Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions for semilinear parabolic PDEs with obstacles, using reflected backward stochastic differential equations for stochastic representation.

Contribution

It introduces a novel stochastic representation for obstacle problems in divergence form PDEs via reflected BSDEs, with new regularity and approximation results.

Findings

Unique solutions exist under natural conditions.

Stochastic representation via reflected BSDEs is established.

Regularity and approximation properties are proven.

Abstract

We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We prove also regularity properties and approximation results for solutions of the problem.