On the Solutions of a Stochastic Control System

The control system considered in this paper is modeled by the stochastic differential equation \[dx(t,\omega ) = f(t,x( \cdot ,\omega ),u(t,\omega ))dt + dB(t,\omega ),\] where B is n-dimensional Brownian motion, and the control u is a nonanticipative functional of $x( \cdot ,\omega )$ taking its values in a fixed set U. Under various conditions on f it is shown that for every admissible control a solution is defined whose law is absolutely continuous with respect to the Wiener measure $\mu $, and the corresponding set of densities on the space C forms a strongly closed, convex subset of $L^1 (C,\mu )$. Applications of this result to optimal control and two-person, zero-sum differential games are noted. Finally, an example is given which shows that in the case where only some of the components of x are observed, the set of attainable densities is not weakly closed in $L^1 (C,\mu )$.