An Averaging Principle for Two-Time-Scale Functional Diffusions
Thursday, May 17, 2018 - 10:00am - 10:30am
Dupire recently developed a functional Ito formula, which has changed the landscape of the study of stochastic functional equations and encouraged a reconsideration of many problems and applications. Delays are ubiquitous, pervasive, and entrenched in everyday life. 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 averaging and 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.