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

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 [S. Peng and M. Xu, Preprint. (2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H. Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Hu and S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121] and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010) 403–426] 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.

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