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

Article Contents

2015, Volume 5, Issue 3: 401-434. Doi: 10.3934/mcrf.2015.5.401

This issue Previous Article Sparse initial data identification for parabolic PDE and its finite element approximations Next Article Strong law of large numbers for upper set-valued and fuzzy-set valued probability

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

Shaokuan Chen^1, and
Shanjian Tang^2,

1.
Institute of Mathematical Finance and Department of Finance and Control Sciences, School of Mathematical Science, Fudan University, Shanghai 200433
2.
Institute of Mathematical Finance and Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: July 2014

Revised: January 2015

Published: September 2015

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Abstract

In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow, we construct the classical solution of the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.

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Mathematics Subject Classification: Primary: 60H10, 60H20; Secondary: 35R09.

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