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In this paper we study optimal stopping problems with respect to distortedexpectations 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 distributionfunction 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-calledtime-consistency property necessary for the dynamic programming approach. Sothe standard approaches are not applicable to optimal stopping under$\mathcal{E}(X).$ In this paper, we prove a novel representation, which relatesthe solution of an optimal stopping problem under distorted expectation to thesequence of standard optimal stopping problems and hence makes the applicationof the standard dynamic programming-based approaches possible. Furthermore, bymeans of the well known Kusuoka representation, we extend our results tooptimal stopping under general law invariant coherent risk measures. Finally,based on our novel representations, we develop several Monte Carloapproximation algorithms and illustrate their power for optimal stopping underAverage Value at Risk and the absolute semideviation risk measures.

Optimal stopping under probability distortions and law invariant coherent risk measures