Extended Backward Stochastic Volterra Integral Equations, Quasilinear Parabolic Equations, and Feynman-Kac Formula

TL;DR

This paper explores the connection between extended backward stochastic Volterra integral equations and non-local quasilinear parabolic PDEs, generalizing the nonlinear Feynman-Kac formula and establishing well-posedness and regularity results.

Contribution

It introduces the concept of extended backward stochastic Volterra integral equations and proves their well-posedness and regularity, linking them to non-local parabolic PDEs.

Findings

Established well-posedness of EBSVIEs.

Derived regularity results using Malliavin calculus.

Connected solutions of EBSVIEs to non-local PDEs, generalizing Feynman-Kac.

Abstract

In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local parabolic partial differential equations (PDEs, for short), which generalizes the famous nonlinear Feynman-Kac formula in Pardoux{Peng [21].