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

• Published in: arXiv.org 2023-03
• Main Authors: Chen, Xingyuan, Goncalo dos Reis
• Format: Article
• Language: English
• Subjects: Convolution; Differential equations; Diffusion; Diffusion coefficient; Granular materials; Granular media
• ISSN: 2331-8422

We consider in this work 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. 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 \(1/2-\varepsilon\) for \(\varepsilon>0\) and an optimal rate \(1/2\) in the non-path-space mean-square error metric. Numerical examples illustrate all findings. 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).