Optimal Stopping of BSDEs with Constrained Jumps and Related Double Obstacle PDEs

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 for 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. In addition, the contraction argument yields existence of a new type of non-linear Snell envelope and extends the theory of probabilistic representation for PDEs.