Reflected BSDEs and continuous solutions of parabolic obstacle problem for semilinear PDEs in divergence form

TL;DR

This paper establishes a connection between continuous solutions of semilinear parabolic PDEs with obstacles and reflected backward stochastic differential equations, providing regularity results and approximation methods.

Contribution

It introduces a stochastic representation for solutions of obstacle problems in divergence form PDEs using reflected BSDEs, under natural conditions.

Findings

Unique continuous solutions exist with stochastic representation.

Regularity properties of solutions are established.

Approximation results for solutions are proved.

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 eqution and mild conditions on the obstacle a unique continuous solution of the problem admits a stochastic representation in terms of reflected backward stochastic differential equations. We derive also some regularity properties of so- lutions and prove useful approximation results.