A flexible split‐step scheme for solving McKean‐Vlasov stochastic differential equations

•Present a flexible implicit Split-Step explicit Euler type scheme for the numericalapproximation of McKean Vlasov stochastic Differential Equations with drifts havingsuperlinear growth in their spatial component.•This is the first implicit type scheme working under the same general conditions as explicitmethods, in particular, no differentability or non-degeneracy assumptions are imposed andstopping-time arguments are fully avoided.•Sufficient conditions for mean-square contractivity of the scheme is presented.•This flexible scheme leverages the structure induced by the interacting particleapproximation system, including parallel implementation and the solvability of the implicitequation.•Four numerical examples are presented including a comparative analysis to other knownalgorithms for this class (taming and adaptive time-stepping) across parallel and non-parallelimplementations. Our schemes computations times compete with those from other explicit algorithms. We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation. The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [1] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations.