朱馳騁 張靜

摘要：本文研究了一類系數(shù)滿足單邊Lipschitz條件的隨機(jī)微分方程隨機(jī)周期解的存在唯一性，利用馴化Euler-Maruyama（EM）方法給出了隨機(jī)周期解的數(shù)值逼近，并證明了數(shù)值逼近在均方意義下以α∈（0，1/2）階收斂到精確解. 數(shù)值算例驗(yàn)證了理論結(jié)果.

關(guān)鍵詞：隨機(jī)周期解; 馴化Euler-Maruyama方法; 單邊Lipschitz條件；數(shù)值逼近

收稿日期： 2022-12-05

基金項(xiàng)目： 國(guó)家自然科學(xué)基金（12161029， 11701127， 11871184）; 海南省自然科學(xué)基金（121RC149， 121QN227）

作者簡(jiǎn)介： 朱馳騁（1997-）， 男， 浙江臺(tái)州人， 碩士研究生， 主要研究方向?yàn)殡S機(jī)微分方程. E-mail： zhu_cc0926@hainnu.edu.cn

通訊作者： 張靜. E-mail： zh_jing0820@hotmail.com

Existence， uniqueness and numerical approximation for random periodic

solutions of the SDEs with one-sided Lipschitz coefficients

ZHU Chi-Cheng， ZHANG Jing

（School of Mathematics and Statistics， Hainan Normal University， Haikou 571158， China）

In this paper， we consider the existence and uniqueness of random periodic solutions of the SDEs with coefficients satisfying the one-sided Lipschitz condition. By using the tamed Euler-Maruyama （EM）method we give a numerical approximation for the random periodic solution and reveal that the numerical approximation converges to the exact solution with an order α∈（0，1/2） in the mean square sense. Examples are given to verify the theoretical result.

Random periodic solution; Tamed Euler-Maruyama method; One-sided Lipschitz condition; Numerical approximation

（2010 MSC 65C20， 60H35）

6 Summary

We have discussed the existence， uniqueness， and numerical approximation of the random periodic solutions of the stochastic differential equations with a drift coefficient satisfying the one-sided Lipschitz condition. We researched the basic properties of the solutions， demonstrated the boundness of the moments， the time-continuity and the relationship between solution and initial condition. Since the random periodic solutions are orbital motions in an infinite time domain， the existence and uniqueness theory for random periodic solutions was obtained by using the properties of random semi-flow.

Furthermore， the tamed EM method was introduced to deduce the numerical approximation of the random periodic solution. This numerical structure can ensure that the drift coefficient is moments-bounded with the one-sided Lipschitz condition. By using the random semi-flow， we discussed the basic properties of the numerical approximation in different time domains and proved that the numerical approximation converged to its exact solution. It was proved that the numerical approximation of the random periodic solution also had the random periodic property.

Finally， we revealed that the convergence rate between the exact random periodic solution and the approximated one was α∈（0，1/2） in the mean square sense.

References：

[1]Zhao H Z， Zheng Z H. Random periodic solutions of random dynamical systems [J]. J Diff Equat， 2009， 246： 2020.

[2]Feng C R， Zhao H Z， Bo Z. Pathwise random periodic solutions of stochastic differential equations [J]. J Diff Equat， 2011， 251： 119.

[3]Chekroun M D， Simonnet E， Ghil M. Stochastic climate dynamics： random attractors and time-dependent invariant measures [J]. Physica D， 2011， 240： 1685.

[4]Franses P H. Periodicity and stochastic trends in economic time series [M]. Oxford： Oxford University Press， 1996.

[5]Poincaré H. Memoire sur les courbes definier par une equation differentiate （I）[J]. J Math Pures Appl， 1881， 7： 375.

[6]Poincaré H. Memoire sur les courbes definier par une equation differentiate （II）[J]. J Math Pures Appl， 1882， 8： 251.

[7]Poincaré H. Memoire sur les courbes definier par une equation differentiate （III）[J]. J Math Pures Appl， 1885， 1： 167.

[8]Poincaré H. Memoire sur les courbes definier par une equation differentiate （IV）[J]. J Math Pures Appl， 1886， 2： 151.

[9]Feng C R， Yu L， Zhao H Z. Numerical approximation of random periodic solutions of stochastic differential equations [J]. Z Angew Math Phys， 2017， 119： 68.

[10]Wei R， Chen C Z. Numerical approximation of stochastic Theta method for random periodic solution of stochastic differential equations [J]. Acta Math Appl Sin E， 2020， 36： 689.

[11]Hutzenthaler M， Jentzen A， Kloeden P E. Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients [J]. Ann Appl Probab， 2012， 22： 1611.

[12]Ji Y T， Yuan C G. Tamed EM scheme of neutral stochastic differential delay equations [J]. J Comput Appl Math， 2017， 326： 335.

[13]Tan L， Yuan C G. Strong convergence of a tamed theta scheme for NSDDEs with one-sided Lipschitz drift [J]. Appl Math Comput， 2018， 338： 607.

[14]Beyn W J， Isaak E， Kruse R. Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes [J]. J Sci Comput， 2016， 67： 955.

[15]Kunita H. Stochastic flows and stochastic differential equation [J]. Stoch Proc Appl， 1990， 18： 230

[16]Deng S N， Fei C， Fei W Y， et al. Tamed EM schemes for neutral stochastic differential delay equations with superlinear diffusion coefficients [J]. J Comput Appl Math， 2021， 388： 113269.

引用本文格式：

中 文： 朱馳騁， 張靜. 系數(shù)滿足單邊Lipschitz條件的隨機(jī)微分方程隨機(jī)周期解的存在唯一性及數(shù)值逼近[J]. 四川大學(xué)學(xué)報(bào)： 自然科學(xué)版， 2023， 60： 061004.

英 文： Zhu C C， Zhang J. Existence， uniqueness and numerical approximation for random periodic solutions of the SDEs with one-sided Lipschitz coefficients [J]. J Sichuan Univ： Nat Sci Ed， 2023， 60： 061004.