Viscosity solutions of systems of variational inequalities with interconnected bilateral obstacles of non-local type

TL;DR

This paper develops a framework for solving complex systems of variational inequalities with interconnected obstacles and non-local terms, using stochastic methods to establish unique viscosity solutions relevant to jump-diffusion games.

Contribution

It introduces a novel approach combining penalized reflected backward SDEs with jumps to construct unique viscosity solutions for non-local variational inequality systems.

Findings

Existence of continuous viscosity solutions for the systems.

Uniqueness of solutions within polynomial growth class.

Application to multiple modes zero-sum switching games.

Abstract

In this paper, we study systems of nonlinear second-order variational inequalities with interconnected bilateral obstacles with non-local terms. They are of min-max and max-min types and related to a multiple modes zero-sum switching game in the jump-diffusion model. Using systems of penalized reflected backward SDEs with jumps and unilateral interconnected obstacles, and their associated deterministic functions, we construct for each system a continuous viscosity solution which is unique in the class of functions with polynomial growth.