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A forward scheme with machine learning for forward-backward SDEs with jumps by decoupling jumps

Reiichiro Kawai, Riu Naito, Toshihiro Yamada

TL;DR

The paper addresses numerical approximation of forward-backward stochastic differential equations with jumps by decoupling the jump component and solving a sequence of linear PDEs that converge to the original jump-augmented FBSDE. It introduces a forward scheme that replaces nested conditional expectations with discretized forward dynamics, and implements the scheme via least-squares Monte Carlo and neural networks to handle both low- and high-dimensional problems. Theoretical contributions include a jump-decoupling result, a recursive linear PDE framework, and exponential convergence with an error bound that vanishes as jump intensity decreases. Numerically, the approach demonstrates accurate, scalable performance up to 100 dimensions across multiple test cases, suggesting practical applicability to high-dimensional stochastic control and finance problems with jumps.

Abstract

Forward-backward stochastic differential equations (FBSDEs) have been generalized by introducing jumps for better capturing random phenomena, while the resulting FBSDEs are far more intricate than the standard one from every perspective. In this work, we establish a forward scheme for potentially high-dimensional FBSDEs with jumps, taking a similar approach to [Bender and Denk, 117 (2007), Stoch. Process. Their Appl., pp.1793-1812], with the aid of machine learning techniques for implementation. The developed forward scheme is built upon a recursive representation that decouples random jumps at every step and converges exponentially fast to the original FBSDE with jumps, often requiring only a few iterations to achieve sufficient accuracy, along with the error bound vanishing for lower jump intensities. The established framework also holds novelty in its neural network-based implementation of a wide class of forward schemes for FBSDEs, notably whether with or without jumps. We provide an extensive collection of numerical results, showcasing the effectiveness of the proposed recursion and its corresponding forward scheme in approximating high-dimensional FBSDEs with jumps (up to 100-dimension) without directly handling the random jumps.

A forward scheme with machine learning for forward-backward SDEs with jumps by decoupling jumps

TL;DR

The paper addresses numerical approximation of forward-backward stochastic differential equations with jumps by decoupling the jump component and solving a sequence of linear PDEs that converge to the original jump-augmented FBSDE. It introduces a forward scheme that replaces nested conditional expectations with discretized forward dynamics, and implements the scheme via least-squares Monte Carlo and neural networks to handle both low- and high-dimensional problems. Theoretical contributions include a jump-decoupling result, a recursive linear PDE framework, and exponential convergence with an error bound that vanishes as jump intensity decreases. Numerically, the approach demonstrates accurate, scalable performance up to 100 dimensions across multiple test cases, suggesting practical applicability to high-dimensional stochastic control and finance problems with jumps.

Abstract

Forward-backward stochastic differential equations (FBSDEs) have been generalized by introducing jumps for better capturing random phenomena, while the resulting FBSDEs are far more intricate than the standard one from every perspective. In this work, we establish a forward scheme for potentially high-dimensional FBSDEs with jumps, taking a similar approach to [Bender and Denk, 117 (2007), Stoch. Process. Their Appl., pp.1793-1812], with the aid of machine learning techniques for implementation. The developed forward scheme is built upon a recursive representation that decouples random jumps at every step and converges exponentially fast to the original FBSDE with jumps, often requiring only a few iterations to achieve sufficient accuracy, along with the error bound vanishing for lower jump intensities. The established framework also holds novelty in its neural network-based implementation of a wide class of forward schemes for FBSDEs, notably whether with or without jumps. We provide an extensive collection of numerical results, showcasing the effectiveness of the proposed recursion and its corresponding forward scheme in approximating high-dimensional FBSDEs with jumps (up to 100-dimension) without directly handling the random jumps.

Paper Structure

This paper contains 18 sections, 3 theorems, 70 equations, 9 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The recursive representation
  4. Decoupling jumps from FBSDEs with jumps
  5. Recursion by linear PDEs
  6. Convergence and error analysis
  7. The forward scheme with machine learning techniques
  8. Discretized recursion without nested conditional expectations
  9. Implementation by least squares Monte Carlo
  10. Implementation by neural networks
  11. Numerical illustrations
  12. First example
  13. One-dimensional case
  14. Multi-dimensional case
  15. Second example
  16. ...and 3 more sections

Key Result

Theorem 3.1

It holds that for every $(t,{\bf x})\in [0,T]\times \mathbb{R}^{d_{\bf x}}$,

Figures (9)

  • Figure 1: Flow of the proposed forward scheme
  • Figure 2: Illustration of the absence of nested conditional expectations in the proposed forward scheme
  • Figure 3: Illustration of the presence of nested conditional expectations in a typical backward Euler scheme
  • Figure 4: The approximation errors $|u_{\rm ref}(0,x)-w_m(0,x)|$ for $m\in \{0,1,\cdots,5\}$ at initial state $x=10.0$ and eleven different strikes $K\in\{5.0,6.0,\cdots,15.0\}$.
  • Figure 5: The approximation errors $|u_{\rm ref}(0,{\bf x})-w_m(0,{\bf x})|$ for $m\in \{0,1,\cdots,5\}$ at initial state ${\bf x}=10.0\mathbbm{1}_d$ and eleven different strikes $K\in\{5.0,6.0,\cdots,15.0\}$.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Remark 4.1
  • proof : Proof of Theorem \ref{['theorem Ytilde']}
  • proof : Proof of Theorem \ref{['theorem section 3.2']}
  • proof : Proof of Theorem \ref{['thm1']}