The optimal control of diffusions

Using a differentiation result of Blagovescenskii and Freidlin calculations of Bensoussan are simplified and the adjoint process identified in a stochastic control problem in which the control enters both the drift and diffusion coefficients. A martingale representation result of Elliott and Kohlmann is then used to obtain the integrand in a stochastic integral, and explicit forward and backward equations satisfied by the adjoint process are derived.This research was partially supported by NSERC under Grant A7964, the U.S. Air Force Office of Scientific Research under Contract AFOSR-86-0332, and the U.S. Army Research Office under Contract DAAL03-87-K-0102.