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

TL;DR

This paper analyzes a split-step Euler scheme for simulating interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth in drift and interaction, achieving near-optimal convergence rates without functional inequalities.

Contribution

It introduces a novel Euler scheme that handles superlinear growth in both drift and interaction terms, avoiding functional inequalities and demonstrating optimal convergence rates.

Findings

Scheme achieves near 1/2 root mean-square error rate.

Numerical tests show limitations of taming methods for these problems.

Method applies to granular media equations with multi-well potentials.

Abstract

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).