A Deep Learning-Based Method for Fully Coupled Non-Markovian FBSDEs with Applications

TL;DR

This paper develops a deep learning approach for solving fully coupled non-Markovian FBSDEs, providing error analysis and demonstrating practical utility in financial utility maximization problems with rough volatility.

Contribution

It introduces a novel deep learning method for fully coupled non-Markovian FBSDEs, including error estimates and applications to complex financial models.

Findings

Effective numerical solutions for non-Markovian FBSDEs

Error bounds and convergence guarantees

Successful application to utility maximization with rough volatility

Abstract

In this work, we extend deep learning-based numerical methods to fully coupled forward-backward stochastic differential equations (FBSDEs) within a non-Markovian framework. Error estimates and convergence are provided. In contrast to the existing literature, our approach not only analyzes the non-Markovian framework but also addresses fully coupled settings, in which both the drift and diffusion coefficients of the forward process may be random and depend on the backward components $Y$ and $Z$. Furthermore, we illustrate the practical applicability of our framework by addressing utility maximization problems under rough volatility, which are solved numerically with the proposed deep learning-based methods.