Elliptic PDEs with distributional drift and backward SDEs driven by a c{\`a}dl{\`a}g martingale with random terminal time

TL;DR

This paper develops a framework for solving semilinear elliptic PDEs with distributional drift by linking them to backward SDEs driven by càdlàg martingales, addressing existence, uniqueness, and boundary conditions.

Contribution

It introduces a generalized notion of elliptic PDEs with distributional drift and connects solutions to backward SDEs driven by càdlàg martingales with random terminal time.

Findings

Existence and uniqueness of $C^1$ solutions for PDEs with generalized drift.

Establishment of a link between PDE solutions and backward SDEs with random terminal time.

Discussion on the uniqueness of BSDEs driven by general càdlàg martingales.

Abstract

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 $C^1$. 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 a BSDE with random terminal time when the driving process is a general c{\`a}dl{\`a}g martingale.