Forward-backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

TL;DR

This paper establishes existence and uniqueness results for fully coupled forward-backward SDEs with jumps by linking them to classical solutions of associated nonlocal quasilinear parabolic PDEs, extending classical PDE methods.

Contribution

It introduces a novel approach to solving coupled FBSDEs with jumps through the classical solutions of associated nonlocal PDEs, expanding existing PDE theory.

Findings

Proves existence and uniqueness of solutions for coupled FBSDEs with jumps.

Extends classical PDE methods to nonlocal quasilinear parabolic PDEs.

Provides explicit forms of FBSDE solutions.

Abstract

We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary value problem for non-local quasilinear parabolic second-order PDEs.