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

• Published in: Journal of Differential Equations 2020-06, Vol.269 (1), p.1037-1077
• Main Authors: Wu, Fuke, Yin, George
• Format: Article
• Language: English
• Subjects: Ergodicity; Functional Itô formula; Martingale method; Path-dependent functional; Stochastic functional differential equation; Two-time scale; Weak convergence
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2019.12.024

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô 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 Itô 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.