This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

中文翻译：

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扩展后向随机 Volterra 积分方程、拟线性抛物方程和 Feynman-Kac 公式

本文关注的是后向随机沃尔泰拉积分方程（简称 BSVIE）与一种非局部拟线性（也可能是退化）抛物线方程之间的关系。作为 BSVIE 的自然扩展，引入和研究了扩展的 BSVIE（简称 EBSVIE）。在一些温和条件下，建立了EBSVIE的适定性，并通过Malliavin演算得到了EBSVIE适应解的一些规律性结果。然后表明，用 EBSVIE 的自适应解表示的给定函数唯一地解决了某个非局部抛物方程组，它推广了 Pardoux-Peng 中著名的非线性 Feynman-Kac 公式 [Backward stochastic Differential equations and quasilinear parabolic equations偏微分方程，