Stochastic Variational Inequalities on Non-Convex Domains

Abstract

The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+{\partial }^{-}\ddot{I}†(x(t))(dt)\ni dm(t),t>0$, $x(0)=x_0$, where $m:{\mathbb{R}}_{+}\to {\mathbb{R}}^d$ is a continuous function and ${\partial }^{-}\ddot{I}†$ is the Fréchet subdifferential of a semiconvex function $\ddot{I}†$; the domain of $\ddot{I}†$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\mapsto x:C([0,T];{\mathbb{R}}^d)\to C([0,T];{\mathbb{R}}^d)$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t=\hat{I}¾+{\int }_0^{t}F(s,X_s)ds+{\int }_0^{t}G(s,X_s)d{B}_s,t\ge 0$, with $d{K}_t(\ddot{I}‰)\in {\partial }^{-}\ddot{I}†(X_t(\ddot{I}‰))(dt)$.

39 pages