}(\rho_k)$ and since almost surely the paths of $\tilde{C}^{\rho_k}$ are elements of $C([0,T], \mathbb{R})$ and a single component of $Q$ hits zero at $\rho_k$, we almost surely do not switch back to the inactive regime at $\rho_k.$ This finishes our induction.\newline

    In summary, we conclude for our candidate $S$ and all $k \geq 1,$
    \begin{align*}
        &\left(\sigma^{(n)}_1, \rho^{(n)}_1, \cdots, \sigma^{(n)}_k, \rho^{(n)}_k, S^{(n)}(\sigma^{(n)}_1), S^{(n)}(\rho^{(n)}_1), \cdots, S^{(n)}(\sigma^{(n)}_k), S^{(n)}(\rho^{(n)}_k)\right)\\
        &\hspace{2cm} \rightarrow \left(\sigma_1, \rho_1,\cdots, \sigma_k, \rho_k, S(\sigma_1), S(\rho_1), \cdots, S(\sigma_k), S(\rho_k)\right)
    \end{align*}
    $\ensuremath\mathbb{P}$-almost surely. By construction of the sequences $(\rho_k)_{k\geq 1}$ and $(\sigma_k)_{k\geq 1}$, observe that each of theses stopping times can be bounded from below by the first hitting time at zero of a Brownian motion and therefore, we must have $\lim_{k\rightarrow \infty} \max\{\rho_k, \sigma_k\} = T.$ Hence, we can apply Theorem \ref{CB:res:convS1} and Theorem \ref{CB:res:convS2} yielding that
    $$S^{(n)} \rightarrow S \quad \mathbb{P}\text{-a.s.}$$
    in the Skorokhod topology on the space $D([0,T], E)$. This proves the stated result.
\end{proof}

\section{Simulation study} \label{CB:sec:simStudyCB}

We discuss the evolution of our market dynamics through simulation studies of different model calibrations. Moreover, we investigate the impact of coupling two markets on price evolution. Therefore, we perform a comparative statistics of the active and inactive market dynamics and discuss a theoretical result on the conditional distribution of duration between price changes (cf. Remark 2 in Cont and de Larrard \cite{CL21}).

\subsection{Simulation of different market situations}

Throughout this section, we choose $\Delta t^{(n)} = n^{-1}$, $\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)} = n^{-1/2}$, and simulate $n = 10,000$ time steps. For simplicity, we assume that the order sizes are independent and simulate the queue sizes after price changes from independent uniform distributions on $\{j\Delta v^{(n)}}\newcommand{\on}{\omega^{(n)}: j = 5,\cdots, 15\}$. Moreover, we choose $\kappa_+ = \kappa_- = 0.5$ and $\delta = 0.1.$\par
In a first simulation, we consider a cross-border market which is \textit{balanced} between cross-border trades in direction $F$ to $G$ and vice versa. Moreover, we choose the frequency of order cancellations higher than the frequency of order placements. This is not unrealistic since in liquid markets the frequency of market orders and cancellations at the best bid and ask prices is known to be much higher than the frequency of order placements. In more detail, we choose $(\sigma^{i,I})^2 = 0.25$ and $\mu^{i,I} = -10,$ for all $(i,I) \in \{b,a\} \times \{F,G\}$. We present one realization of this balanced cross-border market model in Figure \ref{CB:fig:balOB}.  \par
In this first simulation, we observe three regime switches: the first around $6{,}500$, the second around $8{,}000,$ and the last around $9{,}500$ time steps. During the active regimes (white areas) the capacity process moves up and down. This is because the cross-border market model is \textit{balanced}, i.e., the frequency of market orders that leads to an increase of the capacity process (i.e. imports to $F$) and the frequency of orders that leads to a decrease of the capacity process (i.e. exports from $F$) are of comparable size. Moreover, by replicating the above simulation $1,000$ times, we obtain an empirical probability of around $0.58$ to observe at least one regime switch for the chosen model parameters. Since we have chosen the frequency of order cancellations higher than the frequency of order placements, we observe several price changes and a comparable high fluctuation of the capacity process. If we change this relation in favor of order placements, we would observe less price changes and fewer fluctuations of the capacity process. This would also reduce the empirical probability to observe a regime switch. The cross-border market model in this simulation might replicate a market situation in which the actual national best bid prices are on a comparable level. In this setting, the main advantage of the market coupling is therefore the increase of standing volumes which typically leads to fewer price changes (cf. our discussion in Section \ref{CB:sec:simStudy-priceChanges} below).

\begin{figure}[H]
    \centering
    \includegraphics[scale = 0.3]{images/Simulation-1-balancedOB.png}
    \caption{Simulation of the bid queues, the ask queues, and the prices of $F$ (orange) and $G$ (turquoise) as well as the capacity process for $n = 10,000$ (balanced setting). The white areas represent the active regimes whereas the gray ones represent the inactive regimes.}
    \label{CB:fig:balOB}
\end{figure}

In a second simulation, we analyze a so-called \textit{imbalanced} cross-border market in which we choose the frequency of cross-border trades in direction $G$ to $F$ (imports to $F$) smaller than the frequency of cross-border trades in direction $F$ to $G$ (exports from $F$). In more detail, we choose $(\sigma^{i,I})^2 = 0.25,$ for all $(i,I) \in \{b,a\} \times \{F,G\},$ $\mu^{b,F} = \mu^{a,G} = -10,$ and $\mu^{a,F} = \mu^{b,G}=0.$ We present one realization of this imbalanced cross-border market model in Figure \ref{CB:fig:imbOB-1}. \par
We observe a single regime switch from the active to the inactive regime around $2,500$ time steps. Since we have chosen the frequency of market orders corresponding to possible exports from $F$ higher than the frequency of those corresponding to possible imports to $F$, the dynamics of the capacity process has a higher probability to move downward. Since this imbalance is maintained after the regime switch, the best bid price of $F$ decreases whereas the price of $G$ increases. Note that the bid price of $G$ increases after the regime switch since we have chosen market orders at the best ask price to be more likely than market orders at the best bid price. For a similar reason, the bid price of $F$ decreases after the regime switch. Our chosen model assumptions in this second simulation might replicate a cross-border market in which the actual national price in $G$ is much higher than in $F$ and therefore, market participant in $F$ are motivated to sell orders to $G$ for a better price. \par

\begin{figure}[H]
	\centering
	 \includegraphics[scale = 0.3]{images/Simulation-3-imbalancedOB.png}
    \caption{Simulation of the bid queues, the ask queue sizes, and the prices of $F$ (orange) and $G$ (turquoise) as well as the capacity process for $n = 10{,}000$ (imbalanced setting). The white areas represent the active regimes whereas the gray ones represent the inactive regimes.}
    \label{CB:fig:imbOB-1}
\end{figure}

Motivated by empirical observations in real-world markets, not only the market matching mechanism but also the model parameters might change because of the occurrence of a regime switch. Therefore, let us choose in a third simulation the following model parameters: $(\sigma^{i,I})^2 = 0.25$, for all $(i,I) \in \{b,a\} \times \{F,G\},$ $\mu^{b,F} = \mu^{a,G} = -10,$ and $\mu^{a,F} = \mu^{b,G}=0$ during the active regimes; and $(\sigma^{b,F})^2 = (\sigma^{a,F})^2 = 0.1,$ $(\sigma^{b,G})^2 = 0.3,$ $(\sigma^{a,G})^2 = 0.5,$ $\mu^{b,F}=\mu^{a,F} = -2,$ and $\mu^{b,G} = 0,$ $\mu^{a,G} = -5$ during the inactive regimes. We present one realization of this imbalanced cross-border market model with a change of the model parameters in Figure \ref{CB:fig:imbOB-2}.\par
In this last simulation, we observe a single regime switch around 4,000 time steps. While we have chosen the same parameters of the underlying order flow for the active regime as in the second simulation, we changed them in the inactive regime. In particular, the model parameters of the queue size process of $F$ are changed in favor of less order arrivals and price changes during the inactive regimes. For this reason, we observe only a single price change of the best bid price of $F$ during the inactive regime. Moreover, the drift of the ask queue of $G$ is still significantly negative while the drift of the bid queue of $G$ is zero, so that the bid price of $G$ moves upward during the inactive regime. These model assumptions might replicate a cross-border market in which the actual national bid price in $G$ is higher than in $F$ and the high number of trading volume in $F$ during the active regime is only caused due to the possibility to sell goods to $G$.

\begin{figure}[H]
	\centering
	 \includegraphics[scale = 0.3]{images/Simulation-5-imbalancedParameterChange.png}
    \caption{Simulation of the bid queues, the ask queue sizes, and the prices of $F$ (orange) and $G$ (turquoise) as well as the capacity process for $n = 10{,}000$ (imbalanced setting, change point in the model parameters). The white areas represent the active regimes whereas the gray ones represent the inactive regimes.}
    \label{CB:fig:imbOB-2}
\end{figure}

\subsection{Impact of the market coupling on the evolution of bid prices}\label{CB:sec:simStudy-priceChanges}

In this subsection, we study the impact of coupling two markets on the evolution of bid prices in our cross-border market model. For this reason, we simulate both the active dynamics $\tilde{S}^{(n)}$ and inactive dynamics $\dbtilde{S}^{(n)}$ from the same underlying order flow process and compare their empirical mean number of price changes and their empirical mean bid price ranges. Therefore, let us again simulate $n=10,000$ time steps and replicate the simulations $m = 1,000$ times. We assume for simplicity that the order types are independent and that all order types have approximately the same probability, i.e.,  $(\sigma^{i,I})^2  = 0.25$ for all $(i,I) \in \{b,a\} \times \{F,G\}.$

We analyze four scenarios: in a) we study a completely balanced cross-border market model in which the frequency of all market and limit orders is of comparable size, i.e., $\mu^{i,I} = 0$ for all $(i,I) \in \{b,a\} \times \{F,G\}.$ Second, we analyze in b) a cross-border market model in which the frequency of market orders leading to possible imports and those leading to possible exports is balanced, but the frequency of market orders at the bid side of the shared order book is higher than on the ask side, i.e., $\mu^{b,F} = \mu^{b,G} = -10$ and $\mu^{a,F} = \mu^{a,G} =0.$ In scenario c) we analyze a balanced cross-border market in which the frequency of market orders with origin $F$ is higher than the frequency of market orders with origin $G$, i.e., $\mu^{b,F} = \mu^{a,F} = -10$ and $\mu^{b,G} = \mu^{a,G}=0.$ Finally, we study in d) an imbalanced cross-border market in which the frequency of market orders leading to possible exports from $F$ is higher than the frequency of those orders leading to possible imports to $F$, i.e., $\mu^{b,F} = \mu^{a,G} = -10$ and $\mu^{a,F} = \mu^{b,G} = 0.$\par
In the following, let us denote by $N^{shared},$ $N^{F},$ and $N^{G}$ the empirical mean number of price changes and by $\mathcal{R}^{shared}$, $\mathcal{R}^F,$ and $\mathcal{R}^G$ the empirical mean bid price ranges in the shared limit order book obtained from simulating the dynamics of $\tilde{S}^{(n)}$, and in the national limit order book of $F$ respectively $G$ obtained from simulating the dynamics of $\dbtilde{S}^{(n)}.$ We note that $\mathcal{R}^{shared},$ $\mathcal{R}^F,$ and $\mathcal{R}^G$ are stated relative to the tick size, i.e., they describe the mean number of ticks between the maximum and minimum bid prices.

\begin{figure}[H]
	\centering
\begin{tabular}{c||c|c|c}
	scenario & $N^{shared}$ & $N^F$ & $N^G$\\
	\hline \hline
	a) $\mu^{b,F} = \mu^{a,F} = \mu^{b,G} = \mu^{a,G} = 0$ & $6.88$ & $11.91$ & $11.86$ \\
	b) $\mu^{b,F} = \mu^{b,G} = -10,$ $\mu^{a,F} = \mu^{a,G}=0$ &  $27.91$ & $34.99$ & $35.36$\\
	c) $\mu^{b,F} = \mu^{a,F} = -10,$ $\mu^{b,G}= \mu^{a,G} = 0$ & $23.39$ & $50.34$ & $11.72$\\
	d) $\mu^{b,F} = \mu^{a,G} = -10,$ $\mu^{a,F} = \mu^{b,G} = 0$ & $23.6$ & $35.19$ & $35.31$
\end{tabular}
\caption{The mean number of price changes in different simulations with $n = 10,000$ time steps and $m = 1,000$ iterations.}
\label{CB:fig:tableNPC}
\end{figure}

\begin{figure}[H]
	\centering
\begin{tabular}{c||c|c|c}
	scenario & $\mathcal{R}^{shared}$ & $\mathcal{R}^F$ & $\mathcal{R}^G$\\
	\hline \hline
	a) $\mu^{b,F} = \mu^{a,F} = \mu^{b,G} = \mu^{a,G} = 0$ & $3.10$ & $4.48$ & $4.44$ \\
	b) $\mu^{b,F} = \mu^{b,G} = -10,$ $\mu^{a,F} = \mu^{a,G}=0$ &  $18.36$ & $15.03$ & $15.13$\\
	c) $\mu^{b,F} = \mu^{a,F} = -10,$ $\mu^{b,G}= \mu^{a,G} = 0$ & $6.85$ & $10.25$ & $ 4.43$\\
	d) $\mu^{b,F} = \mu^{a,G} = -10,$ $\mu^{a,F} = \mu^{b,G} = 0$ & $6.71$ & $14.93$ & $15.48$
\end{tabular}
\caption{The mean bid price ranges relative to the tick size in different simulations with $n = 10,000$ time steps and $m = 1,000$ iterations.}
\label{CB:fig:tableRPC}
\end{figure}

Recall that a price change in the shared order book always yields price changes in both national limit order books. First, we observe that the coupling of the national markets reduces the number of price changes in the shared limit order book (cf. Table \ref{CB:fig:tableNPC}). Except from scenario b) the coupling of the national markets also reduces the mean price ranges (cf. Table \ref{CB:fig:tableRPC}). This can be explained as follows: on the one hand, the coupling of two limit order books always leads to an increase of the order volume at the best bid and ask queues. On the other hand, however, also the drift, the volatility, and the correlation parameters of the cumulative order flow process differ from the  parameters of order flow process corresponding to the national order books. The difference of the model parameters could amplify or cancel out the effects caused due to the increase in trading volume. For this reason, the bid price range in scenario b) is higher in the shared order book than in the national ones. More precisely, even though the order volume has increased due to the coupling, the drifts and volatilities of the cumulative order flow process have doubled. In particular, the drift of the bid side of the shared order book of size -20 compared to the drift of the bid side of the national order books of size -10 leads to a more extreme price evolution in the shared order book. In contrast, in the scenarios a) and d), the drifts of the cumulative order flow process equals the drifts of the national order flow processes. Hence, the coupling of the two limit order books leads to a decrease of the number of price changes and of the bid price ranges since the standing volumes at the best bid and ask queues have increased. Note that generally in a completely balanced order book, fewer price changes occur as the frequency of market and limit orders is of comparable size. Finally, in scenario c) in which we have a high imbalance of market and limit orders in favor of market orders in only one of the two countries, we observe a kind of balancing effect, i.e., the number of price changes as well as the bid price range in the shared order book are smaller than in the national order book with the high imbalance but higher than in the balanced national limit order book. \newline

In the following, we study the effect of coupling two markets  using a theoretical result presented in Cont and de Larrard \cite{CL21}. It describes the conditional distribution of duration between price changes in a single country limit order book model and is based on the results in \cite{Z01} on the first exit time of a planar Brownian motion with drift. This theoretical result can also be used to formulate the conditional distribution of duration between price changes in the shared limit order book:
let $\mathcal{Q}^b$ and $\mathcal{Q}^a$ be the bid and ask components of the limit of the queue size process. In particular, $\mathcal{Q} := (\mathcal{Q}^b, \mathcal{Q}^a)$ might be the limit process of the cumulative queue sizes of the shared order book or of the national queue sizes of a single order book. Moreover, let us denote by $(\mu_b, \mu_a),$ $(\sigma_b, \sigma_a),$ and $\rho$ the drift,  volatility, and correlation of the corresponding limit process $\mathcal{X}$ of the (cumulative) net order flow process. Furthermore, let the random time $\tau$ denote the duration until the next price change in $\mathcal{Q}.$ Then, the conditional distribution of duration between price changes is given by
\begin{equation}\label{CB:def:condDist-Duration}
\begin{split}
&\ensuremath\mathbb{P}[\tau > t| \mathcal{Q}(0) = (x,y)]\\
 &\quad = \frac{2}{\alpha t}\exp\left(a_1 x + a_2 y + a_t t- \frac{U}{2t}\right)\sum_{j = 1}^{\infty} \sin\left(\frac{j\pi \theta_0}{\alpha}\right)\int_0^{\alpha} \sin\left(\frac{j\pi \theta}{\alpha}\right) g_j(\theta) d\theta,
\end{split}
\end{equation}
where
\begin{align*}
	&\hspace{-3cm} U:= (1-\rho)^{-1}\left(\frac{x^2}{\sigma^2_b} + \frac{y^2}{\sigma^2_a} -\frac{2 \rho x y}{\sigma_b \sigma_a}\right),\\
	\alpha := \begin{cases}
	\pi + \arctan\left(-\frac{\sqrt{1-\rho^2}}{\rho}\right) & \rho > 0\\
	\frac{\pi}{2} & \rho = 0\\
	\arctan\left(-\frac{\sqrt{1-\rho^2}}{\rho}\right) & \rho < 0
	\end{cases},&\qquad
	\theta_0 := \begin{cases}
	\pi + \arctan\left(\frac{y\sqrt{1-\rho^2}}{x-\rho y}\right) & x < \rho y\\
	\frac{\pi}{2} & x = \rho y\\
	\arctan\left(\frac{y\sqrt{1-\rho^2}}{x-\rho y}\right) & x > \rho y
	\end{cases},\\
	&\hspace{-5.5cm} g_j(\theta) := \int_0^{\infty} r \exp\left(-\frac{r^2}{2t}\right)\exp\left(d_1 r \sin(\theta-\alpha) - d_2 r \cos(\theta-\alpha)\right) I_{j\pi/\alpha}\left(\frac{r \sqrt{U}}{t}\right) dr,
\end{align*}
where $I_j$ denotes the $j$-th modified Bessel function of first kind, and
\begin{align*}
	a_1 &:= \frac{\rho \mu_a \sigma_b - \mu_b \sigma_a}{(1-\rho^2)\sigma^2_b \sigma_a},  & a_2&:= \frac{\rho \mu_b \sigma_a - \mu_a \sigma_b}{(1-\rho^2)\sigma^2_a \sigma_b},\\
	d_1&:= a_1 \sigma_b + \rho a_2 \sigma_a, & d_2&:=a_2 \sigma_a \sqrt{1-\rho^2},\\
\end{align*}
and
\[a_t = \frac{a_1^2\sigma_b^2}{2} + \rho a_1 a_2 \sigma_b \sigma_a + \frac{a_2^2 \sigma_a^2}{2} + a_1 \mu_b + a_2 \mu_b.\]

\begin{Rmk}[Qualitative observations of the conditional distribution above] The conditional distribution of $\tau$ in \eqref{CB:def:condDist-Duration} is quite complex and gives only little insights into its dependence on the model parameters $\mu_b,$ $\mu_a$, $\sigma_b$, $\sigma_a$, and $\rho$ and of the initial queue sizes $\mathcal{Q}(0)$. A detailed study of the above distribution, however, reveals the following observations for fixed $t \in [0,T]$:
\begin{itemize}
\setlength\itemsep{0em}
	\item an increase of the initial queue sizes $\mathcal{Q}(0) = (x,y)$ (componentwise) increases the conditional probability,
	\item an increase of the mean $(\mu_b, \mu_a) \in \mathbb{R}^2$ (componentwise) increases the conditional probability,
	\item an increase of the correlation $\rho \in (-1,1)$ increases the conditional probability, and
	\item the impact of the volatility $(\sigma_b, \sigma_a) \in \mathbb{R}^2_+$ on the conditional distribution is more complex and depends on the other model parameters. For example, let $\varepsilon > 0,$ $B^{+}_2(0, \varepsilon) = \{x\in \mathbb{R}^2_+: \|x\| \leq \varepsilon\}$ and $B^{-}_2(0,\varepsilon) = \{x\in \mathbb{R}^2_-: \|x\| \leq \varepsilon\}.$
	\begin{itemize}
	\setlength\itemsep{0em}
	\item[$\circ$] If $(\mu_b, \mu_a) \in \mathbb{R}^2_-\setminus B^-_2(0,\varepsilon)$, then an increase of the volatility (componentwise) leads to an increase of the conditional distribution.
	\item[$\circ$] If $(\mu_b, \mu_a) \in \mathbb{R}^2_+ \setminus B^+_2(0,\varepsilon)$, then an increase of the volatility (componentwise) leads to a decrease of the conditional distributions.
	\item[$\circ$] Inbetween, one of the two observation or a mixture thereof, where no monotonic relation can be observed, occurs.
\end{itemize}
\end{itemize}
\end{Rmk}

In summary, the results of our simulation study in the Table \ref{CB:fig:tableNPC}, Table \ref{CB:fig:tableRPC}, and of studying the theoretical result on the conditional distribution of duration between price changes yield the same conclusions: in many cases, the coupling of two markets leads to a reduction of the number of price changes and of the bid price ranges, since the order volumes at the best bid and ask prices increase. However, coupling the two markets also changes the means, volatilities, and the correlation of the underlying order flow process, which could amplify or cancel out the positive effect caused due to the increase of order volume.

\newpage

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