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

Luca Di Persio, Matteo Garbelli, Adrian Zalinescu

TL;DR

This work develops a probabilistic, path-dependent framework to represent mild solutions of nonlinear PDKEs with Neumann boundary via a decoupled forward–backward SDE system in which the forward component is reflected and the backward component features time-delayed generators. A transition semigroup on path spaces is constructed and a Feynman–Kac representation is established, linking $u(t,\psi)$ to the backward process with $Y^{t,\psi}(s)=u(s,\mathbf X^{t,\psi})$ and $Z^{t,\psi}(s)=\nabla^{\sigma}u(s,\mathbf X^{t,\psi})$ in the generalized gradient sense. The authors introduce a penalization scheme to justify the reflected dynamics and derive the associated infinitesimal generator $\mathcal L_t$, connecting the semigroup to the Kolmogorov operator under Neumann conditions. Overall, the paper provides existence/uniqueness and a robust representation for path-dependent PDEs with boundary constraints, broadening the applicability of FBSDE methods to delayed, boundary-influenced problems. The methodology has potential implications for stochastic control and financial modeling where path dependence, delays, and reflections are essential.

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.

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

TL;DR

This work develops a probabilistic, path-dependent framework to represent mild solutions of nonlinear PDKEs with Neumann boundary via a decoupled forward–backward SDE system in which the forward component is reflected and the backward component features time-delayed generators. A transition semigroup on path spaces is constructed and a Feynman–Kac representation is established, linking to the backward process with and in the generalized gradient sense. The authors introduce a penalization scheme to justify the reflected dynamics and derive the associated infinitesimal generator , connecting the semigroup to the Kolmogorov operator under Neumann conditions. Overall, the paper provides existence/uniqueness and a robust representation for path-dependent PDEs with boundary constraints, broadening the applicability of FBSDE methods to delayed, boundary-influenced problems. The methodology has potential implications for stochastic control and financial modeling where path dependence, delays, and reflections are essential.

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.
Paper Structure (11 sections, 10 theorems, 114 equations)

This paper contains 11 sections, 10 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. The Forward Reflected SDE
  4. Tower Property
  5. Markov property and Transition Semigroup
  6. The Forward Backward system
  7. The Feynman-Kac formula
  8. The generalized directional gradient
  9. Mild solution and connections to the Kolmogorov Equation
  10. The convergence of the penalized PDE via penalization scheme
  11. The infinitesimal generator

Key Result

Proposition 3.1

Let assumption (A$_1$) hold and let the domain $G\subset\mathbb{R}^d$ be open, bounded and convex. Then equation forward2 admits a unique strong solution $\mathbf X^{t,\psi}\in\mathbb{S}^2(\bar{G})\times\mathbb{S}^2(\mathbb{R}^d)$.

Theorems & Definitions (24)

  • Proposition 3.1: Existence and Uniqueness
  • Proposition 3.2
  • proof
  • Proposition 3.3: Tower Property
  • proof
  • Proposition 3.4: Markov Property
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2: Tower property for $Y$ and $Z$
  • ...and 14 more