The obstacle problem for semilinear parabolic partial integro-differential equations

TL;DR

This paper develops a probabilistic framework for the obstacle problem in semilinear parabolic PIDEs, linking solutions to RBSDEs with jumps and applying it to financial derivatives with constraints.

Contribution

It introduces a probabilistic interpretation for weak Sobolev solutions of obstacle PIDEs, connecting them with RBSDEs with jumps and financial applications.

Findings

Probabilistic interpretation for obstacle PIDEs established.

Connection between solutions and RBSDEs with jumps demonstrated.

Application to pricing and hedging in constrained financial markets.

Abstract

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Leandre (1985) concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs. Moreover it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.