Minimum Exit Probabilities and Differential Games

The problem considered is that of minimizing the probability that a controlled diffusion process exits from a given region of “acceptable¹¹ states during a given time interval. The controlled diffusion obeys a stochastic differential equation driven by low intensity white noise. An asymptotic estimate of Ventcel-Freidlin type is given for the minimum exit probability. The crucial parameter in this estimate is the lower value of a certain differential game.