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This paper is dedicated to the analysis of backward stochastic differentialequations (BSDEs) with jumps, subject to an additional global constraintinvolving all the components of the solution. We study the existence anduniqueness of a minimal solution for these so-called constrained BSDEs withjumps via a penalization procedure. This new type of BSDE offers a nice andpractical unifying framework to the notions of constrained BSDEs presented in[19] and BSDEs with constrained jumps introduced in [14]. More remarkably, thesolution of a multidimensional Brownian reflected BSDE studied in [11] and [13]can also be represented via a well chosen one-dimensional constrained BSDE withjumps.This last result is very promising from a numerical point of view for theresolution of high dimensional optimal switching problems and more generallyfor systems of coupled variational inequalities

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