系数满足单边Lipschitz条件的随机微分方程随机周期解的存在唯一性及数值逼近

朱驰骋 张静

摘要：本文研究了一類系数满足单边Lipschitz条件的随机微分方程随机周期解的存在唯一性，利用驯化Euler-Maruyama（EM）方法给出了随机周期解的数值逼近，并证明了数值逼近在均方意义下以α∈（0，1/2）阶收敛到精确解. 数值算例验证了理论结果.

关键词：随机周期解; 驯化Euler-Maruyama方法; 单边Lipschitz条件；数值逼近

收稿日期： 2022-12-05

基金项目： 国家自然科学基金（12161029， 11701127， 11871184）; 海南省自然科学基金（121RC149， 121QN227）

作者简介： 朱驰骋（1997-）， 男， 浙江台州人， 硕士研究生， 主要研究方向为随机微分方程. E-mail： zhu_cc0926@hainnu.edu.cn

通讯作者： 张静. 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.

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引用本文格式：

中 文： 朱驰骋， 张静. 系数满足单边Lipschitz条件的随机微分方程随机周期解的存在唯一性及数值逼近[J]. 四川大学学报： 自然科学版， 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.