Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process

Shaokuan Chen,Shanjian Tang

doi:10.3934/mcrf.2015.5.401

Abstract

In this paper weinvestigate classical solution of a semi-linear system of backwardstochastic integral partial differential equations driven by aBrownian motion and a Poisson point process. By proving anItô-Wentzell formula for jump diffusions as well as an abstractresult of stochastic evolution equations, we obtain the stochasticintegral partial differential equation for the inverse of thestochastic flow generated by a stochastic differential equationdriven by a Brownian motion and a Poisson point process. Bycomposing the random field generated by the solution of a backwardstochastic differential equation with the inverse of the stochasticflow, we construct the classical solution of the system of backwardstochastic integral partial differential equations. As a result, weestablish a stochastic Feynman-Kacformula.

Highlights

Backward stochastic partial differential equations (BSPDEs) are function space-valued backward stochastic differential equations (BSDEs), the theories and applications of which can be found in [2], [4], [7], [9], [30], [31], [42], etc
They appear as the adjoint equations in the stochastic maximum principle of systems governed by stochastic partial differential equations (SPDEs) driven by a Brownian motion or driven by both a Brownian motion and a Poisson random measure
Englezos and Karatzas [10] characterized the value function of a utility maximization problem with habit formation as a classical solution of the corresponding stochastic HJB equation, which gives a concrete illustration of BSPDEs in a stochastic control context beyond the classical linear quadratic case

Summary

Introduction

Backward stochastic partial differential equations (BSPDEs) are function space-valued backward stochastic differential equations (BSDEs), the theories and applications of which can be found in [2], [4], [7], [9], [30], [31], [42], etc. A class of fully nonlinear BSPDEs, the so-called backward stochastic Hamilton-Jacobi-Bellman (HJB) equations, were proposed by Peng [32] in the study of the optimal control problems for non-Markovian cases. Meng and Tang [25] have studied the maximum principle of non-Markovian stochastic differential systems driven only by Poisson point processes and obtained a kind of backward stochastic HJB equations of jump type. As demonstrated in Tang [40], the above two facts play crucial roles in the construction of the classical solutions to BSPDEs. by the analysis of solutions of BSDEs driven by a Brownian motion and a Poisson point process, we generalize.

Notations and Preliminary results

An Ito-Wentzell formula for jump diffusions

An abstract result

Degenerate case

BSDEs with jumps

Classical solutions to BSIPDEs