Mild Solutions for Path-Dependent Parabolic PDEs with Neumann Boundary Conditions via Generalized BSDEs

TL;DR

This paper establishes a link between a class of generalized backward stochastic differential equations with reflection and non-linear path-dependent PDEs with Neumann boundary conditions, providing a new probabilistic representation.

Contribution

It introduces a novel FBSDE system with time delays and reflection, connecting it to path-dependent PDEs with Neumann boundary conditions.

Findings

Representation formula linking FBSDEs and PDEs

Extension of BSDE theory to path-dependent cases

New probabilistic methods for boundary value problems

Abstract

We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form of a Generalized BSDE. We establish the connection between this FBSDE system and non-linear path-dependent PDEs with Neumann boundary conditions by deriving a representation formula.