Stochastic variational inequalities with oblique subgradients

TL;DR

This paper investigates the existence and uniqueness of solutions for a class of stochastic variational inequalities involving oblique subgradients, extending classical stochastic differential equations with oblique reflection.

Contribution

It generalizes stochastic differential equations with oblique reflection by establishing existence and uniqueness results for variational inequalities with oblique subgradients.

Findings

Proves existence and uniqueness of solutions.

Extends classical models to more general oblique subgradient cases.

Uses a deterministic approach for the existence proof.

Abstract

In this paper we will study the existence and uniqueness of the solution for the stochastic variational inequality with oblique subgradients of the following form:{l} dX_{t}+H(X_{t}) \partial \phi (X_{t}) (dt) \ni f(t,X_{t}) dt+g(t,X_{t}) dB_{t},\quad t>0,\smallskip \ X_{0}=x\in \bar{\emph{Dom}(\phi)}.% This problem is the generalization of the stochastic differential equation with oblique reflection considered by Lions and Sznitman in `84. The existence result is based on a deterministic approach; first, we prove the existence and uniqueness of the solution of a differential system with singular input.