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Backward SDEs with constrained jumps and Quasi-Variational Inequalities
Idris Kharroubi1, 2, Jin Ma3, Huyen Pham1, 2, Jianfeng Zhang3

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 we obtain a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.
1:  LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2:  CREST - Centre de Recherche en Économie et Statistique
3:  Department of Mathematics, University of Southern California
Backward stochastic differential equation – jump-diffusion process – jump constraints – penalization – quasi-variational inequalities – impulse control problems – viscosity solutions.