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Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion

Ariel Neufeld, Tuan Anh Nguyen, Philipp Schmocker

TL;DR

The paper tackles numerical approximation of high-dimensional McKean–Vlasov SDEs with nonconstant diffusion. It introduces a multilevel Picard (MLP) scheme that builds a Picard iteration while storing full discrete trajectories to handle distribution-dependent drift and diffusion. Under Lipschitz conditions, the authors prove $L^2$ convergence with computational cost growing polynomially in the dimension $d$ and the inverse error tolerance, thereby mitigating the curse of dimensionality. Two large-scale numerical experiments—a mean-field Ornstein–Uhlenbeck model and a multidimensional geometric Kuramoto model—demonstrate practical feasibility up to $d=1000$, highlighting the method’s potential for complex mean-field problems in physics, neuroscience, and related fields.

Abstract

We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm approximates the solution of the SDE in the $L^2$-sense without the curse of dimensionality. The latter means that its computational cost grows at most polynomially in both the dimension and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate its applicability by approximating McKean--Vlasov SDEs in dimensions up to 1000.

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion

TL;DR

The paper tackles numerical approximation of high-dimensional McKean–Vlasov SDEs with nonconstant diffusion. It introduces a multilevel Picard (MLP) scheme that builds a Picard iteration while storing full discrete trajectories to handle distribution-dependent drift and diffusion. Under Lipschitz conditions, the authors prove convergence with computational cost growing polynomially in the dimension and the inverse error tolerance, thereby mitigating the curse of dimensionality. Two large-scale numerical experiments—a mean-field Ornstein–Uhlenbeck model and a multidimensional geometric Kuramoto model—demonstrate practical feasibility up to , highlighting the method’s potential for complex mean-field problems in physics, neuroscience, and related fields.

Abstract

We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm approximates the solution of the SDE in the -sense without the curse of dimensionality. The latter means that its computational cost grows at most polynomially in both the dimension and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate its applicability by approximating McKean--Vlasov SDEs in dimensions up to 1000.

Paper Structure

This paper contains 7 sections, 2 theorems, 80 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Numerical experiments
  3. Mean-field Ornstein-Uhlenbeck model
  4. Multidimensional geometric Kuramoto model
  5. Conclusion
  6. Picard iteration revisited
  7. Proof of Theorem \ref{['c01']}

Key Result

Theorem 1

Let $T\in (0,\infty)$, $c\in [1,\infty)$, $\Theta=\cup_{n\in {\mathbbm{N}}}({\mathbbm{N}}_0)^n$. For every $d\in {\mathbbm{N}}$ let $\xi_d\in {\mathbbm{R}}^d$, $\mu_d \in C({\mathbbm{R}}^d\times{\mathbbm{R}}^d,{\mathbbm{R}}^d)$, $\sigma_d \in C({\mathbbm{R}}^d\times{\mathbbm{R}}^d,{\mathbbm{R}}^{d\t Let $(\Omega,\mathcal{F},{\mathbbm{P}}, ({\mathbbm{F}}_t)_{t\in [0,T]})$ be a filtered probability

Theorems & Definitions (5)

  • Theorem 1
  • Remark 2
  • Lemma 3
  • proof : Proof of \ref{['c12']}
  • proof : Proof of \ref{['c01']}