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

Published online by Cambridge University Press:  01 July 2014

Romuald Elie  and
Idris Kharroubi
Romuald Elie
Affiliation:
CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, Place du maréchal de Lattre de Tassigny, 75016 Paris, France. elie@ceremade.dauphine.fr,kharroubi@ceremade.dauphine.fr
Idris Kharroubi
Affiliation:
CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, Place du maréchal de Lattre de Tassigny, 75016 Paris, France. elie@ceremade.dauphine.fr,kharroubi@ceremade.dauphine.fr
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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.

Keywords

Stochastic controlswitching problemsBSDE with jumpsreflected BSDE
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 233 - 250
Copyright
© EDP Sciences, SMAI 2014

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