On the solutions of a stochastic control system

Abstract

The control system considered in this paper is modeled by the stochastic differential equation dx(t, to) f(t, x(., o), u(t, to)) dt + dB(t, to), where B is n-dimensional Brownian motion, and the control u is a nonanticipative functional of x(., to) 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 #, and the corresponding set of densities on the space C forms a strongly closed, convex subset of LI(C, I). 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 LI(C, t).