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An Euler scheme for BSDEs via the Wiener chaos decomposition

Pere Díaz Lozano, Giulia Di Nunno

TL;DR

This paper develops a fully implementable Euler-type scheme for backward SDEs by leveraging a finite Wiener chaos decomposition to approximate conditional expectations and martingale terms, enabling non-Markovian terminal conditions. It provides a rigorous convergence analysis that accounts for chaos truncation and Monte Carlo estimation errors, and presents practical guidance on choosing the chaos order, basis size, and sample size. The authors demonstrate the method on linear and nonlinear drivers in up to five dimensions, showing competitive accuracy and runtime compared with a Picard-based chaos approach. Overall, the work offers a robust, scalable alternative for numerically solving BSDEs with general terminal data and explicit error control.

Abstract

The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a forward-backward (Markovian) structure, our approach accommodates arbitrary $\mathcal{F}_T$-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis and illustrate the method on several numerical examples.

An Euler scheme for BSDEs via the Wiener chaos decomposition

TL;DR

This paper develops a fully implementable Euler-type scheme for backward SDEs by leveraging a finite Wiener chaos decomposition to approximate conditional expectations and martingale terms, enabling non-Markovian terminal conditions. It provides a rigorous convergence analysis that accounts for chaos truncation and Monte Carlo estimation errors, and presents practical guidance on choosing the chaos order, basis size, and sample size. The authors demonstrate the method on linear and nonlinear drivers in up to five dimensions, showing competitive accuracy and runtime compared with a Picard-based chaos approach. Overall, the work offers a robust, scalable alternative for numerically solving BSDEs with general terminal data and explicit error control.

Abstract

The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a forward-backward (Markovian) structure, our approach accommodates arbitrary -measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis and illustrate the method on several numerical examples.

Paper Structure

This paper contains 33 sections, 29 theorems, 163 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notation
  4. BSDEs and their numerical approximation
  5. A numerical scheme for BSDEs
  6. Convergence rate of the scheme
  7. The Wiener chaos decomposition
  8. Wiener chaos decomposition and Hermite polynomials
  9. Wiener chaos decomposition and iterated integrals
  10. Error given by truncating the order of the chaos
  11. Error due to the truncation of the basis
  12. Description of the algorithm
  13. Description of the truncated basis
  14. Description of the implementation
  15. Wiener chaos decomposition formulae
  16. ...and 18 more sections

Key Result

Theorem 2.3

Under Assumption assumption 1, the BSDE (BSDE) has a unique solution $(Y,Z)$ in $H_{T}^{2}(\mathbb{R}) \times H_{T}^{2}(\mathbb{R}^{d})$, which actually belongs to $S_{T}^{2}(\mathbb{R}) \times H_{T}^{2}(\mathbb{R}^{d})$.

Figures (5)

  • Figure 1: Histogram that compares Picard vs Euler. Left: comparison of the $Y$ component evaluated at $t=0$. Right: comparison of the $Z$ component evaluated at $t=0$.
  • Figure 2: Dependence on the number of time steps $m$. Left: $Y_0$. Right: $Z_0$.
  • Figure 3: Dependence on the chaos basis size $M$. Left: $Y_0$. Right: $Z_0$.
  • Figure 4: Dependence on the chaos order $P$. Left: $Y_0$. Right: $Z_0$.
  • Figure 5: Sample paths from the numerical approximation of the BSDE. Left: approximation $Y$. Right: approximation of the hedging strategies $H$.

Theorems & Definitions (54)

  • Definition 2.1
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Proposition 2.11
  • Theorem 2.12
  • ...and 44 more