  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.
