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The convergence of the EM scheme in empirical approximation of invariant probability measure for McKean-Vlasov SDEs

Cui Yuanping, Li Xiaoyue

TL;DR

The paper addresses approximating the invariant measure $\mu^{*}$ of McKean–Vlasov SDEs by exploiting a self-interacting process whose coefficients depend on current and historical information. It develops weighted empirical and averaged weighted empirical schemes, proving convergence in $W_2$ and establishing a uniform $1/2$-order EM convergence for the self-interacting dynamics, along with rates $t^{-\eta}$ (with $\eta=\rho/((d+2)(\rho+2))$) for the empirical measures to $\mu^{*}$; it also analyzes multi-particle averaging and provides detailed computational-cost comparisons. The results are supported by rigorous bounds and numerical experiments that validate the theoretical rates and demonstrate practical efficiency, including cases with known invariant Gaussians. Overall, the work offers implementable algorithms for invariant-measure approximation in MV-SDEs with rigorous convergence guarantees and favorable computational trade-offs.

Abstract

Based on the assumption of the existence and uniqueness of the invariant measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a self-interacting process that depends only on the current and historical information of the solution is constructed for MV-SDEs. The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric. Furthermore, under the condition of linear growth, an EM scheme whose uniformly 1/2-order convergence rate with respect to time is obtained is constructed for the self-interacting process. Then, the convergence rate between the weighted empirical measure of the EM numerical solution of the self-interacting process and the invariant measure of MV-SDEs is derived. Moreover, the convergence rate between the averaged weighted empirical measure of the EM numerical solution of the corresponding multi-particle system and the invariant measure of MV-SDEs in the W2-Wasserstein metric is also given. In addition, the computational cost of the two approximation methods is compared, which shows that the averaged weighted empirical approximation of the particle system has a lower cost. Finally, the theoretical results are validated through numerical experiments.

The convergence of the EM scheme in empirical approximation of invariant probability measure for McKean-Vlasov SDEs

TL;DR

The paper addresses approximating the invariant measure of McKean–Vlasov SDEs by exploiting a self-interacting process whose coefficients depend on current and historical information. It develops weighted empirical and averaged weighted empirical schemes, proving convergence in and establishing a uniform -order EM convergence for the self-interacting dynamics, along with rates (with ) for the empirical measures to ; it also analyzes multi-particle averaging and provides detailed computational-cost comparisons. The results are supported by rigorous bounds and numerical experiments that validate the theoretical rates and demonstrate practical efficiency, including cases with known invariant Gaussians. Overall, the work offers implementable algorithms for invariant-measure approximation in MV-SDEs with rigorous convergence guarantees and favorable computational trade-offs.

Abstract

Based on the assumption of the existence and uniqueness of the invariant measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a self-interacting process that depends only on the current and historical information of the solution is constructed for MV-SDEs. The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric. Furthermore, under the condition of linear growth, an EM scheme whose uniformly 1/2-order convergence rate with respect to time is obtained is constructed for the self-interacting process. Then, the convergence rate between the weighted empirical measure of the EM numerical solution of the self-interacting process and the invariant measure of MV-SDEs is derived. Moreover, the convergence rate between the averaged weighted empirical measure of the EM numerical solution of the corresponding multi-particle system and the invariant measure of MV-SDEs in the W2-Wasserstein metric is also given. In addition, the computational cost of the two approximation methods is compared, which shows that the averaged weighted empirical approximation of the particle system has a lower cost. Finally, the theoretical results are validated through numerical experiments.
Paper Structure (10 sections, 183 equations, 3 figures)

This paper contains 10 sections, 183 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. EM scheme in the weighed empirical approximation
  4. The empirical approximation of the self-interacting process
  5. The EM scheme for self-interacting process
  6. EM scheme in the averaged weighted empirical approximation
  7. The averaged weighted empirical approximation
  8. The EM scheme in the averaged weighed empirical approximation
  9. Computational cost
  10. Numerical examples

Figures (3)

  • Figure 1: The functions $\log_2(RMSE)$ of $q=5,6,7,8,9,10,11$ at $t=10, 20, 40$ and $60$.
  • Figure 2: The empirical density functions of self-interacting process at $t=500, 1000, 2000, 4000$ and the normal distribution N(0, 9/4).
  • Figure 3: The empirical density functions of multi-particle system at $t=100$ for $N=1, 50, 100, 200$ and the normal distribution $N(0,4/9)$.

Theorems & Definitions (13)

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