text:
\section{Introduction}
In most articles dealing with stochastic optimization problems one major assumption is that the decision maker has full knowledge of the parameter of the underlying stochastic process. This does not seem to be a realistic assumption in many real world situations. Therefore, different multiple prior models were studied in the economic literature in the last years. Here, we want to mention \cite{DE} and \cite{ES}, and refer to \cite{CR} for an economic discussion and further references.

In this setting it is assumed that the decision maker deals with the uncertainty via a worst-case approach, that is, she optimizes her reward under the assumption that the ``market'' chooses the worst possible prior. This is a natural assumption, and we also want to pursue this approach.

A very important class of stochastic optimization problems is given by optimal stopping problems. These problems arise in many different fields, e.g., in pricing American-style options, in portfolio optimization, and in sequential statistics. Discrete time problems of optimal stopping in a multiple prior setting were first discussed in \cite{R} and analogous results to the classical ones were proved. In this setting a generalization of the classical best choice problem was treated in detail in \cite{CR2}. In continuous time the case of an underlying diffusion with uncertainty about the drift is of special interest. The general theory (including adjusted Hamilton-Jacobi-Bellman equations) is developed in \cite{CR}. Some explicit examples are given there, but no systematic way for finding an analytical solution is described. In \cite{A07} the case of monotonic reward functions for one-dimensional diffusion processes is considered. The restriction to monotonic reward functions simplifies the problem since only two different worst-case measures can arise.

Another class of stochastic optimization problems under uncertainty was dealt with in a series of papers starting with \cite{KW}: Portfolio optimization problems are considered under the assumption that the underlying asset price process may crash down at a certain (unknown) time point. The decision maker is again considered to be ambiguity averse in the sense that she tries to choose the best possible stopping policy out of the worst possible realizations of the crash date. See \cite{KS} for an overview on existing results.

The aim of this article is to treat optimal stopping problems under uncertainty for underlying one-dimensional diffusion processes. These kinds of problems are of special interest since they arise in many situations and often allow for an explicit solution.

The structure of this article is as follows:
In Section \ref{sec:drift} we first review some well-known facts about the solution of ordinary optimal stopping problems for an underlying Brownian motion. These problems can be solved graphically by characterizing the value function as the smallest concave majorant of the reward function. Then we treat the optimal stopping problem under ambiguity about the drift in a similar way: The result is that the value function can be characterized as the smallest majorant of the reward function in a two-parameter class of functions. The main tool is the use of generalized $r$-harmonic functions. After giving an example and characterizing the worst-case measure, we generalize the results to general one-dimensional diffusion processes.

In Section \ref{sec:crashes} we introduce the optimal stopping problem under ambiguity about crashes of the underlying process in the spirit of \cite{KS}. In this situation the optimal strategy can be described by two easy strategies: One pre-crash and one post-crash strategy. These strategies can be found as solutions of a one-dimensional Dynkin game and an ordinary optimal stopping problem, which can both be solved using standard methods. We want to point out that this model is a natural situation where Dynkin games arise and the theory developed in the last years can be used fruitfully.
As an explicit example we study the valuation of American call-options in the model with crashes. Here, the post-crash strategy is the well-known threshold-strategy in the standard Black-Scholes setting. The pre-crash strategy is of the same type, but the optimal threshold is lower.

\section{Optimal stopping under drift ambiguity}\label{sec:drift}

\subsection{Graphical solution of ordinary optimal stopping problems}\label{sec:graph_sol}
Problems of optimal stopping in continuous time are well-studied and the general theory is well-developed. Nonetheless, the explicit solution to such problems is often hard to find and the class of explicit examples is very limited. Most of them are generalizations of the following situation, that allows for an easy geometric solution:\\
Let $(W_t)_{t\geq 0}$ be a standard Brownian motion on a compact interval $[a,b]$ with absorbing boundary points $a$ and $b$. We consider the problem of optimal stopping given by the value function
\[v(x)=\sup_{\tau}\ensuremath{{\mathbb E}}_x(g(W_\tau)\mathds{1}_{\{\tau<\infty\}}),\mbox{~~~~$x\in[a,b]$,}\]
where the reward function $g:[a,b]\rightarrow[0,\infty)$ is continuous and the supremum is taken over all stopping times w.r.t. the natural filtration for $(W_t)_{t\geq0}$. Here and in the following, $\ensuremath{{\mathbb E}}_x$ denotes taking expectation for the process conditioned to start in $x$. In this case it is well-known that the value function $v$ can be characterized as the smallest concave majorant of $g$, see \cite{DyY}. This means that the problem of optimal stopping can be reduced to finding the smallest majorant of $g$ in an easy class of functions. For finding the smallest concave majorant of a function $g$ one only has to consider affine functions, i.e., for each fixed point $x\in[a,b]$ the value of the smallest concave majorant is given by
\[\inf\{h_{c,d}(x):c,d\in\ensuremath{{\mathbb R}}, h_{c,d}\geq g\},\]
where $h_{c,d}$ is an element of the two-parameter class of affine functions of the form $h_{c,d}(y)=cy+d$. This problem can be solved geometrically, see Figure \ref{fig:conc}. We want to remark that this problem is indeed a semi-infinite linear programming problem:
\begin{align*}
\min ! ~~~~~&cx+d\\
\mbox{s.t~~~~~}&cy+d\geq g(y)\mbox{~~~for all $y\in [a,b].$}
\end{align*}
This gives rise to an efficient method for solving these problems, which can be generalized in an appropriate way, see \cite{HS} for an analytical method and \cite{C} for a numerical point of view.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{concave_maj}
\caption{Graph of a function $g$ (black) and its smallest concave majorant (blue)}\label{fig:conc}
\end{center}
\end{figure}
The example described above is important both for theory and applications of optimal stopping since by studying it one can obtain an intuition for more complex situations such as finite time horizon problems and multidimensional driving processes, where numerical methods have to be used in most situations of interest.\\

The goal of this section is to handle optimal stopping problems with drift ambiguity for diffusion processes similarly to the ordinary case discussed above. This gives rise to an easy to handle geometric method for solving optimal stopping problems under drift ambiguity explicitly.

\subsection{Special Case: Brownian motion}\label{subsec:BM}
In the following we use the notation of \cite{CR}:
Let $(X_t)_{t\geq0}$  be a Brownian motion under the measure $Q$, fix $\kappa\geq0$ and denote by $\mathcal{P}^\kappa$ the set of all probability measures, that are equivalent to $Q$ with density process  of the form
\[\exp\left(\int_0^t\theta_sdX_s-1/2\int_0^t\theta^2_sds\right)\]
for a progressively measurable process $(\theta_t)_{t\geq0}$ with $|\theta_t|\leq \kappa$ for all $t\geq0$. We want to find the value function
\[v(x)=\sup_\tau\inf_{P\in\mathcal{P}^\kappa}\ensuremath{{\mathbb E}}_x^P(e^{-r\tau}g(X_\tau)\mathds{1}_{\{\tau<\infty\}})\]
for some fixed discounting rate $r>0$ and a measurable reward function $g:\ensuremath{{\mathbb R}}\rightarrow[0,\infty)$, where $\ensuremath{{\mathbb E}}_x^P$ means taking expectation under the measure $P$ when the process is started in $x$. Instead of taking affine functions as in Subsection \ref{sec:graph_sol} we construct another class of appropriate functions based on the minimal $r$-harmonic functions (introduced below) for the Brownian motion with drift $-\kappa$ resp. $\kappa$ as follows:\\
Denote the roots of the equation
\[1/2z^2-\kappa z-r=0\]
by $\alpha_1<0<\alpha_2$ and the roots of
\[1/2z^2+\kappa z-r=0\]
by $\beta_1<0<\beta_2$. Then $e^{\alpha_ix},i=1,2,$ are the minimal $r$-harmonic functions for a Brownian motion with drift $-\kappa$, and $e^{\beta_ix},i=1,2,$ the corresponding functions for a Brownian motion with drift $\kappa$. Note that $\beta_1\leq\alpha_1\leq0\leq\beta_2\leq\alpha_2$ and $\beta_1=-\alpha_2$ and $\beta_2=-\alpha_1$. For all $c\in \ensuremath{{\mathbb R}}$ define the functions $h_c:\ensuremath{{\mathbb R}}\rightarrow[0,\infty)$ via
\[h_c(x)=\begin{cases}
  \frac{\alpha_2}{\alpha_2-\alpha_1}e^{\alpha_1(x-c)}-\frac{\alpha_1}{\alpha_2-\alpha_1}e^{\alpha_2(x-c)},  & \text{if}~~x> c\\
  \frac{\beta_2}{\beta_2-\beta_1}e^{\beta_1(x-c)}-\frac{\beta_1}{\beta_2-\beta_1}e^{\beta_2(x-c)},  & \text{if}~~x\leq c,
\end{cases}\]
and
\[h_\infty(x)=e^{\beta_1x},~~~h_{-\infty}(x)=e^{\alpha_2x}.\]
For $c\in\ensuremath{{\mathbb R}}$, the function $h_c$ is constructed by smoothly merging $r$-harmonic functions for the Brownian motion with drift $\kappa$ (for $x\leq c$) and $-\kappa$ (for $x> c$) at their minimum in $c$. By taking derivatives and taking into account that $\beta_1=-\alpha_2$ and $\beta_2=-\alpha_1$, one sees that the function $h_c$ is indeed $C^2$. \\
The set $\{\lambda h_c:c\in[-\infty,\infty],\lambda\geq 0\}$ does not form a convex cone for $\kappa>0$. This is the main difference compared to the case without drift ambiguity. Therefore, the  standard techniques for optimal stopping are not applicable immediately. Nonetheless, this leads to the right $\mathcal{P}^\kappa$-supermartingales to work with:

\begin{lemma}\label{lem:harmonic}
\begin{enumerate}[(i)]
\item For all $a,b,x\in\ensuremath{{\mathbb R}}$ with $a\leq x\leq b$, $c\in[-\infty,\infty]$, $P\in\mathcal{P}^\kappa$ and $\tau=\inf\{t\geq 0:X_t\not\in[a,b]\}$ it holds that
\[\ensuremath{{\mathbb E}}_x^{P}(e^{-r\tau}h_c(X_\tau)\mathds{1}_{\{\tau<\infty\}})\geq h_c(x)\mbox{~~and~~}\ensuremath{{\mathbb E}}_x^{P_c}(e^{-r\tau}h_c(X_\tau)\mathds{1}_{\{\tau<\infty\}})=h_c(x),\]
where the measure $P_c$ is such that
\[dX_t=-\kappa \sgn(X_t-c)dt+dW^c_t\]
for a Brownian motion $W^c$ under $P_c$.
\item For all $c\in[-\infty,\infty]$ and all stopping times $\tau$ it holds that
\[\ensuremath{{\mathbb E}}_x^{P_c}(e^{-r\tau}h_c(X_\tau)\mathds{1}_{\{\tau<\infty\}})\leq h_c(x).\]
\item For all $a,b,x\in\ensuremath{{\mathbb R}}$ with $a<x<b$, $P\in\mathcal{P}^\kappa$, and $\tau_a=\inf\{t\geq 0:X_t=a\}$, $\tau_b=\inf\{t\geq 0:X_t=b\}$ it holds that
\begin{align*}
&\ensuremath{{\mathbb E}}_x^{P}(e^{-r\tau_a}h_\infty(X_{\tau_a})\mathds{1}_{\{\tau_a<\infty\}})\geq h_\infty(x),\\
&\ensuremath{{\mathbb E}}_x^{P_\infty}(e^{-r\tau_a}h_\infty(X_{\tau_a})\mathds{1}_{\{\tau_a<\infty\}})=h_\infty(x),
\end{align*}
and
\begin{align*}
&\ensuremath{{\mathbb E}}_x^{P}(e^{-r\tau_b}h_{-\infty}(X_{\tau_b})\mathds{1}_{\{\tau_b<\infty\}})\geq h_{-\infty}(x),\\
&\ensuremath{{\mathbb E}}_x^{P_{-\infty}}(e^{-r\tau_b}h_{-\infty}(X_{\tau_b})\mathds{1}_{\{\tau_b<\infty\}})=h_{-\infty}(x).
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(i)]
\item For $P\in\mathcal{P}$ with density process $\theta$, by Girsanov's theorem, we may write
\[X_t=W_t^P+\int_0^t\theta_sds,\]
where $W^P$ is a Brownian motion under $P$. Since $h_c\in C^2$ we can apply It\^o's lemma and obtain
\[dh_c(X_t)=h_c'(X_t)dW^P_t+(h_c'(X_t)\theta_t+1/2h_c''(X_t))dt.\]
By construction of $h_c$, it holds that
\[1/2h_c''(X_t)-\kappa \sgn(X_t-c)h_c'(X_t)-rh_c(X_t)=0,\]
hence
\begin{align*}
e^{-rt}h_c(X_t)=&h_c(X_0)+\int_0^t e^{-ru}(\kappa\sgn(X_u-c)+\theta_u)h_c'(X_u)du\\
&+\int_0^t e^{-ru}h_c'(X_u)dW^P_u.
\end{align*}
Noting that $(\kappa\sgn(X_u-c)+\theta_u)\geq 0$ iff $h_c'(X_u)\geq 0$, we obtain that the process $(e^{-r(t\wedge \tau)}h_c(X_{t\wedge \tau}))_{t\geq0}$ is a bounded $P$-submartingale. Therefore, by the optional sampling theorem,
\[\ensuremath{{\mathbb E}}_x^P(e^{-r\tau}h_c(X_{\tau}))\geq \ensuremath{{\mathbb E}}_x^P(h_c(X_{0}))=h_c(x).\]
Under $P^c$ we see that $(e^{-r(t\wedge \tau)}h_c(X_{t\wedge \tau}))_{t\geq0}$ is actually a local martingale that is bounded. Therefore, the optional sampling theorem yields equality.
\item By the calculation in $(i)$ the process $(e^{-rt}h_c(X_t))_{t\geq0}$ is a positive local $P^c$-martingale, i.e. also a $P^c$-supermartingale. The optional sampling theorem for non-negative supermartingales is applicable.
\item By noting that $h_\infty$ is decreasing and $h_{-\infty}$ is increasing the same arguments as in $(i)$ apply.
\end{enumerate}
\end{proof}

The following theorem shows that the geometric solution described in Subsection \ref{sec:graph_sol} can indeed be generalized to the drift ambiguity case. Moreover, we give a characterization of the optimal stopping set as maximum point of explicitly given functions.
\begin{theorem}\label{thm:main_brownian}
\begin{enumerate}[(i)]
\item
 It holds that
\[v(x)=\inf\{\lambda h_{c}(x):c\in[-\infty,\infty],\lambda \in[0,\infty], \lambda h_{c}\geq g\}\mbox{~~~~~for all $x\in\ensuremath{{\mathbb R}}$}.\]
Furthermore, the infimum in $c$ is indeed a minimum.
\item
 A point $x\in\ensuremath{{\mathbb R}}$ is in the optimal stopping set $\{y:v(y)=g(y)\}$ if and only if there exists $c\in[-\infty,\infty]$ such that
\[x\in\argmax \frac{g}{h_c}.\]
\end{enumerate}
\end{theorem}
\begin{proof}
For each $x\in\ensuremath{{\mathbb R}}$, $c\in[-\infty,\infty]$ and each stopping time $\tau$ we obtain using Lemma \ref{lem:harmonic} (ii)
\begin{align*}
\inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau}g(X_\tau)\mathds{1}_{\{\tau<\infty\}})&=\inf_P\ensuremath{{\mathbb E}}_x^P\left(e^{-r\tau}h_c(X_\tau)\frac{g}{h_c}(X_\tau)\mathds{1}_{\{\tau<\infty\}}\right)\\
&\leq \sup\left(\frac{g}{h_c}\right)\inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau}h_c(X_\tau)\mathds{1}_{\{\tau<\infty\}})\\
&\leq \sup\left(\frac{g}{h_c}\right)\ensuremath{{\mathbb E}}_x^{P_c}(e^{-r\tau}h_c(X_\tau)\mathds{1}_{\{\tau<\infty\}})\\
&\leq \sup\left(\frac{g}{h_c}\right)h_c(x).
\end{align*}
Since $\lambda h_c\geq g$ holds if and only if $\lambda\geq \sup\left(\frac{g}{h_c}\right)$ we obtain that
\[v(x)\leq \inf\{\lambda h_{c}(x):c\in[-\infty,\infty],\lambda \geq 0, \lambda h_{c}\geq g\}.\]
For the other inequality consider the following cases:\\
\textit{Case 1:}
\[\sup_{y\in\ensuremath{{\mathbb R}}}\frac{g(y)}{h_\infty(y)}=\sup_{y\leq x}\frac{g(y)}{h_\infty(y)}.\]
Take a sequence $(y_n)_{n\in\ensuremath{{\mathbb N}}}$ with $y_n\leq x$ such that ${g(y_n)}/{h_\infty(y_n)}\rightarrow\sup_{y\in\ensuremath{{\mathbb R}}}\frac{g(y)}{h_\infty(y)}$. Then for $\tau_n=\inf\{t\geq 0: X_t=y_n\}$ using Lemma \ref{lem:harmonic} (iii) we obtain
\begin{align*}
v(x)&\geq \inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau_n}g(X_{\tau_n})\mathds{1}_{\{\tau_n<\infty\}})\\
&=\inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau_n}h_\infty(X_{\tau_n})\frac{g}{h_\infty}(X_{\tau_n})\mathds{1}_{\{\tau_n<\infty\}})\\
&=\frac{g}{h_\infty}(y_n)E_x^{P_\infty}(e^{-r\tau_n}h_\infty(X_{\tau_n})\mathds{1}_{\{\tau_n<\infty\}})\\
&=\frac{g}{h_\infty}(y_n)h_\infty(x)\\
&\rightarrow \sup_{y\in\ensuremath{{\mathbb R}}}\frac{g(y)}{h_\infty(y)} h_\infty(x)~~~~~\mbox{for $n\rightarrow\infty$.}
\end{align*}
Therefore, $v(x)\geq \inf\{\lambda h_{c}(x):c\in[-\infty,\infty],\lambda\in[0,\infty], \lambda h_{c}\geq g\}$.\\
Moreover, if $x$ is in the stopping set, i.e. $v(x)=g(x)$, then we see that $g(x)/h_\infty(x)=\sup_{y\in\ensuremath{{\mathbb R}}}\frac{g(y)}{h_\infty(y)}$, i.e. $x$ is a maximum point of the function ${g}/h_{\infty}$, i.e. $(ii)$.\\
\textit{Case 2:} The case $\sup_{y\in\ensuremath{{\mathbb R}}}{g(y)}/{h_{-\infty}(y)}=\sup_{y\geq x}{g(y)}/{h_{-\infty}(y)}$ can be handled the same way.\\
\textit{Case 3:}
\begin{align*}
\sup_{y\leq x}\frac{g(y)}{h_\infty(y)}>\sup_{y\geq x}\frac{g(y)}{h_\infty(y)}~~~\mbox{and}~~~\sup_{y\leq x}\frac{g(y)}{h_{-\infty}(y)}<\sup_{y\geq x}\frac{g(y)}{h_{-\infty}(y)}.
\end{align*}
First we show that there exists $c^*\in\ensuremath{{\mathbb R}}$ such that
\[\sup_{y\leq x}\frac{g(y)}{h_{c^*}(y)}=\sup_{y\geq x}\frac{g(y)}{h_{c^*}(y)}:\]
To this end, write
\begin{align*}
h_{c,1}(y)&=\frac{\alpha_2}{\alpha_2-\alpha_1}e^{\alpha_1(y-c)}-\frac{\alpha_1}{\alpha_2-\alpha_1}e^{\alpha_2(y-c)},\\
h_{c,2}(y)&=\frac{\beta_2}{\beta_2-\beta_1}e^{\beta_1(y-c)}-\frac{\beta_1}{\beta_2-\beta_1}e^{\beta_2(y-c)}.
\end{align*}
By construction of $h_c$ it holds that $h_c=\min(h_{c,1},h_{c,2})$. Therefore,
\begin{align*}
\sup_{y\leq x}\frac{g(y)}{h_c(y)}&=\left[\inf_{y\leq x}\left(\min\left(\frac{h_{c,1}(y)}{g(y)},\frac{h_{c,2}(y)}{g(y)}\right)\right)\right]^{-1}\\
=\bigg[\min\bigg\{&e^{-\alpha_2 c}\inf_{y\leq x}\left(\frac{\alpha_2}{\alpha_2-\alpha_1}\frac{e^{\alpha_1 y}}{g(y)}e^{(\alpha_2-\alpha_1)c}+\frac{\alpha_1}{\alpha_2-\alpha_1}\frac{e^{\alpha_2 y}}{g(y)}\right),\\
& e^{-\beta_2 c}\inf_{y\leq x}\left(\frac{\beta_2}{\beta_2-\beta_1}\frac{e^{\beta_1 y}}{g(y)}e^{(\beta_2-\beta_1)c}+\frac{\beta_1}{\beta_2-\beta_1}\frac{e^{\beta_2 y}}{g(y)}\right)\bigg\}\bigg]^{-1}
\end{align*}
Since the functions
\[z\mapsto \inf_{y\leq x}\left(\frac{\alpha_2}{\alpha_2-\alpha_1}\frac{e^{\alpha_1 y}}{g(y)}z+\frac{\alpha_1}{\alpha_2-\alpha_1}\frac{e^{\alpha_2 y}}{g(y)}\right)\]
and
\[z\mapsto\inf_{y\leq x}\left(\frac{\beta_2}{\beta_2-\beta_1}\frac{e^{\beta_1 y}}{g(y)}z+\frac{\beta_1}{\beta_2-\beta_1}\frac{e^{\beta_2 y}}{g(y)}\right)\]
are continuous as concave functions, we obtain that the function $c\mapsto\sup_{y\leq x}\frac{g(y)}{h_c(y)}$ is continuous. By the same argument, the function $c\mapsto\sup_{y\geq x}\frac{g(y)}{h_c(y)}$ is also continuous. By the intermediate value theorem applied to the function
\[c\mapsto\sup_{y\leq x}\left(\frac{g(y)}{h_c(y)}\right)-\sup_{y\geq x}\left(\frac{g(y)}{h_c(y)}\right)\]
there exists $c^*$ with $\sup_{y\leq x}\frac{g(y)}{h_{c^*}(y)}=\sup_{y\geq x}\frac{g(y)}{h_{c^*}(y)}$ as desired.\\
Now take sequences $(y_n)_{n\in\ensuremath{{\mathbb N}}}$ and $(z_n)_{n\in\ensuremath{{\mathbb N}}}$ with $y_n\leq x\leq z_n$ such that
\[\sup_{y\leq x}\frac{g(y)}{h_{c^*}(y)}=\lim_{n\rightarrow\infty}\frac{g(y_n)}{h_{c^*}(y_n)}=\lim_{n\rightarrow\infty}\frac{g(z_n)}{h_{c^*}(z_n)}=\sup_{y\geq x}\frac{g(y)}{h_{c^*}(y)}.\]
Using $\tau_n=\inf\{t\geq0:X_t\not\in[y_n,z_n]\}$ we obtain by Lemma \ref{lem:harmonic} (i)
\begin{align*}
v(x)&\geq \inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau_n}h_{c^*}(X_{\tau_n})\frac{g}{h_{c^*}}(X_{\tau_n})\mathds{1}_{\{\tau_n<\infty\}})\\
&\geq \left(\frac{g}{h_{c^*}}(y_n)\wedge \frac{g}{h_{c^*}}(z_n)\right)\inf_P\ensuremath{{\mathbb E}}_x^P(e^{-r\tau_n}h_{c^*}(X_{\tau_n})\mathds{1}_{\{\tau_n<\infty\}})\\
&=\left(\frac{g}{h_{c^*}}(y_n)\wedge \frac{g}{h_{c^*}}(z_n)\right)h_{c^*}(x)\rightarrow \sup\left(\frac{g}{h_{c^*}}\right)h_{c^*}(x).
\end{align*}
This yields the result $(i)$. As above we furthermore see that if $x$ is in the optimal stopping set, then it is a maximum point of $g/h_{c^*}$, i.e. $(ii)$.
\end{proof}
\begin{remark}
\begin{enumerate}
\item We would like to emphasize that we do not need any continuity assumptions on $g$. This is remarkable, because even for the easy case described at the beginning of this section most standard techniques do not lead to such a general result.
\item The previous proof is inspired by the ideas first described in \cite{BL}. It seems that other standard methods for dealing with optimal stopping problems for diffusions without drift ambiguity (such as Martin boundary theory as in \cite{sa}, generalized concavity methods as in \cite{dk}, or linear programming arguments as in \cite{HS}) are not applicable with minor modifications due to the nonlinear structure coming from drift ambiguity. A characterization of the optimal stopping points as in Theorem \ref{thm:main_brownian} (ii) for the problem without ambiguity can be found in \cite{CI2}.
\end{enumerate}
\end{remark}

\subsection{Worst-case prior}\label{subsec:worst_case}
Theorem \ref{thm:main_brownian} leads to the value of the optimal stopping problem with drift ambiguity and also provides an easy way to find the optimal stopping time. Another important topic is to determine the worst-case measure for a process started in a point $x$, i.e. we would like to determine the measure $P$ such that $v(x)=\sup_\tau \ensuremath{{\mathbb E}}^P_x(e^{-r\tau}g(X_\tau)\mathds{1}_{\{\tau<\infty\}})$. Using the results described above the worst-case measure can also be found immediately:
\begin{theorem}\label{thm:worst_case_brown}
Let $x\in \ensuremath{{\mathbb R}}$ and let $c$ be a minimizer as in Theorem \ref{thm:main_brownian} (i). Then $P^c$ is a worst-case measure for the process started in $x$.
\end{theorem}

\begin{proof}
This is immediate from the proof of Theorem \ref{thm:main_brownian}.
\end{proof}

\subsection{Example: American straddle in the Bachelier market}
Because it is easy and instructive we consider the example discussed in \cite{CR} in the light of our method: \\
We consider a variant of the American straddle option in a Bachelier market model as follows: As a driving process we consider a standard Brownian motion under $P^0$ with reward function $g(x)=|x|$. Our aim is to find the value in 0 of the optimal stopping problem
\[\sup_\tau \min_{P\in\mathcal{P}^\kappa}\ensuremath{{\mathbb E}}^P(e^{-r\tau}|X_\tau|\mathds{1}_{\{\tau<\infty\}}).\]
Using Theorem \ref{thm:main_brownian} we have to find the majorant of $|\cdot|$ in the set
\[\{\lambda h_{c}:c\in[-\infty,\infty],\lambda \in[0,\infty], \lambda h_{c}\geq g\}.\]
One immediately sees that if $\lambda h_c(\cdot)\geq |\cdot|$, then $\lambda h_0(\cdot)\geq |\cdot|$ and furthermore $\lambda h_0(0)\leq \lambda h_c(0)$. Therefore, we only have to consider majorants of $|\cdot|$ in the set
\[\{\lambda h_{0}:\lambda \in[0,\infty], \lambda h_{0}\geq g\}.\]
This one-dimensional problem can be solved immediately. For $\lambda=\max(|\cdot|/h_0(\cdot))$ one obtains $v(0)=\lambda h_0(0)$. \\
In fact, if $-b,b$ denote the maximum points of $|\cdot|/h_0(\cdot)$ we obtain that $v(x)=\lambda h_0(x)$ for $x\in[-b,b]$. Moreover, for $x\not\in[-b,b]$ one immediately sees that there exists $c\in\ensuremath{{\mathbb R}}$ such that $x$ is a maximum point of $|\cdot|/h_c(\cdot)$ and we obtain
\[v(x)=\begin{cases}
  \lambda h_0(x),  & \text{if}~~x\in[-b,b]\\
  |x|,  & \text{else}.
\end{cases}\]
Moreover, the worst-case measure is $P_0$, i.e. the process $X$ has positive drift $\kappa$ on $(-\infty,0)$ and drift $-\kappa$ on $[0,\infty)$.

\subsection{General diffusion processes}\label{subsec:general}
The results obtained before can be generalized to general one-dimensional diffusion processes. The only problem is to choose appropriate functions $h_c$ carefully. After these functions are constructed the same arguments as in the previous subsections work.\\
Let $(X_t)_{t\geq0}$ be a regular one-dimensional diffusion process on some interval $I$ with boundary points $a<b,a,b\in[-\infty,\infty]$, that is characterized by its generator
\[A=\frac{1}{2}\sigma^2(x)\frac{d^2}{dx^2}+\mu(x)\frac{d}{dx}\]
for some continuous functions $\sigma>0,\mu$. For convenience we furthermore assume that the boundary points $a, b$ of $I$ are natural. For a generalization to other boundary behaviors see the discussion in \cite[Section 6]{BL00}. Again denote by $\mathcal{P}^\kappa$ the set of all probability measures, that are equivalent to $Q$ with density process  of the form
\[\exp\left(\int_0^t\theta_sdX_s-1/2\int_0^t\theta^2_sds\right)\]
for a progressively measurable process $(\theta_t)_{t\geq0}$ with $|\theta_t|\leq \kappa$ for all $t\geq0$. We denote the fundamental solutions of the equation
\[\frac{1}{2}\sigma^2(x)\frac{d^2}{dx^2}\psi+(\mu(x)+\kappa)\frac{d}{dx}\psi=r\psi\]
by $\psi_+^\kappa$ resp. $\psi_-^\kappa$ for the increasing resp. decreasing positive solution, cf. \cite[II.10]{BS} for a discussion and further references. Analogously, denote the fundamental solutions of
\[\frac{1}{2}\sigma^2(x)\frac{d^2}{dx^2}\psi+(\mu(x)-\kappa)\frac{d}{dx}\psi=r\psi\]
by $\psi_+^{-\kappa}$ resp. $\psi_-^{-\kappa}$.
Note that for each positive solution $\psi$ of one of the above ODEs it holds that
\begin{equation}\label{eq:sec_der}
\frac{d^2}{dx^2}\psi(x)=\frac{-(\mu(x)\pm\kappa) \frac{d}{dx}\psi(x)+r\psi(x)}{\frac{1}{2}\sigma^2(x)},
\end{equation}
hence all extremal points are minima, so that $\psi$ has at most one minimum. Therefore, for each $s\in (0,1)$ the function $\psi=s\psi_+^{\pm\kappa}+(1-s)\psi_-^{\pm\kappa}$ has a unique minimum point and each $c\in I$ arises as such a minimum point. Therefore, for each $c\in (a,b)$ we can find constants $\gamma_1,...,\gamma_4$ such that the function
\[h_c:E\rightarrow\ensuremath{{\mathbb R}},x\mapsto\begin{cases}
  \gamma_1\psi^\kappa_+(x)+\gamma_2\psi^{\kappa}_-(x),  & \text{if}~~x\leq c\\
  \gamma_3\psi^{-\kappa}_+(x)+\gamma_4\psi^{-\kappa}_-(x),  & \text{if}~~x>c
\end{cases}\]
is $C^1$ with a unique minimum point in $c$ with the standardization $h_c(c)=1$. More explicitly, $\gamma_1,...,\gamma_4$ are given by
\begin{equation}\label{eq:gamma}
\gamma_1=\frac{{\psi_-^{\kappa}}'(c)}{D^\kappa(c)},\; \gamma_2=\frac{-{\psi_+^{\kappa}}'(c)}{D^\kappa(c)},\;\gamma_3=\frac{{\psi_-^{-\kappa}}'(c)}{D^{-\kappa}(c)},\; \gamma_4=\frac{-{\psi_+^{-\kappa}}'(c)}{D^{-\kappa}(c)},
\end{equation}
where
\[D^{\pm\kappa}(c)=\psi_+^{\pm\kappa}(c){\psi_-^{\pm\kappa}}'(c)-\psi_-^{\pm\kappa}(c){\psi_+^{\pm\kappa}}'(c).\]
Furthermore, write $h_a=\psi^{-\kappa}_+$ and $h_b=\psi^{\kappa}_-$. First, we show that the functions $h_c$ are always $C^2$.

\begin{lemma}
For each $c\in[a,b]$, the function $h_c$ is $C^2$.
\end{lemma}

\begin{proof}
For $c\in\{a,b\}$ the claim obviously holds. Let $c\in(a,b)$. We only have to prove that $h_c''(c-)=h_c''(c+)$. Using equation \eqref{eq:sec_der}, we obtain that
\begin{align*}
h_c''(c-)&=\gamma_1{\psi^\kappa_+}''(c)+\gamma_2{\psi_-^{\kappa}}''(c)\\
&=\frac{-(\mu(c)+\kappa)}{\frac{1}{2}\sigma^2(c)}(\gamma_1{\psi_+^\kappa}'(c)+\gamma_2{\psi_-^\kappa}'(c))+\frac{r}{\frac{1}{2}\sigma^2(c)}(\gamma_1{\psi_+^\kappa}(c)+\gamma_2{\psi_-^\kappa}(c))\\
&=\frac{-(\mu(c)+\kappa)}{\frac{1}{2}\sigma^2(c)}h_c'(c)+\frac{r}{\frac{1}{2}\sigma^2(c)}h_c(c).
\end{align*}
By the choice of $\gamma_1,\gamma_2$, we obtain
\begin{align*}
h_c''(c-)&=\frac{r}{\frac{1}{2}\sigma^2(c)}
\end{align*}
and analogously
\begin{align*}
h_c''(c+)&=\frac{r}{\frac{1}{2}\sigma^2(c)}.
\end{align*}
This proves the claim.
\end{proof}

Now all the arguments given in Subsection \ref{subsec:BM} and \ref{subsec:worst_case} apply and we again obtain the following results (compare Theorem \ref{thm:main_brownian} and Theorem \ref{thm:worst_case_brown}):

\begin{theorem}\label{thm:main_diffusion}
\begin{enumerate}[(i)]
\item
 It holds that
\[v(x)=\inf\{\lambda h_{c}(x):c\in [a,b],\lambda \in[0,\infty], \lambda h_{c}\geq g\}\mbox{~~~~~for all $x\in I$.}\]
\item
A point $x\in I$ is in the optimal stopping set $\{y:v(y)=g(y)\}$ if and only if there exists $c\in [a,b]$ such that
\[x\in\argmax \frac{g}{h_c}.\]
\end{enumerate}
\end{theorem}

\begin{theorem}\label{thm:worst-case_diff}
Let $x\in \ensuremath{{\mathbb R}}$ and let $c$ be a minimizer as in Theorem \ref{thm:main_diffusion} (i). Then $P^c$ is a worst-case measure for the process started in $x$.
\end{theorem}
\subsection{Example: An optimal decision problem for Brownian motions with drift}

The following example illustrates that our method also works in the case of a discontinuous reward function $g$, where differential equation techniques cannot be applied immediately. Furthermore, we see that our approach can be used for all parameters in the parameter space, although the structure of the solution changes.\\
Let $X=\sigma W_t+\mu t$ denote a Brownian motion with drift $\mu\in(-\infty,\infty)$ and volatility $\sigma$ under $P^0$, and let
\[g(x)=\begin{cases}
1,&\;\;x\leq 0,\\
x,&\;\;x>0.
\end{cases}\]
The fundamental solutions are given by
\[\psi_+^\kappa(x)=e^{\alpha_1x},\;\psi_-^\kappa(x)=e^{\alpha_2x},\;\psi_+^{-\kappa}(x)=e^{\beta_1x},\;\psi_-^{-\kappa}(x)=e^{\beta_2x},\]
where $\alpha_1<0<\alpha_2$ and  $\beta_1<0<\beta_2$ are the roots of
\[1/2\sigma^2 z^2+(\mu-\kappa) z-r=0,\mbox{ resp. }1/2\sigma^2 z^2+(\mu+\kappa) z-r=0.\]
Using equation \eqref{eq:gamma} we obtain
\[h_c(x)=\begin{cases}
  \frac{\alpha_2}{\alpha_2-\alpha_1}e^{\alpha_1(x-c)}-\frac{\alpha_1}{\alpha_2-\alpha_1}e^{\alpha_2(x-c)},  & \text{if}~~x> c,\\
  \frac{\beta_2}{\beta_2-\beta_1}e^{\beta_1(x-c)}-\frac{\beta_1}{\beta_2-\beta_1}e^{\beta_2(x-c)},  & \text{if}~~x\leq c.
\end{cases}\]
We consider
\[l_*(c):=\sup_{y\leq 0}\frac{1}{h_c(y)}=\begin{cases}
\frac{1}{h_c(0)},&\;c\geq0,\\
1,&\;c\leq 0
\end{cases}\]
and $l^*(c):=\sup_{y\geq 0}\frac{y}{h_c(y)}=\frac{y_c}{h_c(y_c)}$, where $y_c$ denotes the unique maximum point of $y/h_c(y),y\geq 0$.\\
We first consider the case\ $1=l_*(0)\geq l^*(0)$. By Theorem \ref{thm:main_diffusion} (ii), we obtain that $x=0$ is in the optimal stopping set $S$ as a maximizer of $y\mapsto g(y)/h_0(y)$. Furthermore, by decreasing $c$ to $-\infty$, we see that $(-\infty,0]\subseteq S$. Since $l_*(c)\rightarrow 0$ and $l^*(c)\rightarrow \infty$ for $c\rightarrow\infty$, there exists a unique $c^*\geq 0$ such that $l_*(c^*)=l^*(c^*)$. Therefore, by Theorem \ref{thm:main_diffusion} (ii) again, $x^*:=y_{c^*}\in S$ and by increasing $c$ to $\infty$, we obtain that $S=(-\infty,0]\cup [x^*,\infty)$. Theorem \ref{thm:main_diffusion} (i) yields
\[v(x)=\begin{cases}
1,&\;x\leq 0\\
\frac{h_{c^*}(x)}{h_{c^*}(0)},&\;x\in[0,x^*],\\
x&\;x\geq x^*,
\end{cases}\]
see Figure \ref{amb1} below.
By Theorem \ref{thm:worst-case_diff} we furthermore obtain that $P^{c^*}$ is a worst-case measure for the process started in $x\in(0,x^*)$. That is, under the worst-case measure, the process has drift $\mu+\kappa$ on $[0,c^*)$ and drift $\mu-\kappa$ on $[c^*,x^*]$.\\
Now, we consider the case $1=l_*(0)< l^*(0)$. By a similar reasoning as in the first case, we see that there exists $c^*<0$ such that $l_*(c^*)=l^*(c^*)$. Write $x_*=c^*<0,x^*=y_{c^*}$. Then, $S=(-\infty,x_*]\cup[x^*,\infty)$ is the optimal stopping set and the value function is given by
\[v(x)=\begin{cases}
1,&\;x\leq x^*\\
{h_{c^*}(x)},&\;x\in[x_*,x^*],\\
x&\;x\geq x^*,
\end{cases}\]
see Figure \ref{amb2} below. The worst-case measure is given by $P^{c^*}$, which means that the process has drift $\mu-\kappa$ on $[x_*,x^*]$.

\begin{figure}[h]
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[width=4.5cm]{ambiguity2}
\caption{Value function for $l_*(0)\geq l^*(0)$}\label{amb1}
\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[width=4.5cm]{ambiguity1}
\caption{Value function for $l_*(0)< l^*(0)$}\label{amb2}
\end{center}
\end{minipage}
\end{figure}

\section{Optimal decision for models with crashes} \label{sec:crashes}
Now denote by $Y$ a one-dimensional regular diffusion process on an interval $I$. Denote by $\mathcal{F}$ the natural filtration generated by $Y$. In this section we assume that all parameters of this process are known. This process represents the asset price process of the underlying asset if no crash occurs; therefore for economical plausibility it is reasonable to assume $I=(0,\infty)$.

Now we modify the process such that at a certain random time point $\sigma$ a crash of bounded height occurs.
To be more precise, let $c\in(0,1)$ be a given constant that describes an upper bound for the height of the crash.  For a given stopping time $\sigma$ and an $\mathcal{F}_\sigma$-measurable and $[c,1]$-valued random variable $\zeta$ we consider the modified process $X^{\sigma,\zeta}$ given by
\[X^{\sigma,\zeta}_t=\begin{cases}
 Y_t & t\leq\sigma \\
 \zeta Y_t &  t> \sigma.
\end{cases}\]
Now we consider the optimal stopping problem connected to the pricing of perpetual American options in this market, i.e., let $g:(0,\infty)\rightarrow[0,\infty)$ be a continuous reward function. We furthermore assume $g$ to be non-decreasing, so that a crash always leads to a lower payoff. We fix a constant discounting rate $r>0$ and furthermore assume that the holder of the option does know that the process will crash once in the future. We assume the crash to be observable for the decision maker, so she will specify her action by a pre-crash stopping time $\underline{\tau}$ and a post-crash stopping time $\overline{\tau}$, i.e. given $\sigma$ she takes the strategy
\begin{equation}\label{eq:strategy}
\tau=\tau_\sigma=\begin{cases}
 \underline{\tau}, &  \underline{\tau}\leq\sigma\\
\sigma+ \overline{\tau}\circ\theta_\sigma, & \mbox{ else},
\end{cases}
\end{equation}
where $\theta_\cdot$ denotes the time-shift operator.
As before we assume the holder of the option to be ambiguity averse in the sense that she maximizes her expected reward under the worst-case scenario, i.e. she tries to solve the problem
\begin{equation}\label{eq:value_crashes}
v(x)=\sup_{\underline{\tau},\overline{\tau}}\inf_{\sigma,\zeta}\ensuremath{{\mathbb E}}_x(e^{-r\tau}g(X^{\sigma,\zeta}_\tau)),
\end{equation}
where $\tau=\tau_\sigma$ is given as in \eqref{eq:strategy}.
\begin{remark}\label{rem:c}
Obviously by the monotonicity of the reward function we always have
\[v(x)=\sup_{\underline{\tau},\overline{\tau}}\inf_{\sigma}\ensuremath{{\mathbb E}}_x(e^{-r\tau}g(X^{\sigma}_\tau))\]
where $X^{\sigma}:=X^{\sigma,c}$.
\end{remark}
We obtain the following reduction of the optimal stopping problem under ambiguity about the crashes: It shows that the problem can be reduced into one optimal stopping problem and one Dynkin game for the diffusion process $Y$ (without crashes).

\begin{theorem}\label{thm:reduction}
\begin{enumerate}[(i)]
\item Let $\hat{g}$ be the value function for the optimal stopping problem for $cY$ with reward $g$, i.e.
\begin{equation}\label{eq:stop}
\hat{g}(y)=\sup_{{\overline{\tau}}}\ensuremath{{\mathbb E}}_y(e^{-r{\overline{\tau}}}g(cY_{\overline{\tau}}))\mbox{~for all~}y\in(0,\infty)
\end{equation}
and let $\hat{g}<\infty$. Then it holds that
\begin{equation}\label{eq:game}
v(x)= \sup_{\underline{\tau}}\inf_\sigma\ensuremath{{\mathbb E}}_x(e^{-r\underline{\tau}}g(Y_{\underline{\tau}})\mathds{1}_{\{{\underline{\tau}}\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{{\underline{\tau}}>\sigma\}}).
\end{equation}
\item If $\overline{\tau}$ is optimal for \eqref{eq:stop} and $\underline{\tau},\sigma$ is a Nash-equilibrium for \eqref{eq:game}, then $(\un{\tau},\ov{\tau})$, $(\sigma,c)$ is a Nash-equilibrium for \eqref{eq:value_crashes}.
\end{enumerate}
\end{theorem}

\begin{proof}
\begin{enumerate}[(i)]
\item First fix $\un{\tau}$, $\ov{\tau}$. Then for all $\sigma$ by conditioning on $\mathcal{F}_\sigma$ we obtain
\begin{align*}
\ensuremath{{\mathbb E}}_x(e^{-r{\tau}}g(X^{\sigma}_\tau))=\ensuremath{{\mathbb E}}_x(&e^{-r\un{\tau}}g(Y_{\un{\tau}})\mathds{1}_{\{\un{\tau}\leq \sigma\}}+e^{-r(\sigma+\ov{\tau}\circ\theta_\sigma)}g(cY_{\sigma+\ov{\tau}\circ\theta_\sigma})\mathds{1}_{\{\un{\tau}>\sigma\}})\\
=\ensuremath{{\mathbb E}}_x(&e^{-r\un{\tau}}g(Y_{\un{\tau}})\mathds{1}_{\{\un{\tau}\leq\sigma\}}\\
&+e^{-r\sigma}\ensuremath{{\mathbb E}}_x(e^{-r(\ov{\tau}\circ\theta_\sigma)}g(cY_{\sigma+\ov{\tau}\circ\theta_\sigma})|\mathcal{F}_\sigma)\mathds{1}_{\{\un{\tau}>\sigma\}}).
\end{align*}
By the strong Markov property we furthermore obtain
\[\ensuremath{{\mathbb E}}_x(e^{-r(\ov{\tau}\circ\theta_\sigma)}g(cY_{\sigma+\ov{\tau}\circ\theta_\sigma})|\mathcal{F}_\sigma)=\ensuremath{{\mathbb E}}_{Y_\sigma}(e^{-r{\overline{\tau}}}g(cY_{\overline{\tau}}))\leq \hat{g}(Y_\sigma).\]
Therefore,
\begin{align*}
\ensuremath{{\mathbb E}}_x(e^{-r\tau}g(X^{\sigma}_\tau))\leq \ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}g(Y_{\un{\tau}})\mathds{1}_{\{\un{\tau}\leq\sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{\un{\tau}>\sigma\}}),
\end{align*}
showing that
\begin{equation*}
v(x)\leq \sup_{\underline{\tau}}\inf_\sigma\ensuremath{{\mathbb E}}_x(e^{-r\underline{\tau}}g(Y_{\underline{\tau}})\mathds{1}_{\{{\underline{\tau}}\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{{\underline{\tau}}>\sigma\}}).
\end{equation*}
Now take a sequence of $1/n$-optimal stopping times $(\ov{\tau}_n)_{n\in\ensuremath{{\mathbb N}}}$ for the problem \eqref{eq:stop}, i.e.
\[\hat{g}(y)\leq \ensuremath{{\mathbb E}}_y(e^{-r\ov{\tau}_n}g(cY_{\ov{\tau}_n}))+\frac{1}{n}\mbox{~~for all~$n\in\ensuremath{{\mathbb N}}$ and all $y$}.\]
Then
\[\ensuremath{{\mathbb E}}_{Y_\sigma}(e^{-r{\overline{\tau}_n}}g(cY_{\overline{\tau}_n}))\leq \hat{g}(Y_\sigma)+\frac{1}{n},\]
and hence considering the post-crash strategy $\ov{\tau}_n$ and arbitrary $\un{\tau},\sigma$ we see that
\begin{equation*}
v(x)+\frac{1}{n}\geq \sup_{\underline{\tau}}\inf_\sigma\ensuremath{{\mathbb E}}_x(e^{-r\underline{\tau}}g(Y_{\underline{\tau}})\mathds{1}_{\{{\underline{\tau}}\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{{\underline{\tau}}>\sigma\}}),
\end{equation*}
proving equality.
\item is obvious by the proof of (i).
\end{enumerate}
\end{proof}

\iffalse
\begin{theorem}\label{thm:reduction}
The value function \eqref{eq:value_crashes} is given as the value to the Dynkin game
\begin{equation}\label{eq:game}
v(x)=\inf_{\sigma}\sup_{\tau}\ensuremath{{\mathbb E}}_x(e^{-r\tau}g(Y_\tau)\mathds{1}_{\{\tau\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{\tau>\sigma\}}),
\end{equation}
where $\hat{g}$ is the value function for the optimal stopping problem for $Y$ with reward $g$, i.e.
\begin{equation}\label{eq:stopping_game}
\hat{g}(y)=\sup_{\tau}\ensuremath{{\mathbb E}}_y(e^{-r\tau}g(cY_\tau))\mbox{~for all~}y\in(0,\infty).
\end{equation}
\end{theorem}

\begin{proof}
This immediately holds by Proposition \ref{prop:reduction} together with Remark \ref{rem:c}.
\end{proof}
\fi

\begin{remark}
Note that the arguments used so far have nothing to do with diffusion processes, but can be applied in the same way for general \textit{nice} one-dimensional strong Markov processes, like one-dimensional Hunt processes. Nonetheless we decided to consider this more special setup because of its special importance and since the theory for explicitly solving optimal stopping problems and Dynkin games is well established.
\end{remark}

The previous reduction theorem solves the optimal stopping problem \eqref{eq:value_crashes} since both problems \eqref{eq:game} and \eqref{eq:stop} are well-studied for diffusion processes, see e.g. the references given above for optimal stopping problems and \cite{EV}, \cite{A}, and \cite{Pe} for Dynkin games. It is interesting to see that the optimal stopping problem under crash-scenarios naturally leads to Dynkin games, which were studied extensively in the last years. The financial applications studied so far were based on Israeli options, which are (at least at first glance) of a different nature, see \cite{Ki}.

\subsection{Example: Call-like problem with crashes}
As an example we consider a geometric Brownian motion given by the dynamics
\[dX_t=X_t(\mu dt +\sigma dW_t),~~t\geq 0\]
and we take $g:(0,\infty)\rightarrow[0,\infty)$ given by $g(x)=(x-K)^+$, where $K>0$ is a constant. To exclude trivial cases we assume that $\mu<r$. Then a closed-form solution of the optimal stopping problem
\begin{equation*}
\hat{g}(y)=\sup_{\tau}\ensuremath{{\mathbb E}}_y(e^{-r\tau}(cY_\tau-K)^+)=\sup_{\tau}\ensuremath{{\mathbb E}}_{cy}(e^{-r\tau}(Y_\tau-K)^+)
\end{equation*}
is well known (see e.g. \cite[Chapter VII]{PS}) and is given by
\begin{equation*}
\hat{g}(y)=\begin{cases}
  cy-K,  & cy \geq x^*,\\
  d(cy)^\gamma, & cy<x^*,
\end{cases}
\end{equation*}
where $\gamma$ is the positive solution to
\[\frac{\sigma^2}{2} z^2+\big(\mu-\frac{\sigma^2}{2}\big)z+r=0,\]
and $x^*$ and $d$ are appropriate constants. Moreover, the optimal stopping time is given by $\ov{\tau}:=\inf\{t\geq 0: X_t\geq x^*\}$. By Theorem \ref{thm:reduction} we are faced with the Dynkin game
\begin{equation}\label{eq:crashes_call}
v(x)=\sup_{\un{\tau}}\inf_{\sigma}\ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}(Y_\un{\tau}-K)^+\mathds{1}_{\{\un{\tau}\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{\un{\tau}>\sigma\}}).
\end{equation}
To solve this problem first note that there exists $x'\in(K,x^*/c)$ such that $g(x)\leq \hat{g}(x)$ for $x\in(0,x']$ and $g(x)\geq \hat{g}(x)$ for $x\in[x',\infty)$; indeed, $x'$ is the unique positive solution to
\[d(cx)^\gamma=x-K,\]
see Figure \ref{fig:call_crashes}.

\begin{figure}%
\begin{center}
\includegraphics[width=0.5\columnwidth]{call_crashes}%
\caption{Graphs of $g$ (blue) and $\hat{g}$ (red).}\label{fig:call_crashes}
\end{center}
\end{figure}

We could use the general theory to solve the optimal stopping game \eqref{eq:crashes_call}, but we can also solve it elementary here:\\
First let $x>x'$. Then for all stopping times $\sigma^*$ with $\sigma^*=0$ under $P(\cdot | Y_0=x)$ and each stopping time $\un{\tau}$ we obtain
\begin{align*}
&\ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}(Y_\un{\tau}-K)^+\mathds{1}_{\{\un{\tau}\leq \sigma^*\}}+e^{-r\sigma^*}\hat{g}(Y_{\sigma^*})\mathds{1}_{\{\un{\tau}>\sigma^*\}})\\
=&\ensuremath{{\mathbb E}}_x(g(x)\mathds{1}_{\{\un{\tau}=0\}}+\hat{g}(x)\mathds{1}_{\{\un{\tau}>0\}})\\
\leq& g(x)
\end{align*}
with equality if $\tau=0$ $P(\cdot | Y_0=x)$-a.s. On the other hand for $\tau^*=0$ the payoff is $g(x)$, independent of $\sigma$. \\

For $x\leq x'$ by by taking $\un{\tau}= \inf\{t\geq 0: X_t\geq x'\}$ we have for each stopping time $\sigma$ by definition of $x'$
\begin{align*}
&\ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}(Y_{\un{\tau}}-K)^+\mathds{1}_{\{\un{\tau}\leq \sigma\}}+e^{-r\sigma}\hat{g}(Y_\sigma)\mathds{1}_{\{\un{\tau}>\sigma\}})\\
=&\ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}(x'-K)\mathds{1}_{\{\un{\tau}\leq \sigma\}}+e^{-r\sigma}d(cY_\sigma)^\gamma\mathds{1}_{\{\un{\tau}>\sigma\}})\\
=&\ensuremath{{\mathbb E}}_x(e^{-r\un{\tau}}d(cx')^\gamma\mathds{1}_{\{\un{\tau}\leq \sigma\}}+e^{-r\sigma}d(cY_\sigma)^\gamma\mathds{1}_{\{\un{\tau}>\sigma\}})\\
=&\ensuremath{{\mathbb E}}_x(e^{-r(\sigma\wedge\un{\tau})}d(cY_{\sigma\wedge\un{\tau}})^\gamma)\\
=&dc^\gamma\ensuremath{{\mathbb E}}_x(e^{-r(\sigma\wedge\un{\tau})}(Y_{\sigma\wedge\un{\tau}})^\gamma)\\
=&dc^\gamma x^\gamma=\hat{g}(x),
\end{align*}
where the last equality holds by the fundamental properties of the minimal $r$-harmonic functions, see e.g. \cite[II.9]{BS}. By taking any stopping time $\sigma^*\leq \inf\{t\geq0:y_t\geq x'\}$ and any stopping time $\tau$ the same calculation holds.\\
Putting pieces together we obtain that $\un{\tau},\sigma^*$ is a Nash-equilibrium of the Dynkin game \eqref{eq:crashes_call} for any stopping time $\sigma^*\leq \un{\tau}$.\\
By applying Theorem \ref{thm:reduction} we get
\begin{proposition}
The value function $v$ is given by
\begin{equation*}
v(x)=\begin{cases}
  x-K,  & x\geq x',\\
  d(cx)^\gamma, & x<x',
\end{cases}
\end{equation*}
and for
\[\un{\tau}=\inf\{t\geq 0: X_t\geq x'\}\hspace{2cm} \mbox{'pre-crash strategy'}\]
and
\[\ov{\tau}=\inf\{t\geq 0: X_t\geq x^*\}\hspace{2cm} \mbox{'post-crash strategy'}\]
and any stopping time $\sigma^*\leq \tau$ it holds that $(\un{\tau},\ov{\tau})$, $(\sigma^*,c)$ is a Nash-equilibrium of the problem.
\end{proposition}
The solution to this example is very natural: If the investor expects a crash in the market, then she exercises the option as soon as the asset price reaches the level $x'$ (pre-crash strategy). After the crash, i.e. if the investor does not expect to have more crashes, then she takes the ordinary stopping time, i.e. she stops if the process reaches level $x^*>x'$ (post-crash strategy).

\section*{Acknowledgements}
This paper was partly written during a stay at \AA bo Akademi in the project \textit{Applied Markov processes -- fluid queues, optimal stopping and population dynamics, Project number: 127719  (Academy of Finland)}.  I would like to express my gratitude for the hospitality and support.

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


First published at 23:24 UTC on April 9th, 2018.
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James Cole, former U.S. deputy attorney general, has had a huge impact on the development of the cannabis industry. Now he says banks need to serve marijuana companies and laws need to be reformed.
By Will Yakowicz, Staff writer, Inc.@WillYakowicz
A close up of marijuana growing inside a Denver cultivation center.
Will Yakowicz
If there is one government official who can take credit for helping marijuana businesses emerge from the grey market, it's former U.S. Deputy Attorney General James Cole.
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text:
Integrated Door Systems, Smoke and Fire Curtains
ARC US Conference 2022
Door Systems was happy to be back to in-person events this past year with ARC US Conference. This prestigious Bond Event brings together America's Top 100 Principal Architects and Vendors in one place for 4 days and 3 nights.
The experience centers around a series of one-to-one business meetings between architects and vendors, that each have selected ahead of time, this ensures high quality interactions making the best use of the attendee's time away from the office.
January 12, 2022 /by Door Systems Writer
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Integrated Door Systems
Tallest towers in San Francisco, mapped
The city's skyline has always been swoon-worthy, but over the last decade, the view has become even more grand—not to mention crowded—as skyscrapers have sprouted taller, taller, and taller. From the Salesforce Tower to 555 California, San Francisco's high-rises have set examples of height and grandeur. Read more
June 26, 2021 /by Door Systems Writer
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History of the Twitter Building
How the internet reshaped Market Street's Art Deco monolith
The Twitter building, once known as the Western Furniture Exchange and Merchandise Mart, got its start as a Depression-era furniture showcase
This is a story about adaptive reuse, unplanned obsolescence, and the way one building can straddle two centuries and come to embody the fickle nature of commerce in a modern American city.
The hulking, Mayan-inspired, Art Deco monolith originally known as the Western Furniture Exchange and Merchandise Mart has stood on its Market Street block between Ninth and Tenth streets for 81 years now.
May 5, 2021 /by Door Systems Writer
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The most polarizing buildings in San Francisco
Love 'em or loathe 'em, these structures won't be ignored
Thom Mayne's jarring yet alluring Federal Building, an 18-story structure with a curtain of perforated steel punctuated by a large square cut-out framing a privately owned public park, and the Glen Park BART station, a concrete public transportation dream, divide many. Read more
April 15, 2021 /by Door Systems Writer
https://www.doorsysinc.com/wp-content/uploads/SanFrancisco-MuseumofModern-Art.png 692 1200 Door Systems Writer https://www.doorsysinc.com/wp-content/uploads/DSI-logo-black.png Door Systems Writer2021-04-15 20:27:582021-07-21 20:34:04The most polarizing buildings in San Francisco
Union Bank Plaza undergoes $20 million renovation
The first phase in the historic preservation-minded rehabbing of Union Bank Plaza (UBP) in Downtown Los Angeles has wrapped up. Completed in 1967, the 40-story office tower, designed by Harrison & Abramovitz and Albert C. Martin & Associates in the International style, became the first L.A. skyscraper to be named as a Historic-Cultural Monument (HCM) earlier this year by the city's Cultural Heritage Commission following a unanimous 5–0 vote. The nominating application for HMU designation was put forth by the Los Angeles Conservancy in July 2019.
March 20, 2021 /by Door Systems Writer
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Rising Construction Costs
Labor and materials are arguably the two most critical ingredients of any construction project. No buildings transform skylines, no bridges span rivers and no homes rise from foundations without them. That's why pricing of these two production components can mean cost-cutting is in store for other areas of the job budget or, given steep-enough increases, the scuttling of the project altogether.
February 14, 2021 /by Door Systems Writer
https://www.doorsysinc.com/wp-content/uploads/contruction-site.jpg 484 710 Door Systems Writer https://www.doorsysinc.com/wp-content/uploads/DSI-logo-black.png Door Systems Writer2021-02-14 15:59:372021-03-26 16:00:18Rising Construction Costs
Arts District High Rise Starts Work
San Francisco-based developer Carmel Partners has broken ground on its first Arts District project. The planned 35-story project designed by Works Progress Architecture will rise at 520 S. Mateo St., near the Fourth Street Bridge. The main tower will include 475 live/work apartments, including 50 affordable units.
https://www.doorsysinc.com/wp-content/uploads/SFArtsDistrict.jpg 563 750 Door Systems Writer https://www.doorsysinc.com/wp-content/uploads/DSI-logo-black.png Door Systems Writer2021-01-26 15:41:322021-03-26 15:42:04Arts District High Rise Starts Work
Hewitt Street Getting a Hotel
Yet another hotel has been announced for the Arts District. Hospitality firm the Radisson Hotel Group announced plans to construct a 140-room Radisson RED hotel at 440 S. Hewitt St. A groundbreaking date has not been set, but the company expects it to open in early 2024.
December 16, 2020 /by Door Systems Writer
https://www.doorsysinc.com/wp-content/uploads/hotel-hewitt-street.jpg 450 750 Door Systems Writer https://www.doorsysinc.com/wp-content/uploads/DSI-logo-black.png Door Systems Writer2020-12-16 15:43:272021-03-26 16:00:02Hewitt Street Getting a Hotel
Anaheim Stadium turned into a mega mixed-use district
Last Wednesday, the owners of the Anaheim Ducks hopped on the "professional sports team turning the area around their stadium into a mixed-use district" bandwagon with the announcement of "ocV!BE," a 115-acre neighborhood that will set up shop around the team's Honda Center in Anaheim, California.
The hockey team's owners will realize that vision by developing the (currently unused) landscape of surface parking lots around the Honda Center. In return, the Southern California city will receive 2,800 new apartments with 15 percent set aside for affordable housing, 825,000-square-feet of office space, and 30 acres of public green spaces, including a "wellness park." Read more
July 14, 2020 /by Door Systems Writer
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510 foot, 45-story tower in San Francisco
This 510 foot, 45-story tower with a rising profile near SFMOMA will repurpose the 10-story Aronson Building, which survived the 1906 earthquake and fire, and add a residential skyscraper next to it, care of Handel Architects. The older building will play home to residential lofts on levels five through ten, as well as the upcoming Mexican Museum on levels two through four.
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Smoke and Fire Curtains


Rob van Vlokhoven is a thought leader on Hostmanship - 'The art of making people feel welcome'. He believes that people who feel like guests take with them an experience of better quality and value. Good Hostmanship can thus evoke not only a feeling of satisfaction, but also feelings of enthusiasm, joy and delight.
The Hostmanship choice therefore is all about seeing the people we meet not just as customers, patients, clients, students, co-workers or colleagues, but as dear guests. This ´guest – host´ type of approach is the starting point for developing a truly welcoming attitude.
Rob hosts around 150 presentations, workshops and creative sessions a year based upon the Hostmanship Philosophy. That makes Rob an expert in inspiring companies to achieve true customer, culture and leadership excellence.
Rob's inspiring, thoughtful and entertaining keynotes are about Hostmanship, the art of greeting, meeting and treating everyone with respect, dignity and care and thereby creating value for the customer, our colleagues, the company and for oneself.
In every meeting you have a choice. The choice to show your best self and to really understand others. Within Rob's workshops he makes his audience aware of the Hostmanship choice and shows them how to deliver customer excellence every single day.
One keynote or workshop usually is not enough to create a true welcoming culture. Therefore Hostmanship offers inspiring (incompany) Programs which are based on the 3 dimensions of Hostmanship: welcoming oneself, welcoming others, welcoming guests.
There are multiple ways for people to grasp the essence of Hostmanship. Therefore Hostmanship offers consultancy and tools to inspire the world with 'the art of making people feel welcome'.
Goal: Delivering WOW-service. Activities: Development of the DeLaMar Hostmanship Academy, selection of new hosts (HR cycle), organizing workshops, creating new business tools and promoting Host-Leadership.
Goal: Delivering the finest designer outlet experience. Activities: Development of an inspirational culture-program, based on the renewed collective ambition. Promoting Host-leadership and consistently improving the customer experience based on Mystery Visits. Locations: Roermond, Berlin, Neumunster, Parndorf, Rome.
Goal: Encouraging Host-Leadership. Activities: KickOff with DeLaMar Experience, development of an inspirational cultural excellence program (50 coaches), coaching on the job, organizing interventions and creating new business tools.
Goal: Delivering the hospital's newest proposition. Activities: KickOff with DeLaMar Experience, development of an inspirational culture program (50 coaches), coaching on the job and organizing interventions.
Goal: Delivering the DIY-store's newest proposition. Activities: KickOff with DeLaMar Experience, development of an inspirational culture program (20 shops, 40 coaches), workshops at the store, interviewing guests, coaching on the job and organizing interventions.
Rob van Vlokhoven! ©2017 All Rights Reserved.


Tired of sad, insipid toaster waffles and flavorless instant batters?
Alton Brown sure is and he's going to do something about it. Learn how to build a better batter, pick the right iron and find out what waffles and church have in common.


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Spids øjentrøst (Euphrasia stricta) er en 3-20 cm høj, kortlevende, utreagtig plante, der er halvsnylter på flere arter af de urteagtige planter i eng og overdrev.

Beskrivelse
Spids øjentrøst er en en- eller toårig, urteagtig plante med en opstigende og busket vækst. Stænglerne er næsten hårløse og ofte rødlige med mere eller mindre udstående grene. bladene sidder modsat. De er bredt ægformede med grove tænder langs randen. Oversiden er rynket og mat mørkegrøn, mens undersiden er lysegrøn og håret. Blomstringen foregår i maj-september, hvor man finder blomsterne siddende i små stande fra bladhjørnerne. De enkelte blomster er 5-tallige og uregelmæssige med hvide til svagt violette kronblade af 4-10 mm længde. Frugterne er kapsler med mange frø.

Rodsystemet består af spinkle og fåtallige trævlerødder, men det skal tages i betragtning, at planten snylter på rodnettet af sine værtplanter (bl.a. gul snerre).

Højde x bredde og årlig tilvækst: 0,15 x 0,50 m (15 x 5 cm/år).

Hjemsted

Spids øjentrøst har sin naturlige udbredelse i Lilleasien, Kaukasus og det meste af Europa. I Danmark er den almindelig i det mest af landet. Den er knyttet til lysåbne og ret tørre voksesteder med en veldrænet, næringsfattig jord.

Ved Tueholm Sø i Kongsholmparken, som ligger i Albertslund, findes arten sammen med bl.a. ene, alm. knopurt, knoldet mjødurt, pilealant, rank frøstjerne, stor knopurt, toårig høgeskæg og trenervet snerre

Vanskelig afgrænsning
Afgrænsningen er vanskelig mellem arterne i slægten Øjentrøst. Spids øjentrøst kan opfattes som bestående af flere former, der undertiden opfattes som arter. I Danmark drejer det sig om Svensk øjentrøst, Kirteløjentrøst, Spinkel øjentrøst, Nordisk øjentrøst, Klitøjentrøst og Kort øjentrøst, der alle kan opfattes som former af spids øjentrøst.

Fodnoter

Kilder
 Signe Frederiksen et al., Dansk flora, 2. udgave, Gyldendal 2012. .

Flerårige urter
Hvide blomster

redpajama_set_name: RedPajamaWikipedia


I'm Jeff Ong: First Words of the Day (Guess Who) and Ang Pow & Movie Time with Mum, of the Year!
First Words of the Day (Guess Who) and Ang Pow & Movie Time with Mum, of the Year!
I haven't watched movies with Mum since a long time ago. And after sorting out my stuff much earlier, we decided to catch a movie together. Mum liked (if not loved) the TVB drama series, "Triumph in the Skies", so naturally this would be our pick since there wasn't much selection at that point of time.


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Investing Advice And InformationStock Market NewsGlobal Markets Choppy, as Volatility Rises
The Alpha Investor Letter Hotlines
Global Markets Choppy, as Volatility Rises
Nicholas Vardy
Nicholas Vardy has a unique background that has proven his knack for making money in different markets around the world.
It's been a choppy five days in global markets over the past week. Stocks suffered steep losses yesterday as oil prices surged, renewing worries that higher fuel prices could hobble the global economic recovery. Oil prices jumped 13% last week with a rise in turmoil across North Africa and the Middle East. Americans are now paying roughly $75 million more per day to fill their gas tanks than a week ago.
After a period of extremely low levels, volatility is back, particularly in sectors like technology. Your position in JinkoSolar Holding Co., Ltd. (JKS) rallied as much as 16% in a single day, before actually ending the day lower.
Despite this, most of your positions in your Global Stock Investor portfolio ended the week higher. A couple of investment themes in your Global Stock Investor portfolio continue to perform particularly well, including oil through the Market Vectors Russia ETF (RSX), agriculture through Agrium Inc. (AGU) and Japanese small caps through WisdomTree Japan SmallCap Dividend (DFJ). You should feel comfortable adding to all of these positions. Technology — Universal Display Corp. (PANL) — also continues to outperform, as long as you are willing to endure the volatility.
Overall, our decision to exit almost all of our emerging market positions over the past two months continues to be a good one, as these formerly red-hot markets have fallen out of favor.
Meanwhile, Las Vegas Sands (LVS) announced that it is being investigated by U.S. authorities over its compliance with provisions of the Foreign Corrupt Practices Act related to its operations in Macau, as a result of a lawsuit filed by a disgruntled former employee in Nevada. Not surprisingly, this hit the stock hard.
Exclusive Despite 2020, It's a Benevolent Universe
Although I know nothing about the details of the case, as a former attorney that helped set up businesses in emerging markets early in my career, I can tell you this is a good example of the legal millstones American companies have around their corporate necks when doing business abroad. No Chinese company that ever cut a deal with the government of Nigeria or Sudan — places that are virtually "no-go zones for U.S. companies — will ever be sued by its own government for the way they do business abroad. No matter where you come down on this issue, it's a high price to pay for a naïve America.
The question is whether that price is worth paying…
Portfolio Update
Agrium Inc. (AGU) jumped 4.07% this past week. With agricultural stocks recovering almost all of their falls from the previous week, and with the stock still trading above its 50-day moving average, AGU remains a BUY.
The WisdomTree Dreyfus Chinese Yuan Fund (CYB) rose .20% this past week. The yuan has been essentially flat as a pancake in all of 2011. Now trading back above its 50-day moving average, CYB is a BUY.
WisdomTree Japan SmallCap Dividend (DFJ) was up 2.06% this past week. Watch this trade become "conventional wisdom" over the coming months. Bumbling Japan actually outperformed dynamic China in 2010. DFJ remains a BUY.
Freeport-McMoRan Copper & Gold, Inc. (FCX) rose 2.46% this past week. We've given this stock plenty of room to pull back, as it is one of the most volatile out there. But if it hits its stop of $47.50, you should be out. Trading below its 50-day moving average, this volatile stock is a HOLD.
Exclusive Initial Public Offerings Set a Record Value in 2020
Global X FTSE Nordic 30 ETF (GXF) ended the week 2.71% higher. Trading back above its 50-day moving average, this bet on the most dynamic part of Europe is a BUY.
JinkoSolar Holding Co., Ltd. (JKS) dropped 2.04% this past week. JKS jumped as high as 16.1% in early trading on Monday before swooning 2.3% below Friday's closing price, all on very heavy trading volume. Italy announced that it might cap solar installations at 8 gigawatts total, and JinkoSolar has a large exposure to the Italian market. Still above its 50-day moving average, JKS remains a volatile BUY.
Las Vegas Sands Corp. (LVS) dropped back 4.84% this past week, thanks largely to its announcement that it is being investigated by the Securities and Exchange Commission. Trading back under its 50-day moving average, and near its stop price, Asia's Las Vegas is now a HOLD.
Market Vectors Russia ETF (RSX) jumped 4.8% this past week, as soaring oil prices support the market. Trading above its 50-day moving average for the first time since September, RSX remains a BUY.
Universal Display Corp. (PANL) ended the week 1.86% higher after a very volatile week. With earnings announcements coming up on March 14, this is a terrific time to add to your position in this stock. But strap yourself in for a wild ride. PANL remains a solid BUY.
Vale S.A. (VALE) rose just under 1% this past week. Despite the company's bullish outlook on China, with the stock trading below its 50-day moving average, Vale is a HOLD.
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