Systems of semilinear parabolic variational inequalities with time-dependent convex obstacles

TL;DR

This paper establishes existence, uniqueness, and stochastic representations for systems of semilinear parabolic variational inequalities with time-dependent convex obstacles, using probabilistic methods.

Contribution

It introduces a novel approach combining Markov process theory and backward stochastic differential equations to analyze these inequalities.

Findings

Proved existence and uniqueness of solutions.

Provided a stochastic representation of solutions.

Demonstrated approximation via penalization method.

Abstract

We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.