Weak Solutions and Optimal Control for Multivalued Stochastic Differential Equations

In this paper we first prove the existence of a weak solution to a finite dimensional multivalued stochastic differential equation of the form dX(t) + A(X(t)) dt (sic) b (t, X) dt + sigma(t, X) dBt, t is an element of[0, T], where A is a maximal monotone operator, and the coefficients b and sigma are continuous functionals of the state variable. The main tool used is the martingale problem approach. Secondly we are concerned with a control problem where the system is driven by a similar equation and the control policy takes its values in a compact space. Using the martingale problem formulation, we show the existence of an optimal relaxed control. Under some supplementary hypotheses of convexity on the coefficients, we prove the existence of an optimal control for the initial problem.