Backward Stochastic Differential Equations and Feynman-Kac Formula for Multidimensional L\'{e}vy Processes, with Applications in Finance

TL;DR

This paper establishes the existence and uniqueness of solutions for multidimensional backward stochastic differential equations driven by Lévy processes, extending classical formulas like Feynman-Kac for application in multidimensional Lévy-based financial models.

Contribution

It introduces new mathematical results on BSDEs driven by Lévy processes and applies these to derive multidimensional Feynman-Kac formulas for option pricing.

Findings

Proves existence and uniqueness of solutions for multidimensional BSDEs driven by Lévy processes.

Derives a multidimensional Feynman-Kac formula applicable to Lévy markets.

Provides a PDE integral equation analogue to Black-Scholes for Lévy processes.

Abstract

In this paper we show the existence and form uniqueness of a solution for multidimensional backward stochastic differential equations driven by a multidimensional L\'{e}vy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for multidimensional L\'{e}vy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equations provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a multidimensional L\'{e}vy market.