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

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/Fréchet) 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.