Optimal stopping with nonlinear expectation: geometric and algorithmic solutions

TL;DR

This paper develops geometric and algorithmic methods to solve risk-sensitive Markovian optimal stopping problems, extending classical linear potential theory to nonlinear expectations with practical computational algorithms.

Contribution

It introduces a geometric approach to nonlinear potential theory and provides an algorithm for constructing the value function with manageable computational complexity.

Findings

Value function characterized as infimum of dominating functions

Algorithm enables practical computation of the value function

Extension of classical linear potential theory to nonlinear expectations

Abstract

We use the geometry of suitably generalised potentials to solve risk-sensitive Markovian optimal stopping problems. As in the linear case due to Dynkin and Yushkievich (1967), the value function is the pointwise infimum of those functions which dominate the gain function. An emphasis is placed on geometric and pathwise arguments, rather than exploiting convexity, positive homogeneity or related analytical properties. An algorithm is provided to construct the value function at the computational cost of a two-dimensional search.