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

We propose an explicit drift-randomised Milstein scheme for bothMcKean–Vlasov stochastic differential equations and associated high di-mensional interacting particle systems with common noise. By using a driftrandomisation step in space and measure, we establish the scheme’s strongconvergence rate of1under reduced regularity assumptions on the drift co-efficient: no classical (Euclidean) derivatives in space or measure derivatives(e.g., Lions/Fréchet) are required. The main result is established by enrich-ing the concepts of bistability and consistency of numerical schemes usedpreviously for standard SDE. We introduce certain Spijker-type norms (andassociated Banach spaces) to deal with the interaction of particles present inthe stochastic systems being analysed. A discussion of the scheme’s com-plexity is provided.