Viscosity solutions of second order integral-partial differential equations: A new result

TL;DR

This paper establishes existence and uniqueness of continuous polynomial growth viscosity solutions for a class of second order IPDEs with arbitrary Lévý measures, using backward stochastic differential equations with jumps.

Contribution

It introduces a new approach to solve second order IPDEs without the usual monotonicity assumption, broadening applicability to arbitrary Lévý measures.

Findings

Proves existence and uniqueness of solutions.

Handles arbitrary Lévý measures, not necessarily finite.

Uses backward stochastic differential equations with jumps.

Abstract

We show existence and uniqueness of a continuous with polynomial growth viscosity solution of a system of second order integral-partial differential equations (IPDEs for short) without assuming the usual monotonicity condition of the generator with respect to the jump component as in Barles et al.'s article \cite{BarlesBuckPardoux}. The L\'evy measure is arbitrary and not necessarily finite. In our study the main tool we used is the notion of backward stochastic differential equations with jumps.