Constrained Backward SDEs with Jumps: Application to Optimal Switching

This paper enlarges the class of backward stochastic differential equation (BSDE) with jumps, adding some general constraints on all the components of the solution. Via a penalization procedure, we provide an existence and uniqueness result for these so-called constrained BSDEs with jumps. This new type of BSDE offers a nice and practical unifying framework to represent and generalize the notions of constrained BSDEs studied in [16], BSDEs with constrained jumps introduced recently by [12], as well as multidimensional BSDEs with oblique reflection presented in [11] and [9]. For example, a switching problem, represented by a multidimensional BSDE with oblique reflection, see [11], can be directly solved through a one dimensional constrained BSDE with jumps. This result is very promising from a numerical point of view for the reso-lution of high dimensional switching problems. All the arguments presented here rely on probabilistic tools. This allows in particular to represent non-Markovian switch-ing problems, where, for example, the switching regime influences the dynamics of the underlined diffusion