Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacles: A BSDEs Approach

Abstract

In the paper we apply methods of the theory of backward stochastic differential equations to prove existence, uniqueness and stochastic representation of solutions of the Cauchy problem for semilinear parabolic equation in divergence form with two time-dependent obstacles. We consider two quite different cases: problems with distinct quasi-continuous obstacles and with irregular obstacles satisfying the so called Mokobodzki condition. As an application we also generalize the Lewy-Stampacchia inequality to non-Radon measures and give new existence result for the Dynkin game problem. © 2012 The Author(s).