On the Equality of Solutions of Max-Min and Min-Max Systems of Variational Inequalities with Interconnected Bilateral Obstacles

TL;DR

This paper investigates the solutions of interconnected min-max and max-min PDE systems with bilateral obstacles, showing their equivalence under certain conditions and linking them to stochastic game values and doubly reflected BSDEs.

Contribution

It establishes the equality of solutions for interconnected min-max and max-min systems under regular switching costs and connects these solutions to zero-sum switching game values.

Findings

Solutions of min-max and max-min systems coincide under regular switching costs.

The common solution relates to a doubly reflected BSDE with bilateral obstacles.

The solution characterizes the value of a zero-sum switching game.

Abstract

In this paper, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min-max and max-min systems coincide. Then, this common viscosity solution is related to a multi-dimensional doubly reflected BSDE with bilateral interconnected obstacles. Finally, its relationship with the the values of a zero-sum switching game is studied.