In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209–227.] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.

在本文中，研究了一类非马尔可夫前向后向双随机系统。利用泛函 Itô（或路径相关）微积分技术，建立了系统与相关路径相关拟线性随机偏微分方程（简称 SPDE）之间的关系，并得到著名的非线性随机 Feynman-Kac Pardoux 和 Peng 的公式 [向后双随机微分方程和拟线性 SPDE 系统，Probab。理论相关 Fields 98 (1994), pp. 209–227.] 被发展到非马尔可夫情况。此外，我们获得了前向-后向双随机系统解的可微性和路径相关 SPDE 解的一些性质。