Probabilistic Representation of Viscosity Solutions to Quasi-Variational Inequalities with Non-Local Drivers

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for quasi-variational inequalities with non-local drivers, extending probabilistic representations of PDEs via reflected BSDEs.

Contribution

It introduces a novel contraction-based approach to prove existence and uniqueness for QVIs with non-local drivers and related reflected BSDE systems.

Findings

Proved existence and uniqueness of viscosity solutions for QVIs with non-local drivers.

Extended probabilistic PDE representations to systems of reflected BSDEs.

Applied contraction arguments to both local and non-local settings.

Abstract

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.