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

Romuald Elie (1) , Idris Kharroubi (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the com-ponents 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 [22] and BSDEs with constrained jumps introduced in [17]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [16] and [14] can also be represented via a well chosen one-dimensional con-strained 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 control; Switching problems; BSDE with jumps; Reflected BSDE MSC Classification (2000): 93E20; 60H30; 60J75

Domains: Probability [math.PR]

Romuald Elie, Idris Kharroubi. Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections. ESAIM: Probability and Statistics, 2014, 18, pp.233 - 250. ⟨10.1051/ps/2013036⟩. ⟨hal-01103771v4⟩