Abstracts

Recently, there were series of papers, where the solution of optimal stopping problems for Lévy processes and random walks when the reward is a monotone function were found in terms of the maximum/minimum of the process. In the present paper we use the integral transform method and develop it further to the case of non-monotone functions. We present a constructive method on how to solve optimal stopping problems (i.e. show how to find the optimal stopping boundary) for a fairly general reward function $g$. The key ingredient here is to introduce the integral transform $\mathcal{A}^{\eta(x)}$ based on the random variable $\eta(x) = \argmax g_{0 \leq t \leq e_q}(x+ X_s) - x$.
We find the stopping region as those arguments at which the function $\mathcal{A}^{\eta(x)}\{g\}$ (where $\mathcal{A}^{\eta(x)}$ is a form of an Esscher-Laplace transform) is non-negative, and the continuation region as those arguments at which the function $\mathcal{A}^\eta\{g\}$ is negative. Our algorithm to find the solution to the optimal stopping problem is the following: firstly, we introduce an auxiliary random variable $\eta(x)$ pathwise tracking the value of $X_t$ that achieved the running maximum of $g(x+X)$. Secondly, we use ${\eta(x)}$ to define the integral transform $\mathcal{A}^{\eta(x)}$. The integral transform maps the reward function $g=g(\cdot)$ into function $\mathcal{A}^{\eta(x)}\{g\} (\cdot)$ for each $x$. Finally, we find the stopping region $S$ as those arguments $(x,y)$ at which $\mathcal{A}^{\eta(x)}\{g\}$(y) is non-negative. The optimal strategy is to stop whenever $(x,x) \in S$, and to continue observations while $(x,x) \notin S$. We illustrate this approach with some examples. The proposed method is computationally attractive as it does not require solving differential or integro-differential equations which appear if the traditional methods are used.