text:
\section{Introduction}

Given a stock price process, we analyse the potential of  arbitrage by insiders, when they chose not to sell short. Short-selling stocks when having access to  a value-destroying event before it is known to the public leads to profits, but insiders that are already detaining the stocks may profit without engaging in selling short.  Selling (and buying) of company shares by people who have special information because they are involved with the company is illegal in many jurisdictions, whenever the information is non-public and material.
Insider trading scandals are nevertheless regularly on the front page of newspapers.
For instance, the US justice department is currently investigating alleged insider trading by US lawmakers who sold stocks just before the coronavirus pandemic sparked a major market downturn. Rarely short selling is part of the insiders' strategies in these scandals. Short selling transactions receive additional attention from supervisory authorities in their attempt to detect abnormal trading activity and it is possible that many insiders refrain from such strategies for their trades to remain undetected.   It is not clear nevertheless, from both a practical and purely theoretical perspective if an insider that does not detain the stock has the possibility to first buy and then sell the stock in a profitable manner, thus exploiting the informational advantage without selling short.

The aim of this paper is to propose some first elements in tackling this question. For the notion of no arbitrage profits, we consider the setting of no free lunch with vanishing risks with short sales prohibitions (NFLVRS) introduced in Pulido \cite{Pulido14},  a restriction of the classical notion of no free lunch with vanishing risks (NFLVR) by Delbaen and Schachermeyer \cite{DelbScha94}, \cite{DelbScha98}.  Based on  previous work by Jouini and Kallal \cite{JouiKall95}, Fritelli \cite{Fritelli97}, Pham and Touzi \cite{PhamTouz99}, Napp \cite{Napp03} and Karatzas and Kardaras \cite{KaraKard07}, the paper by  Pulido  \cite{Pulido14} establishes important properties of price processes under short sale prohibitions namely the equivalence between (NFLVRS) and the existence of an equivalent supermartingale measure for the price processes. Structure conditions for a price process under (NFLVRS) are provided in Coculescu and Jeanblanc \cite{CocJean19}. Recently, Platen and Tappe \cite{PlattenTappe} raised a similar question to the one we are analysing here, namely if exploiting arbitrages requires short-selling, however without insider information being specifically analysed. It is shown that without any possibility of selling short, it is not possible to exploit arbitrages. But at the difference of their paper, we allow the investor to borrow risk-free (that is equivalent selling short risk-free assets), and this assumption results in additional possibilities for exploiting possible arbitrages. Hence, the results in \cite{PlattenTappe} do not apply to our framework.

 There is a very vast literature studying arbitrage possibilities and characterising no-arbitrage models, for different types of  inside information and for different notions of arbitrage, sometimes weaker that no free lunch with vanishing risks. Usually,  a model with respect to a given information set (filtration)  is specified  on a probability space, representing the financial market such as common investors perceive it, and the insider information is introduced as a strictly larger information set, via an enlargement of the original filtration. A fundamental question in this context is  whether the additional information allows for arbitrage profits.
We mention the early work by Grorud and Pontier \cite{GrorudPontier},  Amendinger et al. \cite{AmeImkSch98}, Imkeller \cite{Imkeller02} and subsequent work by  Hillairet \cite{Hill05}, Acciaio et al. \cite{AcciFontKard16}, Aksamit et al. \cite{AksaChouJean15}, Fontana et al \cite{FontJeanSong14}, Chau et al \cite{ChauTank18}, among athers.
 In a general setting, the situations where the additional information of an insider is insufficient for obtaining profits of arbitrage in the form of no free lunch with vanishing risk,  were characterised in the literature and linked to the so-called (H) hypothesis holding between the asset price filtration and the larger filtration containing the additional information, under an equivalent martingale measure (see Blanchet-Scalliet and Jeanblanc \cite{bsjeanblanc} and Coculescu et al. \cite{CocJeanNik12}).

 The present paper contributes to the literature on insider trading by  studying specifically the short-sales restrictions.
Within the two-filtration setting above described, with a filtration larger than the other, and where a stock price is an adapted process for both filtrations, we shall  assume that no free lunch with vanishing risks prevails for the common investors having access only to the smallest filtration. We will then exclusively focus on the no arbitrage conditions in the larger filtration, that is, when considering the insider information. As mentioned above, the existence of a supermartingale measure for the stock price is equivalent to (NFLVRS) (Pulido \cite{Pulido14}). After we introduce the notion of minimal supermartingale measure, we analyse some of  its connections with the minimal martingale measure. The results in Section \ref{Sec3} are actually general  and do not depend on the two-filtration setting. In Section \ref{secH}, a further analysis of the connections between the two measures is provided, specific to the the case of an insider, when assuming the smaller filtration has the predictable representation property with respect to a continuous martingale.  The failure of both the minimal martingale measure and the minimal supermartingale measure to exist is obtained when  the (H) hypothesis is satisfied under a probability measure such that the original one is absolutely continuous to it, but not equivalent. The additional null sets of the original measure belong to the inside information: they are not measurable in the smaller filtration that models the information of the common investors. Some new results on the importance of the (H) hypothesis may also be found in Section \ref{secH}, in particular for the case where (NFLVRS) holds but  (NFLVR) fails to hold.  This corresponds to the case where the arbitrages cannot be exploited without selling the stock short. In the Conclusions section, we shall discuss some possible extensions and uses of our results.

\section{The mathematical setting}\label{SectionSetting}
A probability space $(\Omega, \mathcal{G}, \mathbb{P} )$ is fixed, endowed with a right-continuous filtration $\mathbb{F}^0=(\mathcal{F}_t^0)_{t\geq 0}$.  We denote by $\mathbb{F}=(\mathcal{F}_t)_{t\geq 0}$ the augmentation of  $\mathbb{F}^0$ with the null sets of $\mathbb{P}$,  ensuring that $\mathbb{F}$ satisfies the usual assumptions.  We consider a financial market where a risky asset (e.g. a stock) is traded, in addition to a risk-free asset and such that  the price process, denoted by $X=(X_t)_{t\geq 0}$ is adapted.  To keep the message simple, we assume that the price of the risk-free asset is constant. Alternatively, one could consider the discounted asset price.   Through this paper, we make the following assumption:

\begin{Hypothesis}\label{H1} There exists an equivalent local martingale measure for $X$ in the filtration $\mathbb{F}$. \end{Hypothesis}

Hypothesis \ref{H1} more precisely means:  there exists a probability $\mathbb{P}^*\sim \mathbb{P}$ on $(\Omega,\mathcal{F}_\infty)$ such that $X$ is a $(\mathbb{F},\mathbb{P}^*)$ local martingale.
We emphasise that it is only required the equivalence of these measures on $\mathcal{F}_\infty$. Implicit to Hypothesis \ref{H1} is also that  $X$ is a  locally bounded $(\mathbb{F},\mathbb{P})$ semimartingale. We have thus a standard arbitrage free market in the sense of no free lunch with vanishing risks \cite{DelbScha94}, whenever the filtration $\mathbb{F}$ summarises the available information. We shall call \textit{common investors} the $\mathbb{F}$ informed investors.

We now take the point of view of a particular agent, that we name the \textit{insider}. We assume that the insider has access to some information that can potentially impact the risky asset price, should this information be publicly released. Hence, the insider may have access to some forms or arbitrage, even under Hypothesis \ref{H1}, as the asset price does not necessarily incorporate the extra information.  In mathematical terms, there is a right continuous filtration $\mathbb{G}^0$ strictly larger than $\mathbb{F}^0$, i.e., such that
\begin{equation}\label{fg0}
\mathcal{F}^0_t\subset \mathcal{G}^0_t\;, \;t\geq 0\text{ and }\mathbb{F}^0\neq \mathbb{G}^0.
\end{equation}We define $\mathbb{G}$ as the $\mathbb{P}$ augmentation of $\mathbb{G}^0$, so that $\mathbb{G}$ satisfies the usual assumptions.  The insider is assumed to be able to use $\mathbb{G}$ adapted strategies in order to make investments in $X$ and the risk-free asset. We do not assume any particular structure of the filtration $\mathbb{G}$, except being strictly larger than $\mathbb{F}$, in the sense of (\ref{fg0}). The filtrations $\mathbb{F}^0$ and $\mathbb{G}^0$ will still play a role later on, as we consider probabilities to which $\mathbb{P}$ is only absolutely continuous. For now, the framework is that of the probability space $(\Omega,\mathcal{G},\mathbb{P})$ endowed with two filtrations $\mathbb{F}\subset\mathbb{G}$.

Importantly, we shall assume a certain structure for the decomposition of the stochastic process $X$ in the insider filtration $\mathbb{G}$, namely:
\begin{Hypothesis}\label{H2}  The price of the risky asset $X$ is a nonnegative $(\mathbb{G},\mathbb{P})$ semimartingale  of the form:
\be\label{Xdecomp}
X=X_0+M+ \int \alpha d\langle M\rangle
\ee
for some $(\mathbb{G},\mathbb{P})$ local martingale $M$ and a $(\mathbb{G},\mathbb{P})$ predictable process $\alpha$. The process $\langle M\rangle$, the sharp bracket  of $M$,  is  the $(\mathbb{G},\mathbb{P})$ dual predictable projection of the quadratic variation of $M$.
For simplicity, and without loss of generality, we shall assume $\alpha$ to be constantly zero on the intervals where $d\langle M\rangle=0$.
\end{Hypothesis}

Before considering the question of whether the insider can have free lunches, one should clarify the available trading strategies for the insider. The insider is supposed to invest in the risky and risk free asset based on her information. We are interested here in two alternative situations: the one where the insider may sell short the risky asset and the one where she is constrained to have only long positions in the risky asset, i.e., short sales are prohibited. Consequently, we define the following sets for $t\geq0$:
\begin{align*}
\mathcal M_t(\mathbb{G},\mathbb{P}):&=\{\mathbb{Q}\sim\mathbb{P}\ \text{ on }\mathcal{G}_t\;|\; X_{\cdot\wedge t} \text{ is a $(\mathbb{G},\mathbb{Q})$ local martingale}\} \text{ and }\mathcal M(\mathbb{G},\mathbb{P}):=\mathcal M_\infty(\mathbb{G},\mathbb{P});\\
\mathcal{S}_t(\mathbb{G},\mathbb{P}):&= \{\mathbb{Q}\sim\mathbb{P} \text{ on }\mathcal{G}_t\;|\; X_{\cdot\wedge t}  \text{ is a $(\mathbb{G},\mathbb{Q})$ supermartingale}\}\text{ and }\mathcal{S}(\mathbb{G},\mathbb{P}):=\mathcal{S}_\infty(\mathbb{G},\mathbb{P}).
\end{align*}

Obviously, we have that
\[
\mathcal M_t(\mathbb{G},\mathbb{P})\subset \mathcal{S}_t(\mathbb{G},\mathbb{P})\quad t\geq 0.
\]

In this paper we intend to analyse from the insider's perspective the connections between arbitrages involving short selling the risky asset and arbitrages that exclude short selling the risky asset.  The no arbitrage conditions we study are relative  to the insider filtration $\mathbb{G}$, and they are formally the following:
\begin{itemize}
\item[(a)] the no free lunch with vanishing risks (NFLVR) condition of Delbaen and Schachermayer \cite{DelbScha94},  in the case where short positions are permitted along with the long positions in the assets. In our setting, (NFLVR) is equivalent to: $\mathcal M(\mathbb{G},\mathbb{P})\neq\emptyset$.
\item[(b)] the no free lunch with vanishing risk under short sales prohibition (NFLVRS), a condition that was introduced in Pulido \cite{Pulido14}. In our setting,  by short sales prohibitions we mean that long positions in the risky asset are permitted, along with short and long in the risk-free asset, and in this case (NFLVRS) is equivalent to: $\mathcal{S}(\mathbb{G},\mathbb{P})\neq\emptyset$ (cf. Theorem 3.9 in \cite{Pulido14}).
\end{itemize}

\begin{notation}\label{notation}Given a $\mathbb{G}$-adapted process $Y=(Y_t)_{t\geq 0}$, we use the following notation
\begin{itemize}
\item[-] $\;^{o,\mathbb{P}}Y$ stands for the optional projection of the process $Y$ onto the filtration $\mathbb{F}$, with respect to the measure $\mathbb{P}$;
\item[-]  whenever $Y$ is  increasing: $Y^{o,\mathbb{P}}$ stands for its $(\mathbb{F},\mathbb{P})$-dual optional projection; $Y^{p,\mathbb{P}}$ stands for its $(\mathbb{F},\mathbb{P})$-dual predictable projection;   $\{dY> 0\}$ stands for the support of the random measure $dY$.  Hence, there exists a unique subset of $\Omega\times \mathbb R_+$ that we denote   $\{dY=0\}$ and satisfying:  $\{dY=0\}\cup\{dY>0\}=\Omega\times \mathbb R_+$ and  $\{dY=0\}\cap\{dY>0\}=\emptyset$.
\item[-] whenever $Y$ is a local martingale, we denote by $\langle Y\rangle$ its sharp bracket with respect to $\mathbb{G}$. If $Y$ is an $\mathbb{F}$ adapted local martingale, we denote by $\llangle Y\rrangle$ its sharp bracket with respect to $\mathbb{F}$. In other words, we have $\llangle Y\rrangle=([Y])^{p,\mathbb{P}}$. We emphasise that whenever $Y$ is continuous,  we simply have: $\llangle Y\rrangle= \langle Y\rangle=[Y]$.
\item[-] whenever $Y$ is a semimartingale with $Y_0=0$, $\mathcal E (Y)$ denotes the Dol\'eans-Dade exponential.
\end{itemize}
\end{notation}

Some results will be established in the particular settings where the process $X$ has continuous sample paths, or $\mathbb{F}$ enjoys the predictable representation property, that is, we have a complete $\mathbb{F}$ market.
We label these particular settings as follows:
\begin{itemize}
\item[(C)] The stochastic process $X$ has continuous sample path.
\item[$\mathbb{F}$-(PRP)] The filtration $\mathbb{F}$ is the natural filtration of the price process $X$. There is a (one dimensional) $(\mathbb{F},\mathbb{P})$ local martingale $m^{\mathbb{F},\mathbb{P}}$ such that $m^{\mathbb{F},\mathbb{P}}$ satisfies the $(\mathbb{F},\mathbb{P})$ predictable representation property, that is: any $(\mathbb{F},\mathbb{P})$ local martingale vanishing at zero is equal to $H\cdot m^{\mathbb{F},\mathbb{P}}$ for a suitable $\mathbb{F}$ predictable process $H$.
\end{itemize}

\bigskip

In the remaining of this section, we emphasise some implications of our modelling assumptions.
The choice of assuming  a decomposition of $X$ as proposed in (\ref{Xdecomp}) is rooted in the result below:
\begin{Theorem}\label{ThmXdecomp} Suppose that Hypothesis \ref{H1} holds. In addition, suppose that at least one of the following two conditions is satisfied:
\begin{itemize}
\item[1.] $\mathcal M(\mathbb{G},\mathbb{P})\neq\emptyset$
\item[2.] (C) and $\mathcal{S}(\mathbb{G},\mathbb{P})\neq\emptyset$.
\end{itemize}
Then,  Hypothesis \ref{H2} holds true.
\end{Theorem}
\noindent {\it Proof. $\, $}
In the case where $\mathcal M(\mathbb{G},\mathbb{P})\neq\emptyset$ holds, the decomposition in (\ref{Xdecomp}) is due to Ansel and Stricker \cite{AnselStick92}  and Schweizer \cite {Schw92}  for a $\mathbb{G}$ predictable process $\alpha$. The same holds when  $\mathcal{S}(\mathbb{G},\mathbb{P})\neq\emptyset$, as proved by Coculescu and Jeanblanc \cite{CocJean19} (note that should $X$ be a discontinuous price process the decomposition above is not automatic).
\hfill $\Box$ \vskip 5 pt

From Theorem \ref{ThmXdecomp}, we may characterise a framework where the insider can have free lunches with vanishing risk without engaging in short sales:
\begin{Cor}
 Suppose that Hypothesis \ref{H1} and condition (C) hold. Then, if Hypothesis \ref{H2} is not satisfied, it follows that $\mathcal M(\mathbb{G},\mathbb{P})=\emptyset$ and $\mathcal{S}(\mathbb{G},\mathbb{P})=\emptyset$.
\end{Cor}
In Coculescu and Jeanblanc \cite {CocJean19}, examples are provided where the  $(\mathbb{G},\mathbb{P})$ decomposition of  $X$ fails to satisfy  (\ref{Xdecomp}), in the case where $\mathbb{G}$ is obtained by  progressive enlargements of $\mathbb{F}$ with a random time. The corresponding arbitrage portfolios are also discussed.  The current paper is intended to further analyse the question of (NFLVRS) within  the frame of Hypothesis \ref{H2}, that is, the representation (\ref{Xdecomp}) for the stock price holds.

Finally, we can  notice that implicit to Hypothesis \ref{H1} is also a $(\mathbb{F},\mathbb{P})$ decomposition of the price process $X$, as follows: there is a $(\mathbb{F},\mathbb{P})$ local martingale $m^{\mathbb{F},\mathbb{P}}$ such that
\begin{equation}\label{XdecompF}
X_t=X_0+\int_0^ta_sd\llangle m^{\mathbb{F},\mathbb{P}}\rrangle_s +m^{\mathbb{F},\mathbb{P}}_t.
\end{equation}
where we have (recall our notation) $\llangle m^{\mathbb{F},\mathbb{P}}\rrangle=\;^{o,\mathbb{P}} [m^{\mathbb{F},\mathbb{P}}]$, that is the sharp bracket of the  $(\mathbb{F},\mathbb{P})$ local martingale $m^{\mathbb{F},\mathbb{P}}$.  This decomposition follows by the same argument as in the proof of Theorem \ref{ThmXdecomp} (i), but applied to $\mathbb{F}$ instead of $\mathbb{G}$.

Given Hypotheses \ref{H1} and \ref{H2}, it follows that  $m^{\mathbb{F},\mathbb{P}}$ is a $(\mathbb{G},\mathbb{P})$ semimartingale that decomposes:
\[
m^{\mathbb{F},\mathbb{P}}_t:=\left(\int_0^t\alpha_sd\langle M\rangle_s-\int_0^ta_sd\llangle m^{\mathbb{F},\mathbb{P}}\rrangle_s\right) +M_t.
\]

Further analysis in Section \ref{secH} will again give rise to the representations (\ref{Xdecomp}) and (\ref{XdecompF}) by means of Girsanov transformations. It is important to keep in mind that we assume that no free lunch with vanishing risk holds for common investors (that is in the smaller filtration $\mathbb{F}$), and we analyse the properties of the price process when some additional information is available for an insider, that is, in the filtration $\mathbb{G}$.

\section{The minimal supermartingale measure: definition and some sufficient conditions for its existence}\label{Sec3}
In this section we introduce the concept of  minimal supermartingale measure and establish some connections with the minimal martingale measure.  We use the framework from the previous section, in particular the representation of the asset price in Hypothesis \ref{H2} is supposed to be verified.   However, Hypothesis \ref{H1} is not used in establishing the results in this section, so that the results are general and still hold even if it is were not satisfied. We introduce the following $(\mathbb{G},\mathbb{P})$ local martingales:
 \begin{align}\label{R}
R_t:&= \mathcal E_t\left(-\int_0^\cdot \alpha_ud M_u\right), \quad t\geq 0\\\label{Rplus}
R^{(+)}_t:&= \mathcal E_t\left(-\int_0^\cdot (\alpha_u)^{+}d M_u\right), \quad t\geq 0.
\end{align}
We denote
\[
\frac{d\mathbb{Q}^{\mathcal M}}{d\mathbb{P}}\Big|_{\mathcal{G}_t}=R_t,
\] and if $R$ is a strictly positive and uniformly integrable martingale, we have $\mathbb{Q}^{\mathcal M}\in\mathcal M(\mathbb{G},\mathbb{P})$. Similarly, we denote
\begin{equation}\label{defQminsup}
\frac{d\mathbb{Q}^{\mathcal S}}{d\mathbb{P}}\Big|_{\mathcal{G}_t}=R^{(+)}_t,
\end{equation} and if $R^{(+)}$ is a strictly positive and uniformly integrable martingale we have $\mathbb{Q}^{\mathcal S}\in\mathcal{S}(\mathbb{G},\mathbb{P})$ which can be checked using Girsanov's theorem. In general, $\mathbb{Q}^{\mathcal M}$ and $\mathbb{Q}^{\mathcal S}$ are signed measures.

We recall below the notion of  minimal martingale measure  which first appeared in Schweizer \cite{Sch88} (see also  F\"ollmer and Schweizer (\cite{FoelSchw10}) for a survey of this very rich topic).

\begin{Definition}
When $R$ is a strictly positive and square integrable $(\mathbb{G},\mathbb{P})$ martingale, we call $\mathbb{Q}^{\mathcal M}\in\mathcal M(\mathbb{G},\mathbb{P})$ the $(\mathbb{G},\mathbb{P})$ minimal  martingale measure for $X$.
\end{Definition}
Symmetrically, let us introduce the following definition:
\begin{Definition}
When $R^{(+)}$ is a strictly positive and square integrable $(\mathbb{G},\mathbb{P})$ martingale, we call $\mathbb{Q}^{\mathcal S}\in\mathcal{S}(\mathbb{G},\mathbb{P})$, that is defined as in (\ref{defQminsup}),  the $(\mathbb{G},\mathbb{P})$ minimal supermartingale measure for $X$.

\end{Definition}

In what follows, we will skip the specification of the filtration and of the reference probability $(\mathbb{G},\mathbb{P})$ and will be simply referring to $\mathbb{Q}^{\mathcal M}$ resp. $\mathbb{Q}^{\mathcal S}$ as  the minimal martingale (resp. supermartingale) measure.

It can be shown that when $X$ has continuous sample paths, the minimal supermartingale measure can be characterised as the unique
solution of a minimisation problem involving the relative entropy with respect to $\mathbb{P}$, in a parallel manner to the result holding for the minimal martingale measure; in both cases the idea is to obtain a ``minimal modification'' of $\mathbb{P}$ that allows to reach the set of supermartingale versus martingale measures. The formal setting is given below.
\begin{Definition}
Given two probability measures $\mathbb{P}$ and $\mathbb{Q}$ on $(\Omega,\mathcal{G})$, the \textit{relative entropy}, $H(\mathbb{Q}| \mathbb{P})$  is defined as
\begin{equation*}
  H(\mathbb{Q}|\mathbb{P})=\begin{cases}
    \int \log \frac{d\mathbb{Q}}{d\mathbb{P}}d\mathbb{Q} & \text{if $\mathbb{Q} \ll \mathbb{P}$}.\\
    +\infty, & \text{otherwise}.
  \end{cases}
\end{equation*}
\end{Definition}
The relative entropy is  interpreted as a directed divergence measure between two probabilities; it is nonnegative and $  H(\mathbb{Q}|\mathbb{P})=0$ if and only if $\mathbb{Q} =\mathbb{P}.$

\begin{Theorem}\label{Entropy}
We assume that Hypothesis \ref{H2} and (C) hold. Assume also that $\mathbb{Q}^{\mathcal S}\in \mathcal{S}_T(\mathbb{G},\mathbb{P})$ and $ H(\mathbb{Q}^{\mathcal S}|\mathbb{P})<\infty$.
Then, the measure $\mathbb{Q}^{\mathcal S}$ is the unique solution of
\be\label{minentropy}
\text{Minimize }H(\mathbb{Q}| \mathbb{P})-\frac{1}{2}\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}(\alpha^{+}_s)^2d\langle X \rangle_s\right],
\ee
over all supermartingale measures $\mathbb{Q} \in \mathcal{S}_T(\mathbb{G},\mathbb{P})$, satisfying the condition $\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}(\alpha_s^{+})^2d\langle X \rangle_s\right]<\infty$.
\end{Theorem}
\begin{proof}

Note that the condition $\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}(\alpha_s^{+})^2d\langle X \rangle_s\right]<\infty$ implies that $\int \alpha^{+}dM$ is a true martingale.

Let $\mathbb{Q} \in \mathcal{S}_T(\mathbb{G},\mathbb{P})$. There exists a $(\mathbb{G},\mathbb{P})$ local martingale $L^\mathbb{Q}$ such that $\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{G}_{t\wedge T}}=\mathcal E_t(L^\mathbb{Q})$, for $t\geq 0$. Further, by the Kunita-Watanabe decomposition,
\begin{equation} \label{LKW}
L^{\mathbb{Q}}_t=\int_{0}^{t}\beta^{\mathbb{Q}}_sdM_s +N^{\mathbb{Q}}_t.
\end{equation}
In the above, $N^{\mathbb{Q}}$ is a $(\mathbb{G},\mathbb{P})$ local martingale orthogonal to $M$ and $\beta^{\mathbb{Q}}$ is a $\mathbb{G}$ predictable process.
Under $(\mathbb{G},\mathbb{Q})$ and for $t\in[0,T]$, $X$ possesses the following decomposition:

\begin{equation*}
X_t= M^\mathbb{Q}_t+\int_{0}^{t}(\alpha_s+\beta^{\mathbb{Q}}_s)d \langle M\rangle_s ,
\end{equation*}
 and  $M^{\mathbb{Q}}=M-\langle L^{\mathbb{Q}}, M \rangle
= M -\int\beta^{\mathbb{Q}} d\langle M \rangle$ is a $(\mathbb{G},\mathbb{Q})$ local martingale.

As $(X_{t\wedge T})$ is a $(\mathbb{G},\mathbb{Q})$ supermartingale,  for any $t\in [0,T]$ it must hold that
\be\label{alphabetaneg}
(\alpha_t+\beta_t^{\mathbb{Q}})1\!\!1_{d\langle M\rangle_t >0}\leq 0\quad \mathbb{P}\; a.s.
\ee
We use the notation  $\{d\langle M\rangle_t> 0\}$, for the support of the random measure on $\mathcal B([0,T])$,
associated with the nondecreasing process $\langle M\rangle$.

We have that
\begin{align*}
H(\mathbb{Q}|\mathbb{P}) &=H(\mathbb{Q}|\mathbb{Q}^{\mathcal S})+\int \log R^{(+)}_T d\mathbb{Q}\\
 &=H(\mathbb{Q}|\mathbb{Q}^{\mathcal S})+ \int \left(-\int_{0}^{T}\alpha^{+}_sdM_s-\frac{1}{2}\int_{0}^{T}(\alpha^{+}_s)^{2}d\langle M\rangle_s \right)d\mathbb{Q}\\
&=H(\mathbb{Q}|\mathbb{Q}^{\mathcal S})+ \int \left(-\int_{0}^{T}\alpha^{+}_sdM^{\mathbb{Q}}_s -\int_{0}^{T}\alpha^{+}_s\beta^{\mathbb{Q}}_sd\langle M \rangle_s   -\frac{1}{2}\int_{0}^{T}(\alpha^{+}_s)^{2}d\langle M\rangle_s \right)d\mathbb{Q}\\
& =H(\mathbb{Q}|\mathbb{Q}^{\mathcal S})+\mathbb{E}^{\mathbb{Q}}\left[\int_{0}^{T}( -\alpha^{+}_s\beta^{\mathbb{Q}}_s-\frac{1}{2}(\alpha^{+}_s)^{2})d\langle X\rangle_s\right ].
\end{align*}
Therefore,
\begin{align*}
H(\mathbb{Q}|\mathbb{P})-\frac{1}{2}\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}(\alpha^{+}_s)^2d\langle X \rangle_s\right]=  H(\mathbb{Q}|\mathbb{Q}^{\mathcal S}) - \mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}\alpha^{+}_s(\alpha_s+\beta^{\mathbb{Q}}_s)d\langle X\rangle_s\right ].
\end{align*}
and
\begin{align*}
 H(\mathbb{Q}|\mathbb{P})-\frac{1}{2}\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}(\alpha^{+}_s)^2d\langle X \rangle_s\right]\geq \min_\mathbb{Q}   H(\mathbb{Q}|\mathbb{Q}^{\mathcal S}) - \max_\mathbb{Q} \mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}\alpha^{+}_s(\alpha_s+\beta^{\mathbb{Q}}_s)d\langle X\rangle_s\right ],
\end{align*}
where the min and max above are taken over the same set as in the problem (\ref{minentropy}); it can be checked that $\mathbb{Q}^{\mathcal S}$ belongs in this set.

We have that  $\min_\mathbb{Q}   H(\mathbb{Q}|\mathbb{Q}^{\mathcal S})=0$; the minimal value of $0$ is obtained if and only if $\mathbb{Q}=\mathbb{Q}^{\mathcal S}.$
Also, in view of the inequality (\ref{alphabetaneg}), $\mathbb{E}^{\mathbb{Q}}\left [\int_{0}^{T}\alpha^{+}_s(\alpha_s+\beta^{\mathbb{Q}}_s)d\langle X\rangle_s\right ]\leq 0$ and we can check that the maximum value of this expected value is 0, obtained when taking $\beta^{\mathbb{Q}}=-\alpha^+$, that is,  $G^{\mathbb{Q}}=R^{(+)}$, or $\mathbb{Q}=\mathbb{Q}^{\mathcal S}.$ This proves that (\ref{minentropy})  equals  $0$ and this value is attained if and only if $\mathbb{Q}=\mathbb{Q}^{\mathcal S}.$
\end{proof}

Let us now emphasise some first connections between $\mathbb{Q}^{\mathcal M}$ and $\mathbb{Q}^{\mathcal S}$:

\begin{Theorem}\label{TheoremGeneral}We assume that the representation in Hypothesis \ref{H2} holds.
\begin{itemize}
\item[(a)]If the minimal martingale measure $\mathbb{Q}^{\mathcal M}$ exists then,  $\mathbb{Q}^{\mathcal S}\in \mathcal{S}(\mathbb{G},\mathbb{P})$.
 \item[(b)] Suppose that the process $(1\!\!1_{\alpha_t> 0})_{t\geq 0}$ admits a right-continuous modification having bounded total variation.
Under this condition,  if $\mathbb{Q}^{\mathcal M}\in \mathcal M(\mathbb{G},\mathbb{P})$ then $\mathbb{Q}^{\mathcal S}\in \mathcal{S}(\mathbb{G},\mathbb{P})$.
 \end{itemize}
\end{Theorem}

\brem
Before we prove this result, let us illustrate the condition appearing in (b). \\ We consider $W$ is a Brownian motion with respect to $(\mathbb{G},\mathbb{P})$, and let $\alpha:=|W|$. We obtain $1\!\!1_{|W_t|>0}=H_t-1\!\!1_{W_t=0}$ where $H_t:=1\!\!1_{|W_t|\geq 0}=1$. By Proposition 3.12 page 108 in \cite{RevuzYor},  the set $\{t: W(t)=0\}$ is a.s. closed, without any isolated points and has zero Lebesgue measure. Hence, $H$ is a modification of the process $1\!\!1_{|W_t|>0}$. As $H$ is continuous, with null total variation, we conclude that for this example, the condition stated in Theorem \ref{TheoremGeneral} (b) is fulfilled.  \\
Alternatively, let us consider there is a  $(\mathbb{G},\mathbb{P})$ counting process $N$ with c\`adl\`ag sample path and with finite explosion time. Let $\alpha:=(-1)^{N}$. Then, the process $\alpha$ is c\`adl\`ag and has infinite total variation. In this case, the  condition stated in Theorem \ref{TheoremGeneral} (b) is not fulfilled.
\erem

\noindent {\it Proof. $\, $} (of Theorem \ref{TheoremGeneral})
\begin{itemize}

\item[(a)]We need to prove that if $R$ is a strictly positive and square integrable $(\mathbb{G},\mathbb{P})$ martingale then $R^{(+)}$ is a strictly positive and uniformly integrable $(\mathbb{G},\mathbb{P})$ martingale.  Let us introduce the following $(\mathbb{G},\mathbb{P})$ local martingale, which is orthogonal to  $R^{(+)}$:
\[
R^{(-)}_{t}=\mathcal E_t\left(\int_0^\cdot (\alpha_u)^-d M_u\right),\quad t\geq 0,
\]
so that:
\[
R_t:=R^{(+)}_{t}R^{(-)}_{t}.
\]
The processes $R^{(+)}$ and $R^{(-)}$ are strictly positive local martingales: they satisfy $R_0^{(+)}=R_0^{(-)}=1$ and can become negative or null only after a jump, i.e., one at a time at most (as they do not have common jumps). In other words, the product $R$ is strictly positive if and only if $R^{(+)}$ and $R^{(-)}$ are strictly positive. Then, one can show that  $R^{(+)}$ is a uniformly integrable martingale, as a direct application of Proposition \ref{Y=UZ} in Appendix \ref{appendix}.

\item[(b)]
  If $ \mathbb{Q}^{\mathcal M}\in \mathcal M(\mathbb{G},\mathbb{P})$, then  $R=\mathcal E(L)$ is a strictly positive, uniformly integrable $(\mathbb{G},\mathbb{P})$ martingale, where $L=-\int \alpha dM$. We denote by $H=(H_t)_{t\geq 0}$ a right-continuous modification of the process  $(1\!\!1_{\alpha_t>0})_{t\geq 0}$, assumed of bounded variation. We can write $R^{(+)}_t=\mathcal E_t(\int HdL)$ and $R^{(-)}_t=\mathcal E_t(\int (1-H)dL)$.   The result then follows as an application of the Lemma \ref{M=UZ(2)} in Appendix \ref{appendix}.
\end{itemize}
\hfill $\Box$ \vskip 5 pt

Another sufficient condition for $\mathbb{Q}^{\mathcal S}\in \mathcal{S}(\mathbb{G},\mathbb{P})$ is the following:

\begin{Lemma}We assume that the representation in Hypothesis \ref{H2} holds.

Suppose that it exists $ \mathbb{Q}\in  \mathcal{S}_T(\mathbb{G},\mathbb{P})$ such that  $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{G}_t}=\mathcal E_t(L^\mathbb{Q})$  is satisfying $\mathbb{E}^{\mathbb{P}}[e^{\frac{1}{2}\langle L^\mathbb{Q}\rangle_T }]<\infty$. Then, $\mathbb{Q}^{\mathcal S}\in\mathcal{S}_T(\mathbb{G},\mathbb{P})$.
\end{Lemma}

\noindent {\it Proof. $\, $}
We  use the same notation as in the proof of Theorem \ref{Entropy}.; in particular the process $L^\mathbb{Q}$ is a $(\mathbb{G},\mathbb{P})$ local martingale with the Kunita Watanabe decomposition given in (\ref{LKW}).
  In order for $X$ to be a $(\mathbb{G},\mathbb{Q})$-supermartingale, necessarily  the process:
\[
 \int_0^t \left(\alpha_u+\beta^\mathbb{Q}_u\right)d \langle M\rangle _u,\quad t\in[0,T]
\]
is nonincreasing, in particular we need:
\[
0\leq \alpha 1\!\!1_{\{\alpha>0\}}\leq - \beta^\mathbb{Q} 1\!\!1_{\{\alpha>0\}}\quad \mathbb{P}\;a.s.
\](notice that $\{(\omega,t)|\alpha_t(\omega)>0\}$ is supposed to be included in the support of $d \langle M\rangle$ by Hypothesis \ref{H2}).
By assumption, $\mathcal E (L^\mathbb{Q})$ satisfies the Novikov condition. Hence, in view of the above inequality, $R^{(+)}$ is a uniformly integrable martingale satisfying the Novikov condition.
\hfill $\Box$ \vskip 5 pt

\section{The (H) hypothesis as a no arbitrage condition for insiders in presence of short sales restrictions}\label{secH}

In this section, assuming $\mathbb{F}$-(PRP), we further analyse connections between the existence of $\mathbb{Q}^{\mathcal M}$ and $\mathbb{Q}^{\mathcal S}$, in particular under the assumption that the (H) hypothesis under some probability measure on $(\Omega, \mathcal{G})$ holds true. Given two filtrations, $\mathbb{F}\subset \mathbb{G}$, the (H) hypothesis, also called the immersion property, is the property that all $\mathbb{F}$ martingales are also $\mathbb{G}$ martingales; the immersion property holds in relation to a reference probability measure. In the financial literature, the (H) hypothesis under an equivalent martingale measure for the asset prices is known as a classical no arbitrage condition whenever asset prices are $\mathbb{F}$ adapted, but agents may employ $\mathbb{G}$-adapted strategies. The main result is due to Blanchet-Scalliet and Jeanblanc \cite{bsjeanblanc} (see Theorem \ref{HNA} below),  further results involving the (H) hypothesis and no arbitrage can be found in  \cite{CocJeanNik12}. It is to be noted that both references  \cite{bsjeanblanc} and  \cite{CocJeanNik12} are focused on the particular case where the filtration $\mathbb{G}$ is obtained from $\mathbb{F}$  via a progressive enlargement with a random time, but some of their results are general (as we shall emphasize below).  These results are investigating the link between (NFLVR) in $\mathbb{G}$ and (H), more precisely, when  the $\mathbb{G}$ informed agent is able to sell short.  We shall see that  the immersion property is still a key condition for understanding (NFLVRS) in $\mathbb{G}$.

For the reader's convenience, let us first give a definition and recall some results that are needed later on.
\begin{Definition}
We say that the immersion property, or (H) hypothesis, holds between $\mathbb{F}$ and $\mathbb{G}$ under the probability measure $\mathbb{Q}$ if  all $(\mathbb{F},\mathbb{Q})$ local martingales are $(\mathbb{G},\mathbb{Q})$ local martingales. When this property holds, we write $\mathbb{F}\overset{\mathbb{Q}}{\hookrightarrow}\mathbb{G}$. When this property does not hold, we write  $\mathbb{F}\overset{\mathbb{Q}}{\centernot{\hookrightarrow}}\mathbb{G}$.
\end{Definition}

More about the immersion of filtrations can be found in \cite{bremaudyor}, \cite{DelMeyer1} or \cite{aj}. See also \cite{Kusuoka}, \cite{CocJeanNik12} where changes of the probability measure are considered together with the immersion property.

In general, the immersion property is not preserved by changes of the probability measure.  Let us introduce the following notation for the set of probability measures that have the immersion property and are equivalent to $\mathbb{P}$:
\begin{equation*}
 \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P}):=\left \{ \widetilde \mathbb{Q} \text{ probability on }(\Omega,\mathcal{G})\; |\;
  \mathbb{P}\sim \widetilde \mathbb{Q},\;\mathbb{F} \overset{\widetilde \mathbb{Q}}{\hookrightarrow}\mathbb{G}
 \right  \}.
\end{equation*}

We now state the result of Blanchet-Scalliet and Jeanblanc that was mentioned above. Even though in \cite{bsjeanblanc}  the filtration $\mathbb{G}$ is defined as a progressive enlargement of $\mathbb{F}$ with a random time, the proof of their Proposition 1 -  that corresponds to Theorem \ref{HNA} below - holds actually in the generality of our definition for $\mathbb{G}$, as one can easily verify; see also Corollary 4.6 in \cite{CocJeanNik12} for an alternative proof of this result.

\begin{Theorem}\label{HNA} We suppose that $\mathbb{F}$-(PRP) holds and $\mathcal M(\mathbb{G},\mathbb{P})\neq \emptyset$. Then, $\mathcal M(\mathbb{G},\mathbb{P})\subset  \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})$.

\end{Theorem}

We want to understand if a potential of arbitrage is generated by the inside information, and the possible connections with the immersion property. To fully isolate this case of interest, no arbitrage for the common investors will be systematically assumed from now, in the form of  Hypothesis \ref{H1}. This way, any possible $\mathbb{G}$ arbitrage is necessarily arising from the exploitation of the inside information and is not accessible with $\mathbb{F}$ adapted strategies.  With this extra requirement, we can be establish some stronger result than Theorem \ref{HNA}, in the form of  Theorem \ref{ThmGenH} below.

But before, let us also recall some well known financial implication of $\mathbb{F}$-(PRP), namely the market is complete for the common investors.
\begin{rem}\label{remPstar}
Suppose that Hypothesis \ref{H1} and $\mathbb{F}$-(PRP) hold. Then, there exists a unique probability $\mathbb{P}^*$ on $(\Omega,\mathcal{F}_\infty)$ such that:
\begin{itemize}
\item[(i)] $\mathbb{P}^*\sim\mathbb{P}$ on $\mathcal{F}_\infty$, and
\item[(ii)]$X$ is a $(\mathbb{F},\mathbb{P}^*)$ local martingale.
\end{itemize}
This result is known under the name of  second theorem theorem of asset pricing (see Harrison and Pliska \cite{HarrPlisk83}).
Furthermore,  $X$ has the  $(\mathbb{F},\mathbb{P}^*)$ predictable representation property in the filtration $\mathbb{F}$. Indeed,  the predictable representation property is stable under equivalent changes of the probability measure (Theorem 13.12 in \cite{HeWangYan92}). \end{rem}
One question is whether there are extensions of $\mathbb{P}^*$ to the larger sigma-field $\mathcal{G}$ that can serve as equivalent martingale measures for the insider.
\begin{Theorem}\label{ThmGenH}
We suppose that Hypothesis \ref{H1} and $\mathbb{F}$-(PRP) hold. Then, the following are equivalent:
\begin{itemize}
\item[(i)] $\mathcal M(\mathbb{G},\mathbb{P})\neq \emptyset$
\item[(ii)] $\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\neq \emptyset$
\item[(iii)]  $ \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\cap \mathcal M(\mathbb{F},\mathbb{P})= \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\cap \mathcal M(\mathbb{G},\mathbb{P})\neq \emptyset$.
\item[(iv)] $ \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\cap  \mathcal{S}(\mathbb{G},\mathbb{P})\neq \emptyset$.
\end{itemize}
\end{Theorem}
\noindent {\it Proof. $\, $}
\begin{itemize}
\item[-] The implication (i) $\Rightarrow$ (ii) follows from Theorem \ref{HNA}.

\item[-] The  implications (ii) $\Rightarrow$ (iii) follows from Proposition 4.4 in  \cite{CocJeanNik12}. We emphasize that this result holds in the generality of our setting for the filtrations $\mathbb{F}$ and $\mathbb{G}$. We recall this result, for the reader's convenience:  \\
\textit{Given a reference probability measure $\mathbb{P}$, if  $\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\neq \emptyset$, then there is  $ \mathbb{Q}\in\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})$  such that every $(\mathbb{F},\mathbb{P})$ martingale is a $(\mathbb{G}, \mathbb{Q})$ martingale, that is,  $\mathbb{Q}=\mathbb{P}$ on $\mathcal{F}_\infty$. }\\
This result means that if (ii) holds, then there exist equivalent probability measures that  keep the $\mathbb{F}$ martingales unchanged and at the same time  ensure the immersion property holds. We now take $\mathbb{P}^*$ as a reference measure, where $\mathbb{P}^*\in\mathcal M(\mathbb{F},\mathbb{P})$, that is, $X$ is a local martingale in the smaller filtration $\mathbb{F}$ under $\mathbb{P}^*$. Because $\mathbb{P}\sim \mathbb{P}^*$, it follows that $\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P}^*)=\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})$. Thence (ii) is equivalent to: $\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P}^*)\neq \emptyset$.
Then, by Proposition 4.4 in  \cite{CocJeanNik12} stated above, there is $ \mathbb{Q}\in\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P}^*)$  such that $\mathbb{Q}=\mathbb{P}^*$ on $\mathcal{F}_\infty$, in particular  $X$ is a local martingale in both filtrations under $\mathbb{Q}$. Hence (iii)   holds.

\item[-] The implication (iii) $\Rightarrow$ (i) is obvious.

\item[-]The equivalences with the last statement are also obvious: (iv) implies (ii) and also (iii) implies (iv), as if $\mathbb{Q}$ is a local martingale measure in filtration $\mathbb{G}$,  it is also a supermartingale measure in filtration $\mathbb{G}$.
\end{itemize}
\hfill $\Box$ \vskip 5 pt

A few comments on the results in Theorem \ref{ThmGenH} are as follows. For the insider, as soon as there exists a probability measure equivalent to $\mathbb{P}$ under which the (H) hypothesis holds, there are no arbitrages to undertake,  without or even involving short positions: $\emptyset\neq \mathcal M(\mathbb{G},\mathbb{P})\subset  \mathcal{S}(\mathbb{G},\mathbb{P})$. But, on the other hand, if there is not such an equivalent probability measure, then there are automatically arbitrages, possibly involving short-selling, as $\mathcal M(\mathbb{G},\mathbb{P})= \emptyset$.

 An interesting case is the one where there are arbitrage opportunities for the insider, but they all require short-selling by the insider.  This  corresponds to the case $\mathcal M(\mathbb{G},\mathbb{P})= \emptyset$ and $\mathcal{S}(\mathbb{G},\mathbb{P})\neq  \emptyset$.   Theorem \ref{ThmGenH}  tells us that such a situation corresponds to the case where no element of $\mathcal{S}(\mathbb{G},\mathbb{P})$ can ensure for the (H) hypothesis to hold, when taken as a reference measure. Indeed, another formulation for the equivalence between (i) and (iv) is:
 \begin{Cor} We suppose that Hypothesis \ref{H1} and $\mathbb{F}$-(PRP) hold. Under these assumptions,
 $\mathcal M(\mathbb{G},\mathbb{P})=\emptyset$ if and only if $\mathcal S(\mathbb{G},\mathbb{P})\cap  \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})=\emptyset$.
\end{Cor}

We illustrate this situation by a simple example using honest times, inspired by the more general construction available in \cite{Nikegh06a}. We refer to \cite{FontJeanSong14} for an analysis of arbitrages arising from knowledge of honest times (where short-sales restrictions are not studied). These references provide a general setting, surely also suitable for a more systematic study of the implications of short-sale restrictions in progressive enlargements of filtrations with honest times (that is beyond the scope here).

\begin{example*}We assume the filtration $\mathbb{G}$ is the progressive enlargement of the filtration $\mathbb{F}$ with the last passage time of the stock price at its supremum level. More exactly let $X=\mathcal E(\sigma W_t)$ where $W$ is $(\mathbb{F},\mathbb{P})$ Brownian motion. We notice that  $\lim_{t\to\infty } X_t =
0$. Let $S_t = \sup_{s\leq t} X_s$ and
\[
g = \sup\{t\geq 0: X_t=S_\infty\} = \sup\{t\geq 0: S_t-X_t=0\}.
\]We let $\mathbb{G}$ be the smallest filtration that satisfies the usual assumptions and  so that $g$ is a $\mathbb{G}$ stopping time. We thus assume the insider observes $g$ when it occurs.
The random time $g$ is a last passage time within the filtration $\mathbb{F}$ and moreover the decomposition of the price process in the filtration $\mathbb{G}$ is given by (cf. Proposition 2.5 in \cite{NikeghYor06}):
\begin{align*}
X_t&=1+\int_0^{t\wedge g} \frac{1}{X_s}d\langle X\rangle_s -\int_{g}^{t\vee g} \frac{1}{S_s-X_s}d\langle X\rangle_s+M_t\\
&=1+\sigma^2\int_0^{t\wedge g}X_sds -\sigma^2\int_{g}^{t\vee g} \frac{X^2_s }{S_s-X_s}ds+\sigma \int_0^tX_sd\hat W_s.
\end{align*}
Where $M= \sigma \int Xd\hat W$ corresponds to the local martingale in the decomposition (\ref{Xdecomp}) with  $\hat W$ being a $(\mathbb{G},\mathbb{P})$ Brownian motion. We now consider a finite horizon, that is $t\in[0,T]$, so that the process $X$ is stopped at $T$. Then,  there exists a minimal supermartingale measure and it satisfies $d\mathbb{Q}^{\mathcal S}= \mathcal E (- \sigma \hat W_{T\wedge g})\cdot d\mathbb{P}$ on $\mathcal{G}_T$. Indeed, $\mathbb{Q}^{\mathcal S}\in\mathcal S_T(\mathbb{G},\mathbb{P})$ as one can easily verify.  However, there is no martingale measure in the filtration $\mathbb{G}$: $\mathcal I_T(\mathbb{G},\mathbb{P})=\emptyset$ and $\mathcal M_T(\mathbb{G},\mathbb{P})=\emptyset$. Indeed, progressive enlargement with honest times are well known examples of models where insiders can have free lunches with vanishing risk (see for instance Theorem 4.1 in \cite{FontJeanSong14}). In conclusion, in this example the insider needs short-selling the stock in order to realise an arbitrage. Indeed, selling the asset $X$ short at $g\wedge T$ and holding the cash obtained, then unwind the positions at  $T$ provides an arbitrage (NFLVR). In conclusion, short-sales prohibitions can be effective in eliminating arbitrages for this example of insider information.

\end{example*}

Now, we want to shed some additional light on the connections between $\mathbb{Q}^{\mathcal S}$ and $\mathbb{Q}^{\mathcal M}$, namely, we provide conditions under which both fail to be probability measures (in Theorem \ref{ThmH} below).
As we shall only focus on a bounded time horizon $[0,T]$, we introduce the stopped filtrations $\mathbb{F}_T=(\mathcal{F}_{T\wedge t})$ and $\mathbb{G}_T=(\mathcal{G}_{T\wedge t})$, and we suppose that all stochastic processes are also stopped at $T$ without introducing further notation.  We also shall assume that $\mathbb{F}_T\overset{\mathbb{P}}{\centernot{\hookrightarrow}}\mathbb{G}_T$, that is, so that our model is compatible with the (possible) failure of  (NFLVR) in the filtration $\mathbb{G}$ (the terminal value of the portfolio will be nonnegative and strictly positive with positive probability).

 Let us introduce the following set of probability measures:
\begin{equation*}
 \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P}):=\left \{ \widetilde \mathbb{Q} \text{ probability on }(\Omega,\mathcal{G}_T)\;\biggm | \;
 \begin{array}{l}
 \mathbb{F}_T\overset{\widetilde \mathbb{Q}}{\hookrightarrow}\mathbb{G}_T,\\
\mathbb{P}\ll\widetilde \mathbb{Q};\; \mathbb{P}\sim \widetilde \mathbb{Q} \text{  on }\mathcal{F}_T.
 \end{array}
 \right  \}.
\end{equation*}
We have that $ \mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})\subset  \mathcal P_\infty(\mathbb{F},\mathbb{G},\mathbb{P})$.

Absolutely continuous, but not equivalent, changes of measure may lead to arbitrage opportunities in the (NFLVR) sense, see e.g. Delbaen and Schachermayer \cite{DelbScha95}, Osterrieder and Rheinl\"ander \cite{OstRhe06},  Ruf and Runggaldier \cite{RufRung13} and Chau and Tankov \cite{ChauTank13}.

\begin{notation}
Suppose that $\mathbb{Q}\in \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})$.  We denote by $V_T(\mathbb{Q})$ the set of $(\mathbb{G}_T,\mathbb{Q})$  local martingales that admit the representation $(c+\int_0^{\cdot\wedge T} H_s dm_s)$, where $c$ is constant, $m$ is an $(\mathbb{F}_T,\mathbb{Q})$  local martingale  and $H$ is a $\mathbb{G}_T$ predictable process such that $\int_0^{\cdot\wedge T} H_s^2d\langle m\rangle_s $ is locally integrable. As the immersion property holds for any $\mathbb{Q}\in \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})$, the stochastic integrals introduced above are well defined.

\end{notation}

In our next two results we will show that for $\mathbb{Q}^{\mathcal M}$ or $\mathbb{Q}^{\mathcal S}$ to be probabilities on $\mathcal{G}_T$, a certain subset of $ \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})$ plays a central role,  namely
\begin{equation*}
\mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P}):=\left\{\widetilde \mathbb{Q}\in \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})
\; \Big| \;
  \frac{d\mathbb{P}}{d\widetilde\mathbb{Q} }\in V_T(\widetilde\mathbb{Q})
 \right  \}.
\end{equation*}

\begin{Theorem}\label{ThmQstar}
We assume that Hypothesis \ref{H1} and $\mathbb{F}$-(PRP) hold. If $p_T(\mathbb{F},\mathbb{G},\mathbb{P})\neq \emptyset$, then there exists a  unique probability measure $\mathbb{Q}^*$ that satisfies:
\begin{itemize}
\item[(i)] $\mathbb{Q}^*\in \mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$.
\item[(ii)] The price process $X$ is a $(\mathbb{G}_T, \mathbb{Q}^*)$ local martingale.
\end{itemize}
Assume furthermore  Hypothesis \ref{H2}  and assumption (C) hold. Then:
\begin{itemize}
\item[(iii)]  The process
$$
D^*_t:=\mathbb{E}^{\mathbb{Q}^*}\left [\frac{d\mathbb{P}}{d\mathbb{Q}^{*}}\;\Big | \;\mathcal{G}_{t\wedge T}\right],\quad t\geq 0
$$ admits the $(\mathbb{G}_T,\mathbb{Q}^*)$ representation:
\begin{equation}\label{defD}
D^*_t=1+\int_0^t D^*_{s}\alpha_sdX_s.
\end{equation}
It follows that $\mathbb{Q}^*\sim \mathbb{P}$  on $\mathcal{G}_T$ if and only if $\mathbb{Q}^*= \mathbb{Q}^\mathcal M\in\mathcal M_T(\mathbb{G},\mathbb{P})$.
\end{itemize}
\end{Theorem}

\noindent {\it Proof. $\, $}
The set $\mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$ contains equivalent probabilities and therefore we shall introduce $\mathbb{Q}^*$ via a Girsanov transformation,  relatively to an arbitrary element in $\mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$.

Let $ \mathbb{Q}\in \mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$.
Then there is a nonnegative $(\mathbb{G}_T, \mathbb{Q})$ martingale $E$ with $d\mathbb{P}=E_T\cdot  \mathbb{Q}$ on $\mathcal{G}_T$ and a strictly positive $(\mathbb{F}_T, \mathbb{Q})$ martingale $e$ with $d\mathbb{P}=e_T\cdot  \mathbb{Q}$ on $\mathcal{F}_T$.

If Hypothesis \ref{H1} together with $\mathbb{F}$-(PRP) hold, there is a unique local martingale measure for the price process $X$ and in the filtration $\mathbb{F}$, that is, there is a unique  $(\mathbb{F}, \mathbb{P})$ uniformly integrable martingale $\delta$ with $\delta$ being strictly positive $\delta_0=1$, such that the process $X\delta$ is  $(\mathbb{F}, \mathbb{P})$ local martingale.  Furthermore,  $\delta$ admits a representation as $$\delta_t=\mathcal E_t\left(-\int_0^\cdot a_s dm_s^{\mathbb{F},\mathbb{P}}\right)\quad t\geq 0$$ (in line with the representation of the asset price given in (\ref{XdecompF})).
Hence, an equivalent local martingale measure for the common ($\mathbb{F}$ informed) investors may be defined on $\mathcal{G}$ via
$$
d\mathbb{P}^*:=\delta_\infty \cdot d\mathbb{P}\quad \text{ an }\mathcal{G}.$$

We are after a probability $\mathbb{Q}^*\in  \mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$ that may potentially serve as an equivalent martingale measure for the insider, as the price process $X$ is required  to be a $(\mathbb{G}_T, \mathbb{Q}^*)$ local martingale, by statement (ii) of the theorem. We introduce $\mathbb{Q}^*$ via:
$$
d\mathbb{Q}^*=\delta_T e_T\cdot  d\mathbb{Q}\quad \text{ on $\mathcal{G}_T$}.
$$
It can be checked that $\mathbb{Q}^*\sim  \mathbb{Q}$ and furthermore $\mathbb{Q}^*$ concides with $\mathbb{P}^*$ on $\mathcal{F}_T$:
\[
\frac{d\mathbb{Q}^*}{d  \mathbb{Q}}\Big|_{\mathcal{F}_T}=\delta_t \times e_t=\frac{d\mathbb{P}^*}{d \mathbb{P}}\Big|_{\mathcal{F}_T}\times\frac{d\mathbb{P}}{d  \mathbb{Q}}\Big|_{\mathcal{F}_T}=\frac{d\mathbb{P}^*}{d  \mathbb{Q}}\Big|_{\mathcal{F}_T}.
\]
Consequently,  $\mathbb{Q}^*$ is a well defined change of measure on $\mathbb{F}_T$ with the Radon-Nikod\'ym density process $(\delta_t e_t)$ being a strictly positive $(\mathbb{F},  \mathbb{Q})$ martingale. Finally, as the (H) hypothesis holds under $ \mathbb{Q}$, the process $(\delta_t e_t)$ is also a  $(\mathbb{G},  \mathbb{Q})$ martingale, so that the mesure $\mathbb{Q}^*$ is well defined also on $\mathcal{G}_T$.

One can see that $\mathbb{Q}^*\in  \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})$. This is the consequence of the fact that the immersion property is preserved under changes of measure using $\mathcal{F}_\infty$  measurable Radon-Nicod\'ym derivatives (see Lemma 4.3 in\cite{CocJeanNik12}).  Indeed, we have $\mathbb{F} \overset{ \mathbb{Q}}{\hookrightarrow}\mathbb{G}$ and $\frac{d\mathbb{Q}^*}{d \mathbb{Q}}$ is $\mathcal{F}_T$ measurable therefore $\mathbb{F} \overset{ \mathbb{Q}^*}{\hookrightarrow}\mathbb{G}$ and so that indeed
\begin{equation}\label{QstarP}
\mathbb{Q}^*\in  \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P}).
\end{equation}
We can now prove the statements of the theorem:
\begin{itemize}
\item[(ii)] Using Girsanov's theorem in the filtration $\mathbb{F}_T$, we obtain that the following is a $(\mathbb{F}_T,\mathbb{Q}^*)$ martingale:

\[
m_t^{\mathbb{F}, \mathbb{Q}^*}: = m_t^{\mathbb{F}, \mathbb{P}}-\int_0^t \frac{d\llangle\delta, m^{\mathbb{F}, \mathbb{P}}\rrangle_s}{\delta_{s-}}= \int_0^t a_s d\llangle m^{\mathbb{F}, \mathbb{P}} \rrangle_s + m_t^{\mathbb{F}, \mathbb{P}}=X_t-X_0.
\]As $\mathbb{F} \overset{ \mathbb{Q}^*}{\hookrightarrow}\mathbb{G}$, it follows that $m^{\mathbb{F}, \mathbb{Q}^*}$ is also a $(\mathbb{G}_T,\mathbb{Q}^*)$ martingale, so that the $(\mathbb{G}_T, \mathbb{Q}^*)$ semimartingale decomposition of the asset price $X$ is:
\begin{equation}\label{intqpG}
X_t=X_0+m^{\mathbb{F},  \mathbb{Q}^*}_t.
\end{equation}
 that is a $(\mathbb{G}_T,\mathbb{Q}^*)$ martingale.

\item[(i)] In view of (\ref{QstarP}), we only need to show that $\frac{d\mathbb{P}}{d \mathbb{Q}^*}|_{\mathcal{G}_T}=\frac{d\mathbb{P}}{d  \mathbb{Q}}|_{\mathcal{G}_T}\times \frac{d  \mathbb{Q}}{d \mathbb{Q}^*}|_{\mathcal{G}_T}=\frac{E_T}{e_T\delta_T}\in V_T(\mathbb{Q}^*)$. We first provide $(\mathbb{G}_T, \mathbb{Q})$ representations for the processes $e$ and $E$.

Because $\mathbb{F}$-(PRP) holds and $ \mathbb{Q}\sim\mathbb{P}$ on $\mathcal{F}_T$, it follows that in the filtration $\mathbb{F}$ the $ \mathbb{Q}$ martingales also have the predictable representation property, with respect to the  martingale $ m^{\mathbb{F}, \mathbb{Q}}= m^{\mathbb{F}, \mathbb{P}}-\int\frac{d\llangle e_{-},m^{\mathbb{F}, \mathbb{P}}\rrangle}{e_{-}}$ (see \cite{HeWangYan92}, Theorem 13.12]). Then, there exists an $\mathbb{F}$ predictable process $h$ so that
$$
e=\mathcal E\left (\int hdm^{\mathbb{F}, \mathbb{Q}}\right).
$$
As (H) holds under $ \mathbb{Q}$, $m^{\mathbb{F}, \mathbb{Q}}$ is also a $ \mathbb{Q}$ martingale in the larger filtration $\mathbb{G}_T$. Then,   $e$ is also a  $(\mathbb{G}_T, \mathbb{Q})$ martingale,  while  $E$ has the representation
$$
E=\mathcal E\left(N+\int H dm^{\mathbb{F}, \mathbb{Q}}\right)
$$ with $N$ a $(\mathbb{G}_T, \mathbb{Q})$ local martingale orthogonal to $m^{\mathbb{F}, \mathbb{Q}}$, and where $H$ is a $\mathbb{G}$ predictable process. This representation of $E$ is $ \mathbb{Q}-a.s.$ unique on the set $[0,T^0]$ with $T^0=\inf\{t\geq 0: E_t=0\}$.
As $ \mathbb{Q}\in \mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})$, we observe that $N_{\cdot \wedge T_0}\equiv 0$ since  $E\in  V_T( \mathbb{Q})$. This leads to the representation
$$
E=\mathcal E\left (\int H dm^{\mathbb{F}, \mathbb{Q}}\right).
$$
Moreover, the condition $\mathbb{E}_{ \mathbb{Q}}[E_t|\mathcal{F}_t]=e_t$ leads to a relation between $H$ and $h$, (cf.  Br\'emaud and Yor \cite{bremaudyor}, Proposition 1) namely
$$
(h_t)\text{ is the $(\mathbb{F},\mathbb{P})$ predictable  projection of }(H_t).
$$

 Using all the above representations, we obtain:
\[
D^*_T:=\frac{d\mathbb{P}}{d \mathbb{Q}^*}|_{\mathcal{G}_T}=E_T(e_T\delta_T)^{-1}=\mathcal E_T\left (\int_0^\cdot H_t-(h_t+a_t)dX_t\right)\in V_T(\mathbb{Q}^*)
\]
that proves the statement (i).
\end{itemize}
We now assume Hypothesis \ref{H2} and (C) and prove the last statements.
\begin{itemize}
 \item[(iii)]  It is sufficient to prove the relation $R=(D^*)^{-1}$ $\mathbb{P}$ a.s. Let us denote $\beta:= (h-H+a)$ so that $D^*_t:\frac{d\mathbb{P}}{d \mathbb{Q}^*}|_{\mathcal{G}_t}=\mathcal E_t\left (\int_0^\cdot \beta_sdX_s\right)$. Given   $ \mathbb{Q}^*\ll\mathbb{P}$ on $\mathcal{G}_T$,  we may apply a Girsanov transformation and obtain that $\hat M_t:=m^{\mathbb{F},\mathbb{Q}^*}_t-\int_0^t\beta_s d\langle m^{\mathbb{F},\mathbb{Q}^*}\rangle_t$ is a $(\mathbb{G}, \mathbb{P})$  local martingale. Given the assumption  (C),  $\langle m^{\mathbb{F},\mathbb{Q}^*}\rangle=\langle \hat M\rangle$, that is, the quadratic variation.  Consequently, we obtain the following $(\mathbb{G}, \mathbb{P})$ representation of  $X$:
\begin{equation}
X_t=X_0+\int_0^t\beta_s d\langle \hat M\rangle_t+\hat M_t.
\end{equation}
By analysing Hypothesis \ref{H2}, we see that the uniqueness of a Doob Meyer decomposition imposes the following relations $\mathbb{P}$ a.e.: $\hat M=M$, leading to $\langle \hat M\rangle =\langle M\rangle=\langle X\rangle$ and also $\beta1\!\!1_{\{d\langle M\rangle >0\}} =\alpha1\!\!1_{\{d\langle M\rangle >0\}}$. This proves that indeed $D^*_t=\mathcal E_t\left (\int_0^\cdot \alpha_tdX_t\right)=\mathcal E_{t\wedge T^0}\left (\int_0^\cdot \alpha_tdX_t\right)$ holds $\mathbb{P}$-a.s.. The equality also holds  $\mathbb{Q}$-a.s  because the possibly additional null sets of $\mathbb{P}$ are included in $\{D_T^*=0\}=\{E_T=0\}=\{T^0<T\}$.
It follows that indeed $R=(D^*)^{-1}$ $\mathbb{P}$ a.s. and is a true $(\mathbb{G},\mathbb{P})$ martingale, if and only if $\mathbb{Q}^*(T^0<T)=0$, that is $\mathbb{Q}^*\sim\mathbb{P}$.
\end{itemize}

 \hfill $\Box$ \vskip 5 pt

If  $\mathbb{P}$ and $\mathbb{Q}^*$ are not equivalent on $\mathcal{G}_T$, it may still be the case that $\mathcal M_T(\mathbb{G},\mathbb{P})\neq \emptyset$. Indeed, one cannot exclude a priori the existence of a probability $\widetilde \mathbb{Q}$ satisfying $\widetilde \mathbb{Q}\in\mathcal I(\mathbb{F},\mathbb{G},\mathbb{P})$ ( by Theorem \ref{ThmGenH} (ii) this is indeed equivalent to $\mathcal M_T(\mathbb{G},\mathbb{P})\neq \emptyset$).  In such a case the measure $\mathbb{Q}^m$ is not a probability, but there exist equivalent local martingale measures for the price process in $\mathbb{G}$.

In the next theorem we emphasise that quite strong relations between $\mathbb{Q}^\mathcal M$ and $\mathbb{Q}^\mathcal S$ exist whenever the reference measure $\mathbb{P}$ is a local martingale measure for the price process $X$, within the smaller filtration $\mathbb{F}$ (i.e., this situation is obtained for $(a_t)\equiv 0$ in (\ref{XdecompF}), or,  equivalently $\mathbb{P}=\mathbb{P}^*$ on $\mathcal{F}_\infty$, where $\mathbb{P}^*$ is as in Remark \ref{remPstar}).
\begin{Theorem}\label{ThmH}
 We assume Hypotheses \ref{H1} and \ref{H2} together with conditions (C) and $\mathbb{F}$-(PRP).  In addition, we consider $X$ is a $(\mathbb{F},\mathbb{P})$ local martingale (in the representation (\ref{XdecompF}), the process $(a_t)$ is taken to be null). Suppose  $\mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})\neq \emptyset$ and let $\mathbb{Q}^*$ be such as in Theorem \ref{ThmQstar}.
 Then, the following hold:
\begin{itemize}
\item[(a)] If   $\mathbb{P}$ and $\mathbb{Q}^*$ are not equivalent on $\mathcal{G}_T$,    then $\mathbb{Q}^{\mathcal M}\notin \mathcal M_T(\mathbb{G},\mathbb{P})$ and $\mathbb{Q}^{\mathcal S}\notin \mathcal{S}_T(\mathbb{G},\mathbb{P})$. \\
\item[(b)] If $\mathbb{P}\sim\mathbb{Q}^*$  on $\mathcal{G}_T$  and $\frac{d\mathbb{P}}{d\mathbb{Q}^*}|_{ \mathcal{G}_T}\in L^2(\Omega,\mathcal{G},\mathbb{Q}^*)$, then $\mathbb{Q}^{\mathcal S}\in  \mathcal{S}_T(\mathbb{G},\mathbb{P})$ and $\mathbb{Q}^{\mathcal M}\in\mathcal M_T(\mathbb{G},\mathbb{P})$.
\end{itemize}
\end{Theorem}

The proof of this theorem is postponed at the end of this section.

\bigskip

In order to prove Theorem \ref{ThmH},  we take as  reference a probability space $(\Omega, \mathcal{G}, \mathbb{Q}^*)$, endowed with right-continuous filtrations $\mathbb{F}^0\subset \mathbb{G}^0$ satisfying (\ref{fg0}). We suppose that (H) hypothesis holds under $\mathbb{Q}^*$ and our aim is to recover $\mathbb{P}$ form $\mathbb{Q}^*$ with the help of a Girsanov transformation. The key properties of our initial model under $\mathbb{P}$, in particular  $R$ and $R^{(+)}$  being or not strict local martingales, will appear as resulting from whether $\mathbb{P}$ is actually equivalent to $\mathbb{Q}^*$ or only absolutely continuous (see Proposition \ref{ReprXfromQ} below).  The filtrations $\mathbb{F}$ and $\mathbb{G}$ will be defined as  completions with the null sets of $\mathbb{P}$, as before.

The assumptions that are going to be used below are the following:
\begin{itemize}
 \item[(A1)] The filtration $ \mathbb{F}^0$ is the natural, $\mathbb{Q}^*$ augmented filtration of a $\mathbb{Q}^*$-Brownian motion $B$.
 \item[(A2)] The filtrations $\mathbb{F}^0$ and $\mathbb{G}^0$ satisfy the condition (\ref{fg0})  and furthermore $ \mathbb{F}^0\overset{\mathbb{Q}^*}{\hookrightarrow} \mathbb{G}^0$.
\item[(A3)] There is a nonnegative $(\mathbb{G}^0,\mathbb{Q}^*)$ martingale having the representation:
\begin{equation}\label{defD}
D^*_t=1+\int_0^t G_sdB_s,
\end{equation}
where $G$ is a $\mathbb{G}^0$ predictable process satisfying  $\mathbb{E}_{\mathbb{Q}^*}[G_t|\mathcal{F}^0_t]=0$ for all $t\in[0,T]$.
\end{itemize}
We now define the probability $\mathbb{P}$ as:
 \begin{equation}\label{dPdQ}
 \frac{d\mathbb{P}}{d\mathbb{Q}^*}\Big|_{\mathcal{G}^0_T}=D^*_{T},
 \end{equation}
where $D^*$ is the martingale defined in (\ref{defD}).

 \begin{Proposition}\label{ReprXfromQ} We consider the probability space $(\Omega,\mathcal{G},\mathbb{Q}^*)$ with the assumptions (A1)-(A3) and let $\mathbb{P}$ be defined as in (\ref{dPdQ}). The filtration $\mathbb{F}$ is defined as the usual augmentation of  $\mathbb{F}^0$ with the $\mathbb{P}$ null sets and similarly, the filtration $\mathbb{G}$ is defined as the usual augmentation of  $\mathbb{G}^0$ with the  $\mathbb{P}$ null sets.

 Let $X$ be  a $(\mathbb{F},\mathbb{Q}^*)$ local martingale.  The following hold:
 \begin{itemize}
 \item[(a)] The  process   $X$ is a $(\mathbb{F},\mathbb{P})$ local martingale and it is a $(\mathbb{G},\mathbb{P})$ semimartingale with the Doob Meyer decomposition:
 \begin{equation}\label{Xrepr}
 X_t=X_0+M_t + \int_0^t \alpha_s d\langle M\rangle_s\quad t\in[0,T]
 \end{equation}
 for a $(\mathbb{G},\mathbb{P})$ local martingale $M$ and a $\mathbb{G}$-predictable process $\alpha$ having a null $(\mathbb{F},\mathbb{P})$ optional projection. Thus, $X$ satisfies the Hypothesis \ref{H2} from Section \ref{SectionSetting} under $\mathbb{P}$, for $t\in[0,T]$.
 \end{itemize}
 Define $R:= \mathcal E\left(-\int_0^\cdot \alpha_ud M_u\right)$ and $R^{(+)}:= \mathcal E\left(-\int_0^\cdot \alpha^+_ud M_u\right)$.
\begin{itemize}
 \item[(b)] If $\mathbb{P}\sim\mathbb{Q}^*$ on $G_T$ and $\mathbb{E}_{\mathbb{Q}^*}[(D^*_T)^2]<\infty$, then, both $R$ and $R^{(+)}$ are true  $(\mathbb{G}_T,\mathbb{P})$-martingales.
 \item[(c)]  If $\mathbb{P}\ll\mathbb{Q}^*$  but $\mathbb{P}$ is not equivalent to $\mathbb{Q}^*$ on $G_T$, then both $R$ and $R^{(+)}$ are strict  $(\mathbb{G}_T,\mathbb{P})$-local martingales.
 \end{itemize}
 \end{Proposition}

 \noindent {\it Proof. $\, $} For simplicity, all processes are considered as stopped at $T$.\\
\textit{Proof of claim in (a).}  The process $X$ being an  $(\mathbb{F},\mathbb{Q}^*)$ local martingale by assumption, there is an $\mathbb{F}$-predictable process $F$  such that
\begin{equation}\label{XFQ}
X=X_0+\int F dB.
\end{equation}
We shall prove that the assumption (A4) implies:
  \begin{equation}\label{LF}
 \frac{d\mathbb{P}}{d\mathbb{Q}^*}\Big|_{\mathcal{F}_T}=1
 \end{equation}
which then implies that all  $(\mathbb{F},\mathbb{Q}^*)$-local martingales remain  $(\mathbb{F},\mathbb{P})$ local martingales, in particular $X$.

The density process $(\frac{d\mathbb{P}}{d\mathbb{Q}^*}|_{\mathcal{F}_t})_{t\in[0,T]}$ is given by the $(\mathbb{F},\mathbb{Q}^*)$ optional projection of  the process $D^*$. The Brownian motion $B$ being an $(\mathbb{F},\mathbb{Q}^*)$ and a $(\mathbb{G},\mathbb{Q}^*)$ local martingale (by assumption (A2)), we obtain (see \cite{bremaudyor}):
 \[
\frac{d\mathbb{P}}{d\mathbb{Q}^*}|_{\mathcal{F}_t}=\;^{o,\mathbb{Q}^*}D^*_t=1+\int_0^t \;^{o,\mathbb{Q}^*}G_sdB_s.
\]
From assumption (A3), $ \;^{o,\mathbb{Q}^*}G\equiv 0$ and we obtain  $ \;^{o,\mathbb{Q}^*}D^*\equiv 1$, hence (\ref{LF}) holds true. Consequently,  $X$  is also a  $(\mathbb{F},\mathbb{P})$-local martingale (by using Girsanov's theorem in the filtration $\mathbb{F}$).

In the larger filtration $\mathbb{G}$ however, using the Lenglart's extension of the Girsanov's theorem, we obtain that the following is a martingale under $\mathbb{P}$:
\begin{equation*}
M:=X-X_0-\int_0^t\frac{d\langle X,D^*\rangle_s}{D^*_s}= X-X_0-\int_0^{t}\frac{G_sF_s }{D^*_s}ds.
\end{equation*}
Because from (\ref{XFQ}) $\langle X\rangle =\int F^2ds$, we can write
\begin{equation*}
X=X_0+M+\int_0^t \alpha_sd\langle M,M\rangle_s
\end{equation*}
with
\begin{equation}\label{alpha}
\alpha_t:=1\!\!1_{F_t\neq 0} 1\!\!1_{D^*_t\neq 0}\frac{G_t}{D^*_tF_t}.
\end{equation}  For all $t\geq 0$:
\begin{align*}
\mathbb{E}_\mathbb{P}[\alpha_t|\mathcal{F}_t]&=1\!\!1_{F_t\neq 0}\mathbb{E}_\mathbb{P}\left [ 1\!\!1_{D^*_t\neq 0}\frac{G_t}{D^*_t}|\mathcal{F}_t\right ] \frac{1}{F_t}=1\!\!1_{F_t\neq 0}\frac{\mathbb{E}_{\mathbb{Q}^*}\left[\frac{d\mathbb{P}}{d\mathbb{Q}^*}\frac{G_t}{ D^*_t}|\mathcal{F}_t\right]}{\mathbb{E}_{\mathbb{Q}^*}\left[\frac{d\mathbb{P}}{d\mathbb{Q}^*}|\mathcal{F}_t\right]F_t} \\
&=1\!\!1_{F_t\neq 0} \frac{\mathbb{E}_{\mathbb{Q}^*}\left[G_t|\mathcal{F}_t\right]}{\mathbb{E}_{\mathbb{Q}^*}\left[D^*_t|\mathcal{F}_t\right]F_t}=0,
\end{align*}
thus, $\alpha$ has a null $(\mathbb{F},\mathbb{P})$ optional projection. The claim in (a) is thus proved.
\bigskip\\
\textit{Proof of claims (b) and (c)}.
We write $D^*$ as product of orthogonal  $(\mathbb{G},\mathbb{Q}^*)$ local martingales in two different ways. First, we write $D^*=LZ$ with
 \begin{align*}
d L_t &= L_t \frac{G_t}{D^*_t}1\!\!1_{\{F_t\neq 0\}\cap\{D^*_t>0\}}dB_t\\
&= L_t\alpha_tF_tdB_t\\
L_0&=1
 \end{align*}
 and
  \begin{align*}
d  Z_t&=Z_t\frac{G_t}{D^*_t}1\!\!1_{\{F_t= 0\}\cap \{D^*_t>0\}}dB_t\\
Z_0&=1
  \end{align*}
Second, we write $D^*=L^{(+)} Z^{(-)}$ with
 \begin{align*}
d L^{(+)}_t &= L^{(+)}_t \frac{(G_t)^+}{D^*_t}1\!\!1_{\{F_t\neq 0\}\cap\{D^*_t>0\}}dB_t\\
&= L^{(+)}_t (\alpha_t)^+F_tdB_t\\
L^{(+)}_0 &=1
 \end{align*} and
 \begin{align*}
d  Z^{(-)}_t&=Z^{(-)}_t \frac{1\!\!1_{\{D^*_t>0\}} }{D^*_t}\left((G_t)^+1\!\!1_{\{F_t= 0\}}- (G_t)^- \right)dB_t\\
Z^{(-)}_0 &=1.
  \end{align*}
  The following relations can be easily verified (using Ito's formula and the expression of the processes $X$ and  $\alpha$ in (\ref{Xrepr}) resp (\ref{alpha})):
 \begin{align*}
R_t &=\frac{1}{L_t}\\
R^{(+)}&=\frac{1}{L^{(+)}_t}.
 \end{align*}
We denote:
 \begin{align}
\label{t0}  T_0:&=\inf\{t\geq 0:D^*_t= 0\}\wedge T\\
\label{ta}\tau:&=\inf\{t\geq 0:L_t= 0\}\wedge T\\
\label{ta+} \tau^{(+)}:&=\inf\{t\geq 0:L^{(+)}_t= 0\}\wedge T.
\end{align}

We notice that the processes $L$, $L^{(+)}$, $Z$ and $Z^{(-)}$ are  such  that the measure $d\langle L\rangle$  is singular with respect to the measure $d\langle Z\rangle$ and also  the measure $d\langle L^{(+)}\rangle$  is singular with respect to the measure $d\langle Z^{(-)}\rangle$.  These local martingales being continuous, their constancy intervals are determined by the constancy intervals of their sharp bracket; hence $L$ and $Z$ do not co-move and $L^{(-)}$ and $Z^{(+)}$ do not co-move.  Additionally the process $D^*$, a nonnegative martingale, is absorbed at the level 0, once this level attained, and consequently the processes $L$, $L^{(+)}$, $Z$ and $Z^{(-)}$  are also stopped at $T_0$. This shows that the following equalities hold $\mathbb{Q}^*$ a.s.:
\begin{align}\label{tauequiv}
\{\tau<T\}&=\{L_T=0\}=\{Z_T>0\}\cap\{T_0<T\};\\\label{tauplusequiv}
\{\tau^{(+)}<T\}&=\{L^{(+)}_T=0\}= \{Z^{(-)}_T>0\}\cap\{T_0<T\}.
 \end{align}

We know that $R_t =\frac{1}{L_t}$. Therefore:
\begin{align*}
\mathbb{E}_\mathbb{P}\left[R_T\right]&=\mathbb{E}_{\mathbb{Q}^*}\left[D^*_T\frac{1}{L_T}\right]=\mathbb{E}_{\mathbb{Q}^*}\left[D^*_T\frac{1}{L_T}1\!\!1_{L_T>0}\right]=\mathbb{E}_{\mathbb{Q}^*}\left[Z_T1\!\!1_{L_T>0}\right]\\
&=\mathbb{E}_{\mathbb{Q}*}[Z_T]-\mathbb{E}_{\mathbb{Q}^*}\left[Z_T1\!\!1_{L_T=0}\right]
\end{align*}
and from this we deduce that
\begin{itemize}
\item[-]  if   $\mathbb{Q}^*(\tau<T)>0$ we have $\mathbb{E}_\mathbb{P}\left[R_T\right]<1$ i.e., $R$ is a strict  $(\mathbb{G},\mathbb{P})$-local martingale. The strict inequality is justified using (\ref{tauequiv}).
 Indeed, these equalities imply that
 $$\mathbb{Q}^*(\tau<T)=\mathbb{Q}^*(L_T=0)>0$$ and  $\{L_T=0\}\subset \{Z_{T}>0\} $ which leads to $\mathbb{E}_\mathbb{Q}^*\left[Z_T1\!\!1_{L_T=0}\right]>0$, hence the result.
\item[-]  if   $\mathbb{Q}^*(\tau<T)=0$ and $Z$ a true $\mathbb{Q}^*$ martingale, we have $\mathbb{E}_\mathbb{P}\left[R_T\right]=1$ i.e., $R$ is a true  $(\mathbb{G},\mathbb{P})$-martingale.
\end{itemize}
Identical steps as above  for  $R =\frac{1}{L}$ can be used now for $R^{(+)} =\frac{1}{L^{(+)}}$. Indeed, the characterisation of $\tau^{(+)}$ in  (\ref{tauplusequiv}), can be used to prove the following:
\begin{itemize}
\item[-]  if   $\mathbb{Q}^*(\tau^{(+)}<T)>0$ we have $\mathbb{E}_\mathbb{P}\left[R^{(+)}_T\right]<1$ i.e., $R^{(+)}$ is a strict  $(\mathbb{G},\mathbb{P})$-local martingale.
\item[-]  if   $\mathbb{Q}^*(\tau^{(+)}<T)=0$ and $Z^{(-)}$ a true $\mathbb{Q}^*$ martingale, we have $\mathbb{E}_\mathbb{P}\left[R^{(+)}_T\right]=1$ i.e., $R^{(+)}$ is a true  $(\mathbb{G},\mathbb{P})$-martingale.
\end{itemize}
The claims (b) and (c) can now be proved using the above analysis and the fact that $\mathbb{Q}^*(\tau^{(+)}<T)>0$ if and only if $\mathbb{Q}^*(\tau<T)>0$. For more clarity, the poof of this result is stated as a separate result below (Lemma \ref{tau}).

In particular, for (b):  As $D^*$ is  by assumption a square integrable  $(\mathbb{G},\mathbb{Q}^*)$-martingale, using Proposition \ref{Y=UZ}, we deduce that, $L$, $L^{(+)}$, $Z$ and $Z^{(+)}$ are  strictly positive and uniformly integrable  $(\mathbb{G},\mathbb{Q}^*)$-martingales. We use Lemma \ref{tau} below together with the above analysis to conclude. For (c) we see that there is no need to have $Z$ and $Z^{(-)}$ be true $\mathbb{Q}^*$ martingales, so that no more constraints on the process $D^*$ are needed.
   \hfill $\Box$ \vskip 5 pt
\begin{Lemma}\label{tau} Recall the definitions of $T_0,$ $\tau$ and $\tau^{(+)}$ in equations ~(\ref{t0}),~(\ref{ta}) and ~(\ref{ta+}), respectively. The following are equivalent: (i) $\mathbb{Q}^*(T_0<T)>0$;  (ii) $\mathbb{Q}^*(\tau^{(+)}<T)>0$;  (iii) $\mathbb{Q}^*(\tau<T)>0$.
\end{Lemma}
\noindent {\it Proof. $\, $}  To begin with, we give an alternative characterisation of the sets $\{\tau<T\}$ and $\{\tau^{(+)}<T\}$, that uses the fact that  the martingale $L$ is constant on the set $\{(\omega,t): F_t(\omega)=0\}$; while $L^{(+)}$ is constant on the set $\{(\omega,t): F_t(\omega)=0,G_t(\omega)\leq 0\}$ (these properties can be easily seen from an inspection of their quadratic variation).
\begin{align*}
\{\tau<T\}&=\{T_0=\tau\}\cap\{T_0<T\}=\{F_{T_0}\neq 0\}\cap\{T_0<T\};\\
\{\tau^{(+)}<T\}&= \{T_0=\tau^{(+)}\}\cap\{T_0<T\}\\
&=\{F_{T_0}\neq 0\}\cap\{G_{T_0}>0\}\cap\{T_0<T\}.
 \end{align*}
 Therefore, denoting by $A:=(1\!\!1_{T_0<t})_{t\in[0,T]}$:
 \begin{align}\nonumber
 \mathbb{Q}^*(\tau^{(+)}<T)&=\mathbb{E}_{\mathbb{Q}^*}[1\!\!1_{F_{T_0}\neq 0}1\!\!1_{G_{T_0}>0}1\!\!1_{T_0<T}]=\mathbb{E}_{\mathbb{Q}^*}\left[\int_0^T1\!\!1_{F_t\neq 0}1\!\!1_{G_t>0}dA_t\right]\\\label{tauplus1}
 &=\mathbb{E}_{\mathbb{Q}^*}\left[\int_0^T1\!\!1_{F_t\neq 0}\mathbb{Q}^*(G_t>0|\mathcal{F}_t)dA^{o,\mathbb{Q}^*}_t\right]
 \end{align}
 and
 \begin{align*}
 \mathbb{Q}^*(\tau<T)&=\mathbb{E}_{\mathbb{Q}^*}[1\!\!1_{F_{T_0}\neq 0}1\!\!1_{T_0<T}]=\mathbb{E}_{\mathbb{Q}^*}\left[\int_0^T1\!\!1_{F_t\neq 0}dA_t\right]. \end{align*}
 We notice that the measure $dA_t$ does not charge the set $\{(\omega,t):G_t(\omega)= 0\}$, that is:
 $$
 dA_t=1\!\!1_{G_t\neq 0}dA_t.
 $$
 This property can be seen as following from  Dubins-Schwarz theorem, which allows to write
 \[
 T_0=\inf\left\{t:W_{\int_0^tG_s^2ds}=0\right\}
 \]
  where $W$ is a Brownian motion (alternatively,  the hitting time of 0 by $D^*$ cannot occur on the constancy intervals of the process $D^*$).

  Hence:
 \begin{align}\nonumber
 \mathbb{Q}^*(\tau<T)&=\mathbb{E}_{\mathbb{Q}^*}\left[\int_0^T1\!\!1_{F_t\neq 0}1\!\!1_{G_t\neq 0}dA_t\right]\\\label{tau1}
 &=\mathbb{E}_{\mathbb{Q}^*}\left[\int_0^T1\!\!1_{F_t\neq 0}\mathbb{Q}^*(G_t\neq0|\mathcal{F}_t)dA^{o,\mathbb{Q}^*}_t\right]
 \end{align}
We remark that because $\mathbb{E}_{\mathbb{Q}^*}[G_t|\mathcal{F}_t]=0$, we have that $ \mathbb{Q}^*(G_t<  0|\mathcal{F}_t)>0$ if and only if $ \mathbb{Q}^*(G_t>  0|\mathcal{F}_t)<0$. Therefore:
 \begin{equation}\label{probaeqsets}
  \mathbb{Q}^*(G_t\neq  0|\mathcal{F}_t)>0\text{ if and only if } \mathbb{Q}^*(G_t>  0|\mathcal{F}_t)>0.
 \end{equation}

From  (\ref{tauplus1}), we see that $\mathbb{Q}^*(\tau^{(+)}<T)>0$ if and only if the measure $dA^{o,\mathbb{Q}^*}_t$ charges a subset of $$\mathcal A^{(+)}:= \{(\omega,t)\in\Omega\times [0,T]|F_t(\omega)\neq 0; \mathbb{Q}^*(G_t>  0|\mathcal{F}_t)(\omega)>0 \}.$$
From  (\ref{tau1}), we see that $\mathbb{Q}^*(\tau<T)>0$ if and only if the measure $dA^{o,\mathbb{Q}^*}_t$ charges a subset of
$$\mathcal A:=\{(\omega,t)\in\Omega\times [0,T]|F_t(\omega)\neq 0; \mathbb{Q}^*(G_t\neq  0|\mathcal{F}_t)(\omega)>0 \}.$$

But because of (\ref{probaeqsets}) we have $\mathcal A^{(+)}=\mathcal A$. Thus, we have proved that $\mathbb{Q}^*(\tau^{(+)}<T)>0$ if and only if $\mathbb{Q}^*(\tau<T)>0$. Obviously, each one of these implies $\mathbb{Q}^*(T_0<T)>0$.
 \hfill $\Box$ \vskip 5 pt

\begin{rem}\label{rem}The results in Proposition \ref{ReprXfromQ} remain true if we replace the Brownian motion $B$ by the continuous martingale $X$ having the predictable representation property in a filtration $\widetilde \mathbb{F}$ under $\mathbb{Q}^*$. More exactly, the statements in Proposition \ref{ReprXfromQ} are true if we can replace the assumptions (A1), (A2), (A3) with the assumptions (A1'), (A2), (A3'), where
\begin{itemize}
\item[(A1')] There is a continuous martingale $X$ having the predictable representation property with respect to $(\mathbb{F}^0,\mathbb{Q}^*)$.
\item[(A3')] There is a nonnegative $( \mathbb{G}^0,\mathbb{Q}^*)$ martingale having the representation:
\begin{equation}\label{defDbis}
D^*_t=1+\int_0^t G_sdX_s,
\end{equation}
where $G$ is a $\mathbb{G}$ predictable process satisfying  $\mathbb{E}_{\mathbb{Q}^*}[G_t|\mathcal{F}^0_t]=0$ for all $t\in[0,T]$.
\end{itemize}
The proof is identical; we simply have to consider that $F\equiv 1$, where $F$ is the process appearing in the proof of Proposition \ref{ReprXfromQ}, see equation (\ref{XFQ}) and to replace $B$ by $X$ everywhere.
\end{rem}

\noindent {\it Proof. $\, $} (of Theorem \ref{ThmH})
 We consider  the probability space is $(\Omega,\mathcal{G},\mathbb{P})$. We assume Hypotheses \ref{H1} and \ref{H2} together with conditions (C) and $\mathbb{F}$-(PRP).  Additionally, we suppose that   $ \mathcal p_T(\mathbb{F},\mathbb{G},\mathbb{P})\neq \emptyset$. By Theorem \ref{ThmQstar}, there is also a probability  $\mathbb{Q}^*\in \mathcal P_T(\mathbb{F},\mathbb{G},\mathbb{P})$, such that  $D^*_t:=\mathbb{E}_{\mathbb{Q}^*}[\frac{d\mathbb{P}}{d\mathbb{Q}^*}| \mathcal{G}_{t\wedge T}]$  has the representation introduced in (\ref{defD}). Using the Remark \ref{rem} and Proposition \ref{ReprXfromQ}, we deduce that:
\begin{itemize}
\item[-] If $\mathbb{P}\sim\mathbb{Q}^*$ and $D^*_T\in L^2(\Omega,\mathcal{F},\mathbb{Q}^*)$, then, both $R$ and $R^{(+)}$ are true  $(\mathbb{G},\mathbb{P})$ martingales. This proves (b).
\item[-] If $\mathbb{P}\ll\mathbb{Q}^*$ but not equivalent,  then, both $R$ and $R^{(+)}$ are strict  $(\mathbb{G},\mathbb{P})$ local martingales. This proves (a).
\end{itemize}
\hfill $\Box$ \vskip 5 pt
\section{Conclusions}
This paper represents a first step in analysing the potential of arbitrage arising from inside information, in presence of short sales prohibitions. Our analysis may be extended in several directions.

The concept of minimal supermartingale measure that we have introduced is independent of the two filtrations setting with inside information. Hence, it may be used for studying the effect of short-sales restrictions in a general market model, without inside information. An important such direction is the study of  local risk minimisation for investors in presence of short sales restrictions. For this purpose, the connections between minimal martingale measures and minimal supermartingale measures may be interesting to further analyse.

The two filtration setting that we used for the analysis of the inside information is very general; the study of particular enlargements of filtrations, or filtering models can possibly lead to additional, more specific results. We also expect analogous results to those in Section 4 to hold for multidimensional assets, when using the conditions (C) and $\mathbb{F}$-(PRP). The framework developed for Proposition \ref{ReprXfromQ} is voluntarily more general than is needed for proving Theorem \ref{ThmQstar}, the filtration $\mathbb{F}$ being larger that the natural filtration of the process $X$. This allows for an easier generalisation to a multidimensional setting.

Finally, we only were focused on arbitrages in the form of free lunches with vanishing risks; other definitions for arbitrages may be considered and  Hypothesis \ref{H1} may be replaced by a weaker condition,  the existence of a supermartingale measure for the common investors.

redpajama_set_name: RedPajamaArXiv


text:
Laws of South Pacific Island Nations
Republic of the Marshall Islands (RMI)
Laws of South Pacific Island Nations: Republic of the Marshall Islands (RMI)
Guide to legal materials for South Pacific Island Nations available at UH Manoa.
Federated States of Micronesia (FSM)
Archival and Special Collections
Court Structure
The Supreme Court is a superior constitutional court of record having appellate jurisdiction with final authority to adjudicate all cases and controversies properly brought before it. The Supreme Court consists of a Chief Justice and two Associates Justices. Historically, the Associate Justices are pro tem judges from other jurisdictions, e.g., the United States Ninth Circuit Court, the Republic of Palau, the Commonwealth of the Northern Mariana Islands, and Canada.
The High Court is a superior constitutional court of record having general jurisdiction over controversies of law and fact in the Marshall Islands. The High Court consists of a Chief Justice and an Associate Justice. The High Court has original jurisdiction over all cases properly filed with it, appellate jurisdiction over cases originally filed in subordinate courts, and, unless provided by law otherwise, jurisdiction to review the legality of any final decision of a government agency.
The District Court is a statutory court of record. It consists of a Presiding Judge and two Associate Judges. The District Court has original jurisdiction concurrently with the High Court: in all civil cases where the amount claimed or the value of the property involved does not exceed $10,000, except matters vested by the Constitution in the High Court, admiralty and maritime matters, and cases of adjudication of title to land or interest in land (other than the right to immediate possession); provided, however the District Court has jurisdiction to award alimony and support for children in divorce cases and for the children of unmarried parents, regardless of the $10,000 limitation, and to include in any such award land or an interest in land owned by any party to the case; provided, however, that this jurisdiction does not include jurisdiction to adjudicate the validity of a claim to ownership of the land or interest therein. The District Court also has original jurisdiction concurrently with the High Court in all criminal cases involving offenses against any law of the Republic for which the maximum penalty does not exceed a fine of $4,000 or imprisonment for a term of less than three years, or both. Finally, the District Court has appellate jurisdiction to review any decision of a Community Court.
Community Court
A Community Court is a statutory court of record for a local government area of which there are 24. Each Community Court consists of a Presiding Judge and such number of Associate Judges, if any, as may be appointed. A Community Court has original jurisdiction concurrently with the High Court and the District Court in all civil cases where the amount claimed or the value of the property involved does not exceed $200, except matters vested by the Constitution in the High Court, admiralty and maritime matters, and cases of adjudication of title to land or interest in land (other than the right to immediate possession). A Community Court also has original jurisdiction concurrently with the High Court and the District Court in all criminal cases involving offenses against any law of the Republic for which the maximum penalty does not exceed a fine of $400 or imprisonment for a term not exceeding six months, or both.
Traditional Rights Court
The Traditional Rights Court ("TRC") is a constitutional court of record consisting of three or more judges selected to include a fair representation of all classes of land rights: Iroijlaplap (high chief); where applicable, Iroijedrik (lower chief); Alap (head of commoner/worker clan); and Dri Jerbal (commoner/worker). The jurisdiction of the TRC is limited to questions relating to titles to land rights or other legal interests depending wholly or partly on customary law and traditional practices. The jurisdiction of the TRC may be invoked as of right upon application by a party to a pending judicial proceeding, if the court in which the proceeding is pending certifies that a substantial question has arisen within the jurisdiction of the TRC. Decisions of the TRC are to be given substantial weight, but are not binding unless the certifying court concludes that justice so requires. The Supreme Court has held that this means the certifying court is to review and adopt the decision of the TRC unless that decision is clearly erroneous or contrary to law. See Abija v. Bwijmaron, 2 MILR 6, 15 (RMI Sup. Ct. 1994).
Flag of the Republic of the Marshall Islands (RMI). The deep blue background represents the Pacific Ocean. The white and orange bands represent the Ratak (Sunrise) and Ralik (Sunset) chains, respectively. The customary symbolism of orange as the color of bravery and white as the color of peace are also recognized. The star represents the cross of Christianity, with each of the 24 points signifying a municipal district of the RMI. The four main points represent the major centers of Majuro, Ebeye, Jaluit and Wotje. From http://www.rmiembassyus.org/Government.htm#constit.
The RMI legal system is a mixed legal system of US and English common law, customary law, and local statutes.
Pacific Islands Legal Information Institute
There are more links to RMI laws on this site.
Nitijela - Legislative Body
RMI - Nitijela (Parliament) Rules
The Nitijela is the legislative body of the Marshall Islands.
RMI - Constitution (2012)
RMI - Revised Code (2012)
2012 Revision of the Marshall Islands Revised Code
2004 Revised Code of the Republic of the Marshall Islands
Law Library has this in print: KVS1.9 2004
RMI - Court Rules (All)
Law Library has Traditional Rights Court Rules in print: KNQ1056 .F8 2010
RMI - Cases (Supreme Court)
RMI - Cases (High Court)
RMI - Cases (Traditional Rights Court)
RMI - Cases (MILR v. 1)
Marshall Islands Law Reports (vol. 1)
The 2004 revision of Volume 1 incorporates decisions included in the original MILR Vol. 1 and decisions issued prior to mid-1993 that were not located for the MILR's publication.
Cases 1982-1993
Law Library has this in print: KVS2.2.A2 M37
Volume 2 incorporates decisions issued after mid-1993 publication through mid-2004.
This is an Interim Report. Marshall Islands Supreme Court decisions for 2005 through 2007. Plans are set to close this volume at the end of 2014.
Cases 2005-
RMI - US Compact of Free Association (RMI Site)
When the Compact of Free Association first entered into force, in 1986, it marked the beginning of a new relationship between the RMI and US. No longer were the Marshall Islands a "United Nation's Strategic Trust Territory" for the Compact granted the RMI sovereignty in domestic and foreign affairs in return for granting the United States defense rights in the islands.
RMI - US Compact of Free Association (US Site)
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Next: About >>
Last Updated: Mar 16, 2017 12:28 PM
URL: https://law-hawaii.libguides.com/pacificislands
Subjects: Pacific Islands - Law
Tags: american samoa, federated states of micronesia, marshall islands, pacific islands legal research, palau

redpajama_set_name: RedPajamaCommonCrawl


The 2016-17 season was a roller coaster ride for the Philadelphia Eagles' franchise quarterback Carson Wentz. In his rookie season, Wentz threw for 3,782 yards with 16 touchdowns and 14 interceptions. That's an average season for a rookie quarterback in the NFL but with what Wentz was working with, it was pretty good.
Wentz got out to a stellar start leading the Eagles' to a 3-0 record to start the season off. Unfortunately for the Eagles' and their rookie quarterback, offensive tackle Lane Johnson was suspended for 10 games for violating the NFL's substance abuse policy. Once that happened, the Eagles fell back down to earth along with Wentz. After starting 3-0, the Eagles managed to lose 9 of their last 13 games finishing with a 7-9 record.
This offseason the Eagles' knew it was important to get Wentz some weapons and wide receivers that he could fully depend on. The Eagles were able to land the top free agent wide receiver in Alshon Jeffery along with a deep threat in Torrey Smith. With the additions of Jeffery and Smith, Eagles' wide receiver Jordan Matthews will now be able to slide into his role, which is a number 2 receiver at best.
In the 2017 NFL Draft the Eagles' were able to land Mack Hollins, wide receiver from North Carolina and also Shelton Gibson, wide receiver from West Virginia. Hollins accounted for 1,667 yards and 20 touchdowns while at North Carolina. Hollins is known as a deep threat that can stretch the field. Gibson had over 1,850 yards receiving while finding the end zone 17 times while at West Virginia.
The one thing that the Eagles did not address was the running back position. Even though they drafted Donnel Pumphrey, he is not a 3 down back that can carry the team. The Eagles' look to go into the 2017-18 season using the running back by committee method.
Carson Wentz can flat out play. There is no denying that and if you think otherwise, you don't know football. He has the arm, accuracy, can make all the throws and most importantly he's smart. The only thing Wentz needs to work on is when the pocket breaks down, him panicking and throwing off his back foot, even though that rarely happens.
Despite not fully addressing the running back position, the Eagles' have surrounded Wentz with weapons that were not there last season. Jeffery is considered to be a top wide receiver in the league if he can stay healthy. Smith still has the speed to take the top off of a defense and open up the field for guys like Matthews and Jeffery.
They say that the sophomore slump is expected in most quarterbacks. Even the elite quarterbacks have gone through this slump. It's hard to believe that Wentz will be a victim of this. He is surrounded by talent that he wasn't blessed with last season. Wentz will have a full off season under his belt compared to only having 8 days to prepare to start against the Cleveland Browns. Wentz is set up to have a breakout season in only his second season. Will he avoid the sophomore slump? I believe he does.


Thirty-one shots on goal were not enough for the Lightning to get past Boston's Tim Thomas.
The Bruins would open up the scoring just a minute and nine seconds into the game as David Krejci would go backhand on Dwayne Roloson off of a pass by Milian Lucic. Halfway through the period Bruins Gregory Campbell and Lightning's Simon Gagne would sit for coincidental roughing penalties; but the game would hit the first intermission in Tampa with Boston leading by one.
Second period would see each team play with the man advantage twice; but neither goalie would give up a goal in the period.
The third period would see the Bruins increase their lead to two when Andrew Ference would tip-in a shot from Michael Ryder just over eight minutes into the period.
Are the Atlanta Thrashers on the verge of moving to Winnipeg?
Detroit Red Wings prospect Ilari Filppula has signed a contract with Jokerit of the SM-liiga after fulfilling a one-year contract with Detroit (played entire season with the Grand Rapids Griffins AHL).
Speaking of Finns and hockey Minnesota Wild (and Finnish National Team) captain Mikko Koivu is spending his time celebrating World Gold a little different that usual.
Bob Boughner resigns as Assistant Coach for the Columbus Blue Jackets.
Fredrik Modin had decided to retire at age 36; Modin is one of only 24 members of the Triple Gold Club.
USA Hockey had an increase of nearly 10,000 registered players 8 and under players in 2010-11 over 2009-10.
Three Young Goalies on Similar Quest for Stardom?


text:
Ready, Set, Script!
Automate Repeatable Processes Without Coding
More than ever, auditors are taking on the role of programmers; writing scripts to run automated analysis and tests on a regular basis for the organization. Luckily, tools like IDEA offer a graphical user interface (GUI) and built-in compiler to design scripts while reducing or eliminating the need for manually typing code. IDEAScripts work much like Microsoft® Excel macros, but with greater speed and flexibility.
This article provides some tips for creating IDEAScripts, examples of how fellow IDEA users are creating their own to meet audit and business goals, and information about available resources (including access to free pre-written IDEAScripts).
What is an IDEAScript?
IDEAScript is a programming language, similar to Visual Basic for Applications (VBA). Like other VBA languages, IDEAScript can only be used inside the IDEA application or an application developed in a higher programming language like Visual Basic, C++ or .NET. Simple routines that users run on a regular basis can be scripted without any VBA knowledge. There are three easy ways to create an IDEAScript:
Use the Recorder
The step-by-step process begins with clicking once on the "Record Macro" button, which activates the recorder.
Once you've finished the steps you want to make repeatable, click once again on the "Record Macro" button to stop the recording. All of the code for the recorded steps will appear in the IDEAScript editor as a new script. The editor will be live to make adjustments and refinements as needed.
Copy from History
First, in the database library, select a file to repeat or use as a template. In the Properties window of the selected file, select the History view.
In the History view, click on the expand button to view all actions related to the database and any parent databases that were used to create it.
You can then right-click on the desired IDEAScript and on the menu select the second option "Copy the IDEAScript for the selected task(s)".
This will copy the selected script into the "Editor". Then click on "Save As" to save the script.
Use Project Overview
Go to the Home tab and on the left side, find "Project Overview." Select the routines to script within the Project Overview workflow.
IDEA will copy the VBA to a new script. After selecting the steps you want to repeat, click "Finish." When you exit the Project Overview screen, IDEA will automatically write the script. Accept the warning notice, then bind to the Ribbon to run the script.
To bind the macro to the IDEA Ribbon, follow these steps:
On the Macros tab, in the Macro group, click Bind to Ribbon. The IDEA Options dialog box appears and the Customize Ribbon tab is displayed.
Click New….The New dialog box appears.
Optionally, from the Symbol box, select a symbol to associate with the macro.
Click the Browse button adjacent to the File name field.
Navigate to and select the required file.
Click Open. The file name and path is added to the File name field and the file name is added to the Command name field.
In the Command name field, enter the name you want to appear on the IDEA Ribbon. The default is the file name with the .iss extension.
Click OK. The macro is added to the Choose command list in the Customize Ribbon tab in the IDEA Options dialog box.
From the Choose command list, select the macro you added.
In the Customize the Ribbon list, select a custom group to which you want to add the macro. If a custom group has not been created, you must add a custom group to the IDEA Ribbon.
Click Add >> The macro name appears under the selected group in the Customize the Ribbon list.
The Script Editor will automatically launch, simply put the "Play" button, which will launch the IDEAScript.
IDEAScripts in Action
IDEAScripts can be customized to repeat just about any task or process. Here are just a few examples of how IDEA users are using scripts to verify data, automate complex processes and meet regulatory requirements:
Government: Verify Data & Identify Exceptions
A federal agency with a staff of approximately 100 people tasked with processing detailed information about seized assets into the division's investigative database system and keeping the information accurate throughout the entire forfeiture lifecycle. Detailed information must be manually recorded and maintained in two non-integrated systems. It was impossible to analyze 100% of the data to search for errors and exceptions, and the entry process required more than 30 staff to examine up to 25,000 entries each year.
A custom IDEAScript was developed to import raw data from the asset tracking system and compare it to data stored in the separate database. The script compares data from 74 fields to produce exceptions in minutes. The IDEAScript streamlined comparisons between the two disparate systems to verify all data entries and search for exceptions. The new process saved hundreds of hours, and the agency was able to reallocate three quality control staff and 25 data entry personnel. More importantly, it significantly increased the accuracy of both systems and allowed the team to quickly identify and correct errors detected.
Energy: Automate a Verification Process
Stream, one of the largest direct-selling companies globally and a leading provider of energy, wanted to check customer billing and commission information for accuracy, and create an automated workflow to perform checks regularly.
IDEA is used to check more than 100,000 monthly commission checks, each containing at least 25 data points that must be verified for accuracy. Files from the commission engine are imported into IDEA for analysis. Using the Join feature in IDEA, the commission payment is compared to the invoice generating commission. Qualification rules are programmed into IDEA utilizing @functions and IDEAScripts to ensure adherence.
IDEA is also used to test for duplicate checks, recalculation of Associate point requirements to ensure they meet eligibility requirements, and accuracy of level payouts. Field statistics assist in completing monthly reporting of verification activities. The automated process helps Stream perform approximately millions of data points for commission checks in minutes. Using exception management techniques, the company is able to proactively identify missed commissions, identify defects in the system that require correction and ensure accuracy when calculating commission payouts.
Each customer bill includes up to 30 data points, and with more than 450,000 customers nationwide, the invoice review process validates hundreds of millions of data points. The company also uses IDEA to analyze hundreds of millions of customer billing records for correctness and completeness. An IDEAScript is used to test the accuracy of up to 30 data points on each customer invoice to be sure inputs are accurate. IDEA is also used to look for instances of high, low or negative usage.
Gaming: Track the Status of Chips
IDEAScripts can be used to improve just about any process. In the gaming industry, computerized slot machines use electronic chips, which are regulated and inspected by the state. Some machines have multiple chips, all of which management must be able to track at any given time. A GLI listing is used to reference whether a chip is approved, revoked or not used. The gaming organization can be fined for using chips with a revoked status.
An IDEA user-developed an IDEAScript to run the GLI listing against the Slot Master file to append every chip's status, approved, revoked, not used, or blank where there may be a chip name discrepancy. In one instance of results, the Slot Machine Management Team found a previously unidentified revoked chip and proactively reported it to the State. The State was unaware of the issue until notified, creating goodwill for the company with the regulators.
Access Free IDEAScripts & Resources
The IDEAScript Vault has over 80 IDEAScripts available to any supported user. They are intended as samples and instructional materials, so you should be sure to test them carefully before using them in a production environment. But they can be extremely useful, particularly if you just need to perform something complex on an IDEA database on a one-time basis.
There are two ways you can get to the IDEAScript Vault, either from inside IDEA or by going straight to the CaseWare Analytics Support Portal (Passport). Here's how to get there from inside IDEA.
Click the "File" tab, then click the "IDEAScript Vault" tile.
When writing or working with IDEAScripts, you don't have to go it alone. The 20-minute rule still applies where all supported IDEA users are encouraged to call or email the Help Desk for assistance:
1-888-641-2800 x4
Hours of Operation: 7:00 AM – 7:00 PM (CST) M-F
Custom IDEAScript Development
Our Solutions Development experts can help you automate virtually any task to repeat analysis and testing on new data, or perform the same analytics across different systems and locations. Using Object Linking and Embedding (OLE) technology facilitated by IDEA, we can also perform calculations within text or spreadsheet applications. They can also assist with integration of non-IDEA applications. Contact [email protected] for more information.
Best Practices , CaseWare IDEA
What a Good FCPA Audit Looks Like On A Global Scale
Apr 20 Enforcing anti-corruption policies among employees and third parties is one of the foremost concerns for compliance departments, but developing a robust process...
Explore the IDEAScript Vault
Oct 20 The IDEAScript Vault has over 80 IDEAScripts available to any supported user. They are intended as samples and instructional materials, so you should be sure to...
Jan 01 Retailers have many of the standard ledgers, purchasing and payroll can be key concerns. However, the main business areas are sales and inventories. Retailers o...


text:
import Users from '../users/collection';
import LWevents from './collection';

const membersActions = [
  'events.new.own',
  'events.view.own',
];
Users.groups.members.can(membersActions);

const adminActions = [
  'events.new',
  'events.edit.all',
  'events.remove.all',
  'events.view.all',
];
Users.groups.admins.can(adminActions);

LWevents.checkAccess = async (user: DbUser|null, document: DbLWEvent, context: ResolverContext|null): Promise<boolean> => {
  if (!user || !document) return false;
  return Users.owns(user, document) ? Users.canDo(user, 'events.view.own') : Users.canDo(user, `events.view.all`)
};

redpajama_set_name: RedPajamaGithub


"Great community feeling and a great event..." stated Micha Berman, Director at Fireside, regarding the annual Summer BBQ. Families and seniors all enjoyed tasty food, fun activities, lots of laughter and good conversation. It was a privilege to volunteer and be with our neighbors on a beautiful evening in July! Here's to next year!


The first proper school in Emmer Green was built from brick and flint on the corner of Grove Road and School Lane in October 1877. Its name changed over the years, but the two-room school was finally closed in 1933 due to lack of support. Children from Emmer Green then had to attend schools in Caversham.
In the 1950s, the current Emmer Green Primary School was built in Grove Road. The building was extended in the 1980s, and had temporary classrooms added in the 1990s and 2010-11 to cope with demand for places.