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

Hasib Uddin Molla, Matthew Backhouse, Ankit Banarjee, Jinniao Qiu

TL;DR

This work extends deep BSDE methods to fully coupled non-Markovian forward-backward stochastic differential equations (FBSDEs), where the forward coefficients can be random and depend on backward components $Y$ and $Z$ as well as an exogenous process $V$. It provides error estimates and convergence analysis for the discretized scheme, and frames the solution as a stochastic optimization problem solvable by neural networks that approximate the decoupling field in a non-Markovian setting. Two algorithms are proposed to solve the non-Markovian coupled FBSDEs, using forward Euler dynamics and time-dependent neural networks to approximate $Y$ and $Z$, with theoretical guarantees on convergence. The framework is demonstrated on utility maximization under rough volatility, illustrating practical performance and scalability in high-dimensional, non-Markovian environments.

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.

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

TL;DR

This work extends deep BSDE methods to fully coupled non-Markovian forward-backward stochastic differential equations (FBSDEs), where the forward coefficients can be random and depend on backward components and as well as an exogenous process . It provides error estimates and convergence analysis for the discretized scheme, and frames the solution as a stochastic optimization problem solvable by neural networks that approximate the decoupling field in a non-Markovian setting. Two algorithms are proposed to solve the non-Markovian coupled FBSDEs, using forward Euler dynamics and time-dependent neural networks to approximate and , with theoretical guarantees on convergence. The framework is demonstrated on utility maximization under rough volatility, illustrating practical performance and scalability in high-dimensional, non-Markovian environments.

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 and . 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.

Paper Structure

This paper contains 17 sections, 9 theorems, 175 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Assumptions
  4. Solutions to FBSDEs
  5. Neural Network Approximation
  6. Numerical Scheme and Simulation Error for Non-Markovian Coupled-FBSDE
  7. A Convergence Analysis
  8. Algorithm for Coupled non-Markovian FBSDEs
  9. Algorithm 1
  10. Algorithm 2
  11. Numerical Examples with Applications
  12. Example 1
  13. Example 2: Utility Maximization with Fixed Consumption Rate
  14. Conclusion
  15. Proof of Lemma \ref{['R3_fw_est']}
  16. ...and 2 more sections

Key Result

Theorem 2.2

Suppose all the processes involved are one-dimensional and that Assumptions A1-wellp -ass-1 hold. Then it holds that

Figures (6)

  • Figure 1: Train and Test Loss During Epochs ($\gamma=0.8, \theta=0.3,a=0.025$)
  • Figure 2: Sample Paths with $\gamma=0.8, \theta=0.3,a=0.0$
  • Figure 3: Sample Paths with $\gamma=0.8, \theta=0.3,a=0.025$
  • Figure 4: Sample Paths with $\gamma=0.2, \theta=0.3,a=0.025$
  • Figure 5: Sample Paths with $\gamma=0.8, \theta=0.1,a=0.025$
  • ...and 1 more figures

Theorems & Definitions (20)

  • Example 2.1
  • Theorem 2.2: Wellpossedness of FBSDE
  • Remark 2.1
  • Theorem 2.3: Continuity of Solution
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.2
  • Theorem 2.6
  • Lemma 2.7
  • ...and 10 more