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: $dx(t)+\partial ^-\varphi (x(t))(dt)\ni dm(t),\ t>0$, $x(0)=x_0$, where $m:\mathbb{R}_+\rightarrow\mathbb{R}^d$ is a continuous function and $\partial^-\varphi$ is the Fr\'{e}chet subdifferential of a semiconvex function $\varphi$; the domain of $\varphi$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\mapsto x:C([0,T];\mathbb{R}^{d})\rightarrow C([0,T] ;\mathbb{R}^{d})$, 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: $X_t+K_t = \xi+\int_0^t F(s,X_{s})ds + \int_0^t G(s,X_s) dB_s,\; t\geq0$, $\;$ with $dK_{t}(\omega)\in\partial^-\varphi( X_t (\omega))(dt)$.