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

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].

References in corpus (4)

• On viscosity solutions of path dependent PDEs
• Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps
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• Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

Cited by in corpus (1)

• Time-Inconsistent Stochastic Optimal Control Problems and Backward Stochastic Volterra Integral Equations