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Euler-Maruyama method for distribution dependent stochastic differential equation driven by multiplicative fractional Brownian motion

Guangjun Shen, Jiangpeng Wang, Xuekang Zhang

TL;DR

This work investigates distribution-dependent SDEs driven by multiplicative fractional Brownian motion, a non-Markovian setting challenging for analysis and simulation. It develops a rigorous framework using Lions derivatives and a Hölder space for probability-measure paths to establish well-posedness and propagation of chaos for mean-field interactions. An Euler–Maruyama scheme is formulated for the interacting particle system with strong convergence guarantees that depend on the time-step and the Hurst parameter. Numerical experiments across various H values corroborate the theoretical rates and demonstrate the method's effectiveness for simulating mean-field systems under fractional noise.

Abstract

In this paper, we establish the propagation of chaos and Euler-Maruyama method of DDSDE driven by multiplicative fractional Brownian motion with Hurst parameter $H\in (\frac{\sqrt{5}-1}{2},1)$. We have not only obtained an upper bound for the error of the Euler-Maruyama method but also verified the correctness of this result via systematic numerical simulation experiments.

Euler-Maruyama method for distribution dependent stochastic differential equation driven by multiplicative fractional Brownian motion

TL;DR

This work investigates distribution-dependent SDEs driven by multiplicative fractional Brownian motion, a non-Markovian setting challenging for analysis and simulation. It develops a rigorous framework using Lions derivatives and a Hölder space for probability-measure paths to establish well-posedness and propagation of chaos for mean-field interactions. An Euler–Maruyama scheme is formulated for the interacting particle system with strong convergence guarantees that depend on the time-step and the Hurst parameter. Numerical experiments across various H values corroborate the theoretical rates and demonstrate the method's effectiveness for simulating mean-field systems under fractional noise.

Abstract

In this paper, we establish the propagation of chaos and Euler-Maruyama method of DDSDE driven by multiplicative fractional Brownian motion with Hurst parameter . We have not only obtained an upper bound for the error of the Euler-Maruyama method but also verified the correctness of this result via systematic numerical simulation experiments.

Paper Structure

This paper contains 11 sections, 11 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation.
  4. Fractional integral and derivative
  5. fractional Brownian motion
  6. The Lions derivative
  7. Hölder space of probability measure paths
  8. Well-posedness
  9. Propagation of chaos
  10. EM method
  11. Numerical examples

Key Result

Lemma 2.2

Let $(\Omega,\mathscr{F},\mathbb{P})$ be an atomless probability space and $X,Y\in L^2(\Omega\rightarrow\mathbb R^d,\mathbb{P})$. If $f\in C^{(1,0)}(\mathscr{P}_2(\mathbb R^d))$, then

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 4.1
  • ...and 4 more