Minimizing escape probabilities: A large deviations approach

Abstract : This document considers the problem of controlling a possibly degenerate diffusion process so as to minimize the probability of escape over a given time interval. It is assumed that the control acts on the process through the drift coefficient, and that the noise coefficient is small. By developing a large deviations type theory for the controlled diffusion, the authors obtain several results. The limit of the normalized log of the minimum exit probability is identified as the value I of an associated (deterministic) differential game. Furthermore, the authors identify a deterministic (and epsilon independent) mapping g from the sample values epsilon w (s), 0 or = S or = t, into the control space such that if we define the control used at time t by u(t) = g(epsilon w(s), 0 or = S , or = t), then the resulting control process is progressively measurable and delta optimal (in the sense that the limit of the normalized log of the exit probability is within delta of I).