Minimizing escape probabilities: A large deviations approach

This paper 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. Developing a large deviations type of theory for the controlled diffusion produces 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, we identify a deterministic (and $\varepsilon $-independent) mapping g from the sample values $\varepsilon w(s)$, $0 \leqq s \leqq t$, into the control space such that if we define the control used at time t by $u(t) = g(\varepsilon w(s),0 \leqq s \leqq t)$, 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).