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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dX_{t} +A(X_{t}) dt \ni b (t, X) dt + \sigma (t, X) dB_{t}, t \ni [0, T]$$\end{document}, where A is a maximal monotone operator, and the coefficients b and σ are continuous functionals of the state variable. The main tool used is the martingale problem approach.