Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator L has a generalized drift. We investigate existence and uniqueness of generalized solutions of class C1. The generator L is associated with a Markov process X which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is X. Since X is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.