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Numerical scheme for delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion

Shuaibin Gao, Qian Guo, Zhuoqi Liu, Chenggui Yuan

TL;DR

The paper studies delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion with $H\in(0,1)\setminus\{1/2\}$ and polynomial delay terms. It develops a Banach fixed point framework to prove wellposedness and uses a stochastic particle approach to establish propagation of chaos in $\mathcal{L}^p$, showing that the unmodified Euler-Maruyama scheme provides convergent approximations for the interacting system. It derives convergence rates for the EM scheme across $H\in(0,1)$, detailing how memory and delay influence the rates, and validates the theory through a stochastic opinion dynamics example that incorporates both extrinsic and intrinsic memory. The results enable efficient, particle-based simulation of mean-field systems with memory effects, with explicit rate assurances for practical implementations.

Abstract

This paper focuses on the numerical scheme for delay-type stochastic McKean-Vlasov equations (DSMVEs) driven by fractional Brownian motion with Hurst parameter $H\in (0,1/2)\cup (1/2,1)$. The existence and uniqueness of the solutions to such DSMVEs whose drift coefficients contain polynomial delay terms are proved by exploting the Banach fixed point theorem. Then the propagation of chaos between interacting particle system and non-interacting system in $\mathcal{L}^p$ sense is shown. We find that even if the delay term satisfies the polynomial growth condition, the unmodified classical Euler-Maruyama scheme still can approximate the corresponding interacting particle system without the particle corruption. The convergence rates are revealed for $H\in (0,1/2)\cup (1/2,1)$. Finally, as an example that closely fits the original equation, a stochastic opinion dynamics model with both extrinsic memory and intrinsic memory is simulated to illustrate the plausibility of the theoretical result.

Numerical scheme for delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion

TL;DR

The paper studies delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion with and polynomial delay terms. It develops a Banach fixed point framework to prove wellposedness and uses a stochastic particle approach to establish propagation of chaos in , showing that the unmodified Euler-Maruyama scheme provides convergent approximations for the interacting system. It derives convergence rates for the EM scheme across , detailing how memory and delay influence the rates, and validates the theory through a stochastic opinion dynamics example that incorporates both extrinsic and intrinsic memory. The results enable efficient, particle-based simulation of mean-field systems with memory effects, with explicit rate assurances for practical implementations.

Abstract

This paper focuses on the numerical scheme for delay-type stochastic McKean-Vlasov equations (DSMVEs) driven by fractional Brownian motion with Hurst parameter . The existence and uniqueness of the solutions to such DSMVEs whose drift coefficients contain polynomial delay terms are proved by exploting the Banach fixed point theorem. Then the propagation of chaos between interacting particle system and non-interacting system in sense is shown. We find that even if the delay term satisfies the polynomial growth condition, the unmodified classical Euler-Maruyama scheme still can approximate the corresponding interacting particle system without the particle corruption. The convergence rates are revealed for . Finally, as an example that closely fits the original equation, a stochastic opinion dynamics model with both extrinsic memory and intrinsic memory is simulated to illustrate the plausibility of the theoretical result.
Paper Structure (5 sections, 13 theorems, 86 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Propagation of chaos
  4. Numerical scheme
  5. Numerical experiments

Key Result

Lemma 2.1

For any $\check{p}>0$ and $H\in (0,1/2)\cup (1/2,1)$, there exist two constants $c_{H,\check{p}},C_{H,\check{p}}>0$ such that for any stopping time $\tau$ of $B_t^H$.

Figures (1)

  • Figure 1: Convergence rates of classical EM scheme for (\ref{['cccexa1']})

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.3
  • Lemma 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.4
  • Lemma 4.5
  • ...and 3 more