Viscosity solutions for second order integro-differential equations without monotonicity conditions: The Probabilistic Approach

TL;DR

This paper proves a new uniqueness result for viscosity solutions of certain integro-partial differential equations without requiring the traditional monotonicity assumption, using a probabilistic approach via backward stochastic differential equations with jumps.

Contribution

It introduces a novel method that relaxes the monotonicity condition on the driver in IPDEs, leveraging the link with BSDEs with jumps for establishing uniqueness.

Findings

Established uniqueness of viscosity solutions without monotonicity assumptions.

Extended results to IPDEs with obstacle.

Validated the probabilistic approach for non-monotone IPDEs.

Abstract

In this paper, we establish a new uniqueness result of a (continuous) viscosity solution for some integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver, assumption which is classically assumed in the literature of viscosity solution of equation with a non local term. Our method strongly relies on the link between IPDEs and backward stochastic differential equations (BSDEs in short) with jumps for which we already know that the solution exists and is unique. In the second part of the paper, we deal with the IPDE with obstacle and we obtain similar results.