Optimal stopping of BSDEs with constrained jumps and related double obstacle PDEs

Abstract We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman–Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope, arising from an optimal stopping problem—where the underlying process is the first component in the solution to a stopped backward stochastic differential equation (BSDE) with jumps and a constraint on the jumps process—to a viscosity solution for the PDE. Leveraging this Feynman–Kac representation, we subsequently prove existence and uniqueness of viscosity solutions in the non-local setting by employing a contraction argument. This approach also introduces a novel form of non-linear Snell envelope and expands the probabilistic representation theory for PDEs.