TY - JOUR
T1 - Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully super-linear growth drifts in space and interaction
AU - Chen, Xingyuan
AU - dos Reis, Gonçalo
N1 - Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 and UIDP/00297/2020 (Centro de Matemática e Aplicações CMA/FCT/UNL) to G.d.R.
PY - 2024/3
Y1 - 2024/3
N2 - 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 − ε for ε > 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).
AB - 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 − ε for ε > 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).
KW - McKean-Vlasov equations
KW - Split-step Euler methods
KW - Stochastic interacting particle systems
KW - Super-linear growth in measure
KW - Super-linear growth in space
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U2 - 10.1093/imanum/drad022
DO - 10.1093/imanum/drad022
M3 - Article
SN - 0272-4979
VL - 44
SP - 751
EP - 796
JO - Ima Journal Of Numerical Analysis
JF - Ima Journal Of Numerical Analysis
IS - 2
ER -