Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:336

Article

Zong, Xiaofeng^1,2  Wu, Fuke³  Xu, Guiping⁴ 

[1] China Univ Geosci, Sch Automat, Wuhan 430074, Hubei, Peoples R China

[2] Hubei Key Lab Adv Control & Intelligent Automat C, Wuhan 430074, Hubei, Peoples R China

[3] Huazhong Univ Sci & Technol, Sch Math & Stat, Wuhan 430074, Hubei, Peoples R China

[4] Huazhong Univ Sci & Technol, Sch Comp Sci & Technol, Wuhan 430074, Hubei, Peoples R China

关键词: SDEs;    Strong convergence rate;    Exponential mean-square stability;    Stochastic theta-Milstein scheme;    Split-step theta-Milstein scheme;   

DOI  :  10.1016/j.cam.2017.12.025

来源: Elsevier

【 摘 要 】

This paper examines convergence and stability of the two classes of theta-Milstein schemes for stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients: the split-step theta-Milstein (SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For theta is an element of[1/2, 1], this paper concludes that the two classes of theta-Milstein schemes converge strongly to the exact solution with the order 1. For theta is an element of[0, 1/2], under the additional linear growth condition for the drift coefficient, these two classes of the theta-Milstein schemes are also strongly convergent with the standard order. This paper also investigates exponential mean-square stability of these two classes of the theta-Milstein schemes. For theta is an element of(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For theta is an element of[0, 1/2], similar to the convergence, under the additional linear growth condition, these two theta-Milstein schemes can also reproduce the exponential mean-square stability of the exact solution. (C) 2018 Elsevier B.V. All rights reserved.

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