概率失真下的最优停止和律不变相干风险测度
Optimal stopping under probability distortions and law invariant coherent risk measures
摘要：
在本文中，我们研究最优停止问题的扭曲的期望的形式\begin{eqnarray*} \mathcal{E}(X)=\int_{-\infty}^{\infty} x\,dG(F_X(x)), \end{eqnarray*}，其中$F_X$是分布函数的$X$和$G$是一个凸分布函数的$[0,1].$作为事实上，除了$G$是身份的$[0,1],$动态版本的$\mathcal{E}(X)$不具有所谓的时间一致性属性所需的动态规划方法。本文证明了一种新的表示方法，它将一个扭曲期望下的最优停止问题的解与标准最优停止问题的序列联系起来，从而使基于标准动态规划的方法的应用成为可能。此外，通过著名的Kusuoka表示，我们将我们的结果推广到一般规律不变的相干风险测度下的最优停止。最后，基于我们的新的表示，我们开发了几个Monte Carlo近似算法，并说明了他们的权力下的最佳停止平均值在风险和绝对半确定性风险措施。
英文摘要：
In this paper we study optimal stopping problems with respect to distorted expectations of the form \begin{eqnarray*} \mathcal{E}(X)=\int_{-\infty}^{\infty} x\,dG(F_X(x)), \end{eqnarray*} where $F_X$ is the distribution function of $X$ and $G$ is a convex distribution function on $[0,1].$ As a matter of fact, except for $G$ being the identity on $[0,1],$ dynamic versions of $\mathcal{E}(X)$ do not have the so-called time-consistency property necessary for the dynamic programming approach. So the standard approaches are not applicable to optimal stopping under $\mathcal{E}(X).$ In this paper, we prove a novel representation, which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. Furthermore, by means of the well known Kusuoka representation, we extend our results to optimal stopping under general law invariant coherent risk measures. Finally, based on our novel representations, we develop several Monte Carlo approximation algorithms and illustrate their power for optimal stopping under Average Value at Risk and the absolute semideviation risk measures.
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