Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles

TL;DR

This paper develops a framework for solving complex nonlinear variational inequalities with interconnected obstacles, using backward SDEs to construct and verify unique viscosity solutions with polynomial growth.

Contribution

It introduces a novel approach employing penalized backward SDEs to establish existence and uniqueness of viscosity solutions for interconnected variational inequalities.

Findings

Constructed continuous viscosity solutions with polynomial growth.

Established a comparison principle ensuring uniqueness.

Linked variational inequalities to a multiple modes switching game.

Abstract

We study a general class of nonlinear second-order variational inequalities with interconnected bilateral obstacles, related to a multiple modes switching game. Under rather weak assumptions, using systems of penalized unilateral backward SDEs, we construct a continuous viscosity solution of polynomial growth. Moreover, we establish a comparison result which in turn yields uniqueness of the solution.