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On modified Euler methods for McKean-Vlasov stochastic differential equations with super-linear coefficients

Jiamin Jian, Qingshuo Song, Xiaojie Wang, Zhongqiang Zhang, Yuying Zhao

TL;DR

This work addresses numerical solutions for McKean-Vlasov SDEs with distribution dependence and super-linear growth by introducing a flexible class of modified Euler schemes that preserve moment bounds. The authors formulate the interacting-particle framework and prove moment boundedness, then establish a strong convergence rate of order $\tfrac{1}{2}$ for these schemes, leveraging propagation of chaos to transfer particle-system results to the MV-SDE. They provide concrete examples (modified Euler, tanh Euler, sin Euler) that satisfy the required operator conditions, and demonstrate, both analytically and numerically, that the proposed methods remain stable and accurate on finite horizons. The numerical experiments corroborate the theory, showing the tanh Euler method often offers superior stability, while implicit methods maintain robustness for difficult initial data, highlighting practical implications for mean-field simulations in high-growth regimes.

Abstract

We introduce a new class of numerical methods for solving McKean-Vlasov stochastic differential equations, which are relevant in the context of distribution-dependent or mean-field models, under super-linear growth conditions for both the drift and diffusion coefficients. Under certain non-globally Lipschitz conditions, the proposed numerical approaches have half-order convergence in the strong sense to the corresponding system of interacting particles associated with McKean-Vlasov SDEs. By leveraging a result on the propagation of chaos, we establish the full convergence rate of the modified Euler approximations to the solution of the McKean-Vlasov SDEs. Numerical experiments are included to validate the theoretical results.

On modified Euler methods for McKean-Vlasov stochastic differential equations with super-linear coefficients

TL;DR

This work addresses numerical solutions for McKean-Vlasov SDEs with distribution dependence and super-linear growth by introducing a flexible class of modified Euler schemes that preserve moment bounds. The authors formulate the interacting-particle framework and prove moment boundedness, then establish a strong convergence rate of order for these schemes, leveraging propagation of chaos to transfer particle-system results to the MV-SDE. They provide concrete examples (modified Euler, tanh Euler, sin Euler) that satisfy the required operator conditions, and demonstrate, both analytically and numerically, that the proposed methods remain stable and accurate on finite horizons. The numerical experiments corroborate the theory, showing the tanh Euler method often offers superior stability, while implicit methods maintain robustness for difficult initial data, highlighting practical implications for mean-field simulations in high-growth regimes.

Abstract

We introduce a new class of numerical methods for solving McKean-Vlasov stochastic differential equations, which are relevant in the context of distribution-dependent or mean-field models, under super-linear growth conditions for both the drift and diffusion coefficients. Under certain non-globally Lipschitz conditions, the proposed numerical approaches have half-order convergence in the strong sense to the corresponding system of interacting particles associated with McKean-Vlasov SDEs. By leveraging a result on the propagation of chaos, we establish the full convergence rate of the modified Euler approximations to the solution of the McKean-Vlasov SDEs. Numerical experiments are included to validate the theoretical results.

Paper Structure

This paper contains 12 sections, 75 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Notations, assumptions, and preliminaries
  3. Notations
  4. Well-posedness of MV-SDEs
  5. The interacting particle system and propagation of chaos
  6. Modified Euler methods with moment bound
  7. Modified Euler approximations
  8. Examples of modified Euler approximations
  9. Boundedness of moments of modified Euler approximations
  10. Strong convergence rate of modified Euler approximations
  11. Numerical experiments
  12. Proof of Lemma \ref{['l:moment_bound_euler_discrete_time']} in Section \ref{['sub:bound-moment-modi-method']}

Figures (14)

  • Figure 1: Several choices for the operators $\mathcal{T}_i$, $i=1,2$.
  • Figure 2: Strong errors for Example \ref{['ex:numerical_ex1_nonlinear_diff']} and \ref{['ex:numerical_ex2_nonlinear_diff']}
  • Figure 3: Density with $X_0 \sim \mathcal{N}(0,\,1)$
  • Figure 4: Density with $X_0 \sim \mathcal{N}(3,\,9)$
  • Figure 5: Density with $X_0 \sim \mathcal{N}(0,\,1)$
  • ...and 9 more figures