Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs

Abstract

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.

arXiv comment28 pages

References in corpus (5)

• Functional Itô calculus and stochastic integral representation of martingales
• On viscosity solutions of path dependent PDEs
• Stationary Solutions of SPDEs and Infinite Horizon BDSDEs
• Super-Brownian motion as the unique strong solution to an SPDE
• Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions