Averaging Principles for Mixed Fast-Slow Systems Driven by Fractional Brownian Motion

Abstract

We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised Riemann-Stieltjes integral using the tools of fractional calculus. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a noise to be averaged out in the limit is established. It is shown that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion. The implication is that one can ignore the complex original systems and concentrate on the averaged systems instead. This averaging principle paves the way for reduction of computational complexity.

arXiv comment23 pages accepted by Kyoto Journal of Mathematics

References in corpus (3)

• Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion
• Averaging dynamics driven by fractional Brownian motion
• Mixed fractional stochastic differential equations with jumps

Cited by in corpus (3)

• Averaging principle for fast-slow system driven by mixed fractional Brownian rough path
• Pathwise unique solutions and stochastic averaging for mixed stochastic partial differential equations driven by fractional Brownian motion and Brownian motion
• Large deviations of slow-fast systems driven by fractional Brownian motion