Risk-sensitive and robust escape control for degenerate diffusion processes

This paper considers the problem of controlling a possibly degenerate small noise diffusion so as to prevent it from leaving a prescribed set. The cri terion of interest is a risk-sensitive version of the mean escape time crit erion. Using a general representation formula, this criterion is expressed as the upper value of a stochastic differential game. It is shown that in t he small noise limit this upper value converges to the value of an associat ed deterministic differential game. Our approach differs from standard FDE 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 f act, strategies that are nearly maximizing for the robust problem can be us ed to define nearly minimizing controls for the risk-sensitive control prob lem for sufficiently small epsilon > 0. The result provides a canonical exa mple of the use of variational representations in connecting risk-sensitive and robust control.