Résumé en anglais

In this paper, we introduce a new class of BSDE generalizing and offering a unifying framework to represent the constrained ones presented in [16] or [12] as well as the oblique reflected ones studied by [11] and [9]. Via a penalization procedure, we provide an existence and uniqueness result for this new class of so-called constrained BSDEs with jumps. Remarkably, these BSDEs appear to be very convenient to represent the solution to eventually non-Markovian switching problems. As a by-product, we enlarge the class of obliquely reflected BSDE's, allowing to represent switching problems with controlled underlined diffusion.