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Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift

Anh-Dung Le

TL;DR

This work analyzes a McKean–Vlasov SDE with density‑dependent drift and constant diffusion, establishing weak well‑posedness and a convergence rate for the Euler–Maruyama scheme in a weighted $L^1$ norm. The authors develop stability estimates for the scheme using Hölder regularity, heat‑kernel bounds, and a Duhamel framework to control marginal densities, yielding convergence to a weak solution characterized by a Fokker–Planck equation. Under mild moment and Hölder assumptions, they prove a quantitative rate of $n^{-\alpha/2}$ in the weighted $L^1$ metric for the marginal densities when $p=1$. The results extend numerical analysis of mean‑field SDEs to settings with both distribution‑ and density‑dependence and provide explicit guarantees for approximating the law of the process.

Abstract

In this paper, we study weak well-posedness of a McKean-Vlasov stochastic differential equations (SDEs) whose drift is density-dependent and whose diffusion is constant. The existence part is due to Hölder stability estimates of the associated Euler-Maruyama scheme. The uniqueness part is due to that of the associated Fokker-Planck equation. We also obtain convergence rate in weighted $L^1$ norm for the Euler-Maruyama scheme.

Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift

TL;DR

This work analyzes a McKean–Vlasov SDE with density‑dependent drift and constant diffusion, establishing weak well‑posedness and a convergence rate for the Euler–Maruyama scheme in a weighted norm. The authors develop stability estimates for the scheme using Hölder regularity, heat‑kernel bounds, and a Duhamel framework to control marginal densities, yielding convergence to a weak solution characterized by a Fokker–Planck equation. Under mild moment and Hölder assumptions, they prove a quantitative rate of in the weighted metric for the marginal densities when . The results extend numerical analysis of mean‑field SDEs to settings with both distribution‑ and density‑dependence and provide explicit guarantees for approximating the law of the process.

Abstract

In this paper, we study weak well-posedness of a McKean-Vlasov stochastic differential equations (SDEs) whose drift is density-dependent and whose diffusion is constant. The existence part is due to Hölder stability estimates of the associated Euler-Maruyama scheme. The uniqueness part is due to that of the associated Fokker-Planck equation. We also obtain convergence rate in weighted norm for the Euler-Maruyama scheme.
Paper Structure (16 sections, 8 theorems, 100 equations)

This paper contains 16 sections, 8 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wasserstein space $(\sP_p (\bR^d), W_p)$
  4. The scheme is well-defined
  5. Heat kernel estimates
  6. Duhamel representation
  7. Stability estimates of the scheme
  8. Bound supremum norm of marginal density
  9. Hölder continuity in space
  10. Hölder continuity in time
  11. Existence of a weak solution
  12. Convergence of marginal densities
  13. Existence of a weak solution
  14. Rate of convergence
  15. Decomposition of error
  16. ...and 1 more sections

Key Result

theorem 1

Let main_assmpt1 hold. There exist constants $c_1>0$ (depending on $\Theta_1$) and $c_2>0$ (depending on $\Theta_1, \nu$) such that for all $0 \le s < t \le T$: If we assume, in addition, that $\int_{\bR^d} (1+|x|^p) \sqrt{\ell_\nu (x)} \diff x \le C$, then

Theorems & Definitions (13)

  • theorem 1
  • theorem 2
  • theorem 3
  • definition 1
  • lemma 1
  • lemma 2
  • proof
  • lemma 3
  • lemma 4
  • proof
  • ...and 3 more