Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Switching Problem without Monotonicity Condition

TL;DR

This paper proves the existence and uniqueness of continuous viscosity solutions for systems of PDEs with interconnected obstacles and switching problems, without relying on the usual monotonicity assumptions, by linking PDEs to reflected backward stochastic differential equations.

Contribution

It introduces a novel approach that removes the monotonicity condition on the driver function, expanding the applicability of viscosity solutions to more general systems.

Findings

Established existence and uniqueness of solutions without monotonicity assumptions

Linked PDE solutions to reflected backward stochastic differential equations

Extended the theoretical framework for interconnected obstacle problems

Abstract

We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamad\`ene and Morlais's article \cite{hamadene2013viscosity}. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.