An averaging principle for two-time-scale stochastic functional differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Wu, Fuke¹  Yin, George² 

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

[2] Wayne State Univ, Dept Math, Detroit, MI 48202 USA

关键词: Path-dependent functional;    Two-time scale;    Weak convergence;    Martingale method;    Ergodicity;    Functional Ito formula;    Stochastic functional differential equation;   

DOI  :  10.1016/j.jde.2019.12.024

来源: Elsevier

【 摘 要 】

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Ito formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Ito formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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