A projected Euler method for random periodic solutions of semi-linear SDEs with non-globally Lipschitz coefficients

Yujia Guo; Xiaojie  Wang; Yue Wu

doi:10.48550/arXiv.2406.16089

A projected Euler method for random periodic solutions of semi-linear SDEs with non-globally Lipschitz coefficients

Yujia Guo, Xiaojie Wang^*, Yue Wu

^*Corresponding author for this work

Mathematics And Statistics

Research output: Working paper › Working Paper/Preprint

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Abstract

The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method, to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate is proved to be order 0.5 for SDEs with multiplicative noise and order 1 for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.

Original language	English
Place of Publication	Ithaca, NY
Number of pages	25
DOIs	https://doi.org/10.48550/arXiv.2406.16089
Publication status	Published - 23 Jun 2024

Funding

This work was supported by Natural Science Foundation of China (12071488, 12371417, 11971488). YW would like to acknowledge the support of the Royal Society through the International Exchanges scheme IES\R3\233115.

Keywords

Projected Euler method
random periodic solution
stochastic differential equations
pull-back
mean square convergence order

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10.48550/arXiv.2406.16089Licence: CC BY 4.0

Guo-etal-arXiv-2024-A-projected-Euler-method-for-random-periodic-solutions-of-semi-linear-SDEsFinal published version, 360 KBLicence: CC BY 4.0

Cite this

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abstract = "The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method, to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate is proved to be order 0.5 for SDEs with multiplicative noise and order 1 for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.",

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AU - Guo, Yujia

AU - Wang, Xiaojie

AU - Wu, Yue

PY - 2024/6/23

Y1 - 2024/6/23

N2 - The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method, to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate is proved to be order 0.5 for SDEs with multiplicative noise and order 1 for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.

AB - The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method, to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate is proved to be order 0.5 for SDEs with multiplicative noise and order 1 for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.

KW - Projected Euler method

KW - random periodic solution

KW - stochastic differential equations

KW - pull-back

KW - mean square convergence order

U2 - 10.48550/arXiv.2406.16089

DO - 10.48550/arXiv.2406.16089

M3 - Working Paper/Preprint

BT - A projected Euler method for random periodic solutions of semi-linear SDEs with non-globally Lipschitz coefficients

CY - Ithaca, NY

ER -

A projected Euler method for random periodic solutions of semi-linear SDEs with non-globally Lipschitz coefficients

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