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Explicit numerical approximations for McKean-Vlasov stochastic differential equations in finite and infinite time

Yuanping Cui, Xiaoyue Li, Yi Liu, Fengyu Wang

TL;DR

This work tackles numerical approximation of McKean–Vlasov SDEs with superlinear coefficients by introducing an explicit truncated Euler-type scheme (TEM) that leverages propagation of chaos via an interacting particle system. It proves a strong convergence rate of order $1/2$ and shows TEM preserves long-time dynamics, including ergodicity, while ensuring convergence of the numerical invariant measure to the true one in the $L^2$-Wasserstein distance. The analysis establishes uniform-in-time propagation of chaos, non-asymptotic error bounds between numerical and true invariant measures, and exponential ergodicity for the numerical scheme. Numerical experiments in scalar and higher-dimensional settings corroborate the theory and demonstrate TEM’s robustness against particle-corruption, outperforming certain existing explicit methods in long-time simulations of MV-SDEs.

Abstract

Inspired by the stochastic particle method, this paper establishes an easily implementable explicit numerical method for McKean-Vlasov stochastic differential equations (MV-SDEs) with superlinear growth coefficients. The paper establishes the theory on the propagation of chaos in the Lq sense. The optimal uniform-in-time strong convergence rate 1/2-order of the numerical solutions is obtained for the interacting particle system. Furthermore, it is proved that the numerical solutions capture the long-term dynamical behaviors of MV-SDEs precisely, including moment boundedness, stability, and ergodicity. Moreover, a unique numerical invariant probability measure is yielded, which converges to the underlying invariant probability measure of MV-SDEs in the L2-Wasserstein distance. Finally, several numerical experiments are carried out to support the main results.

Explicit numerical approximations for McKean-Vlasov stochastic differential equations in finite and infinite time

TL;DR

This work tackles numerical approximation of McKean–Vlasov SDEs with superlinear coefficients by introducing an explicit truncated Euler-type scheme (TEM) that leverages propagation of chaos via an interacting particle system. It proves a strong convergence rate of order and shows TEM preserves long-time dynamics, including ergodicity, while ensuring convergence of the numerical invariant measure to the true one in the -Wasserstein distance. The analysis establishes uniform-in-time propagation of chaos, non-asymptotic error bounds between numerical and true invariant measures, and exponential ergodicity for the numerical scheme. Numerical experiments in scalar and higher-dimensional settings corroborate the theory and demonstrate TEM’s robustness against particle-corruption, outperforming certain existing explicit methods in long-time simulations of MV-SDEs.

Abstract

Inspired by the stochastic particle method, this paper establishes an easily implementable explicit numerical method for McKean-Vlasov stochastic differential equations (MV-SDEs) with superlinear growth coefficients. The paper establishes the theory on the propagation of chaos in the Lq sense. The optimal uniform-in-time strong convergence rate 1/2-order of the numerical solutions is obtained for the interacting particle system. Furthermore, it is proved that the numerical solutions capture the long-term dynamical behaviors of MV-SDEs precisely, including moment boundedness, stability, and ergodicity. Moreover, a unique numerical invariant probability measure is yielded, which converges to the underlying invariant probability measure of MV-SDEs in the L2-Wasserstein distance. Finally, several numerical experiments are carried out to support the main results.
Paper Structure (10 sections, 37 theorems, 345 equations, 11 figures)

This paper contains 10 sections, 37 theorems, 345 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. TEM Scheme and Strong Convergence
  4. Strong Convergence Rate
  5. Approximation of Invariant Measure
  6. The non-asymptotic error estimate
  7. The Exponential Ergodicity of Numerical Solution
  8. The uniform-time convergence rate
  9. Examples
  10. Proof of Counterexample

Key Result

Lemma 2.3

Let Assumptions ass1-ass3 and $X_0\in L^{p}_{0}$ hold. eq3.1 has a unique strong solution $X_t$ on $[0,\infty)$ such that

Figures (11)

  • Figure 1: The sample paths of the numerical solutions for the IPS \ref{['Ne1']} by the EM scheme for the initial value $X_0=18$, $\Delta=0.05$ and $M=2000$.
  • Figure 2: The numerical error v.s. time step size $\Delta$ at $t=1$.
  • Figure 3: The sample paths of the numerical solution by the TEM scheme for the initial value $X_0=18$, $\Delta=0.05$ and $M=2000$.
  • Figure 4: Strong error v.s. step size $\Delta$ for different dimension $d$.
  • Figure 5: The blue one is the $\log_2(RMSE)$ as a function of $q\in\{-10,-11,-12,-13,-14\}$, while the red one is the reference line with slope ${1}/{2}$.
  • ...and 6 more figures

Theorems & Definitions (70)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: Fournier
  • Lemma 2.5: MR4667613
  • Theorem 2.6
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 60 more