On a stochastic control problem with exit constraints

In this paper the logarithmic transformation of Fleming [1] is used to discuss a specific problem of controlled diffusions. The problem is to minimize a certain quadratic functional of the applied drift while satisfying the requirement that the place where the process exits a domain is not in a specified subset of its boundary. The main result is that the solution of this problem is given by the logarithm of a related exit probability.This research was supported in part by the Air Force Office of Scientific Research under AF-AFOSR 76-3063.