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.