Existence for nonlinear finite dimensional stochastic differential equations of subgradient type

Abstract

One proves via variational techniques the existence and uniqueness of a strong solution to the stochastic differential equation \begin{document}$dX+{\partial} {\varphi} (t,X)dt\ni \sum\limits^N_{i = 1}σ_i(X)d{β}_i, X(0) = x,$\end{document} where \begin{document}${\partial}{\varphi} :{\mathbb{R}}^d\to2^{{\mathbb{R}}^d}$\end{document} is the subdifferential of a convex function \begin{document}${\varphi}:{\mathbb{R}}^d\to{\mathbb{R}}$\end{document} and \begin{document}$σ_i∈ L({\mathbb{R}}^d,{\mathbb{R}}^d)$\end{document} , \begin{document}$1≤ d .