Backward SDEs with constrained jumps and quasi-variational inequalities

TL;DR

This paper introduces a class of constrained backward stochastic differential equations driven by Brownian motion and Poisson jumps, establishing their unique solutions and linking them to quasi-variational inequalities for impulse control problems.

Contribution

It provides existence, uniqueness, and a probabilistic representation of solutions to constrained BSDEs and QVIs, including a new stochastic formula for impulse control value functions.

Findings

Proved existence and uniqueness of minimal solutions for constrained BSDEs.

Characterized solutions as viscosity solutions of QVIs.

Suggested a numerical scheme via penalized BSDE simulation.

Abstract

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.