Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully super-linear growth drifts in space and interaction

This work addresses the convergence of a split-step Euler type scheme (SSM) for the numerical simulation of interacting particle Stochastic Differential Equation (SDE) systems and McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with full super-linear growth in the spatial and the interaction component in the drift, and non-constant Lipschitz diffusion coefficient. Super-linearity is understood in the sense that functions are assumed to behave polynomially but also satisfy a so-called one-sided Lipschitz condition.
The super-linear growth in the interaction (or measure) component stems from convolution operations with super-linear growth functions allowing in particular application to the granular media equation with multi-well confining potentials.
From a methodological point of view, we avoid altogether functional inequality arguments (as we allow for non-constant non-bounded diffusion maps).
The scheme attains, in stepsize, a near-optimal classical (path-space) root mean-square error rate of 1/2 −ε for ε > 0 and an optimal rate 1/2 in the non-path space (pointwise) mean-square error metric. All findings are illustrated by numerical examples. In particular, the testing raises doubts if taming is a suitable methodology for this type of problem (with convolution terms and non-constant diffusion coefficients).