An explicit Milstein-type scheme for interacting particle systems and McKean--Vlasov SDEs with common noise and non-differentiable drift coefficients

TL;DR

This paper introduces an explicit Milstein-type scheme for McKean--Vlasov SDEs and particle systems with common noise, achieving strong convergence with minimal regularity assumptions on the drift.

Contribution

It develops a drift-randomised Milstein scheme that does not require classical derivatives, extending numerical methods to less regular McKean--Vlasov systems.

Findings

Achieves strong convergence rate of 1 under reduced regularity

Introduces Spijker-type norms for analysis of particle interactions

Provides complexity discussion of the scheme

Abstract

We propose an explicit drift-randomised Milstein scheme for both McKean--Vlasov stochastic differential equations and associated high-dimensional interacting particle systems with common noise. By using a drift-randomisation step in space and measure, we establish the scheme's strong convergence rate of $1$ under reduced regularity assumptions on the drift coefficient: no classical (Euclidean) derivatives in space or measure derivatives (e.g., Lions/Fr\'echet) are required. The main result is established by enriching the concepts of bistability and consistency of numerical schemes used previously for standard SDE. We introduce certain Spijker-type norms (and associated Banach spaces) to deal with the interaction of particles present in the stochastic systems being analysed. A discussion of the scheme's complexity is provided.