A flexible split-step scheme for MV-SDEs

Goncalo Dos Reis; Xingyuan Chen

doi:10.1016/j.amc.2022.127180

A flexible split-step scheme for MV-SDEs

Goncalo Dos Reis, Xingyuan Chen

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.
The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, "Simulation of McKean-Vlasov SDEs with super-linear growth" in IMA Journal of Numerical Analysis, 01 2021. draa099] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations.

Original language	English
Article number	127180
Number of pages	30
Journal	Applied Mathematics and Computation
Volume	427
Early online date	28 Apr 2022
DOIs	https://doi.org/10.1016/j.amc.2022.127180
Publication status	Published - 15 Aug 2022

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2105.09688Accepted author manuscript, 5.31 MB

https://arxiv.org/abs/2105.09688

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title = "A flexible split-step scheme for MV-SDEs",

abstract = "We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, {"}Simulation of McKean-Vlasov SDEs with super-linear growth{"} in IMA Journal of Numerical Analysis, 01 2021. draa099] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations. ",

author = "{Dos Reis}, Goncalo and Xingyuan Chen",

year = "2022",

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doi = "10.1016/j.amc.2022.127180",

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T1 - A flexible split-step scheme for MV-SDEs

AU - Dos Reis, Goncalo

AU - Chen, Xingyuan

PY - 2022/8/15

Y1 - 2022/8/15

N2 - We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, "Simulation of McKean-Vlasov SDEs with super-linear growth" in IMA Journal of Numerical Analysis, 01 2021. draa099] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations.

AB - We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, "Simulation of McKean-Vlasov SDEs with super-linear growth" in IMA Journal of Numerical Analysis, 01 2021. draa099] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented, including a comparative analysis to other known algorithms for this class (Taming and Adaptive time-stepping) across parallel and non-parallel implementations.

U2 - 10.1016/j.amc.2022.127180

DO - 10.1016/j.amc.2022.127180

M3 - Article

SN - 0096-3003

VL - 427

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 127180

ER -

A flexible split-step scheme for MV-SDEs

Abstract

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