Strong solutions of semilinear parabolic equations with measure data and generalized backward stochastic differential equations

TL;DR

This paper establishes a link between strong solutions of certain semilinear parabolic equations with measure data and solutions of generalized backward stochastic differential equations, including applications to obstacle problems.

Contribution

It introduces a stochastic representation for strong solutions of semilinear parabolic equations with measure data using generalized backward stochastic differential equations.

Findings

Representation of solutions via backward stochastic differential equations

Stochastic homographic approximation for reflected backward equations

Application to obstacle problem solutions

Abstract

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem be means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.