Euler simulation of interacting particle systems and McKean–Vlasov SDEs with fully super-linear growth drifts in space and interaction
Xingyuan Chen, Gonçalo dos Reis
Abstract
Abstract 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 nonconstant 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 nonconstant nonbounded diffusion maps). The scheme attains, in stepsize, a near-optimal classical (path-space) root mean-square error rate of $1/2-\varepsilon $ for $\varepsilon>0$ and an optimal rate $1/2$ in the nonpath-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 nonconstant diffusion coefficients).