Risk-Sensitive and Robust Escape Control for Degenerate Diffusion Processes

Abstract

This paper considers the problem of controlling a possibly degenerate small noise diffusion so as to prevent it from leaving a prescribed set. The criterion of interest is a risk-sensitive version of the mean escape time criterion. Using a general representation formula, this criterion is expressed as the upper value of a stochastic differential game. It is shown that in the small noise limit this upper value converges to the value of an associated deterministic differential game. Our approach differs from standard PDE approaches in a number of ways. For example, the upper game representation allows one to relate directly the prelimit and the limit controls and, in fact, strategies that are nearly maximizing for the robust problem can be used to define nearly minimizing controls for the risk-sensitive control problem for sufficiently small e>0. The result provides a canonical example of the use of variational representations in connecting risk-sensitive and robust control.