  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
[19] and BSDEs with constrained jumps introduced in [14]. More remarkably, the
solution of a multidimensional Brownian reflected BSDE studied in [11] and [13]
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
