Adding constraints to BSDEs with Jumps: an alternative to multidimensional reflections

TL;DR

This paper introduces a new class of constrained BSDEs with jumps, establishing existence and uniqueness, and provides a unifying framework that connects various types of constrained BSDEs, with promising implications for high-dimensional problems.

Contribution

It develops a novel penalization approach for constrained BSDEs with jumps, unifies different constrained BSDE frameworks, and links multidimensional reflected BSDEs to one-dimensional constrained BSDEs with jumps.

Findings

Existence and uniqueness of minimal solutions for constrained BSDEs with jumps.

Representation of multidimensional reflected BSDEs via one-dimensional constrained BSDEs with jumps.

Potential for improved numerical methods for high-dimensional optimal switching problems.

Abstract

This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the components of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [19] and BSDEs with constrained jumps introduced in [14]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [11] and [13] can also be represented via a well chosen one-dimensional constrained BSDE with jumps.This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities