Stochastic Variational Inequalities on Non-Convex Domains

TL;DR

This paper establishes existence, uniqueness, and stability of solutions for stochastic variational inequalities involving non-convex domains and semiconvex functions, extending classical results to more general settings.

Contribution

It proves the well-posedness of stochastic variational inequalities with non-convex domains and semiconvex functions, including solution continuity and tightness criteria.

Findings

Existence and uniqueness of solutions for the deterministic differential inclusion.

Continuity of the solution map with respect to the input function.

Extension of deterministic results to stochastic variational inequalities driven by Brownian motion.

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 $dK_{t}(\omega)\in\partial^-\varphi( X_t (\omega))(dt)$.