Generating Diffusions with Fractional Brownian Motion

Abstract

We study fast/slow systems driven by a fractional Brownian motion B with Hurst parameter $$H\in (\frac{1}{3}, 1]$$ H ∈ ( 1 3 , 1 ] . Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $$Y^\varepsilon $$ Y ε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $$\varepsilon \ll 1$$ ε ≪ 1 , the solutions of the equation $$\begin{aligned} dX^\varepsilon = {\varepsilon }^{\frac{1}{2}-H} F(X^\varepsilon ,Y^\varepsilon )\,dB+F_0(X^\varepsilon ,Y^{\varepsilon })\,dt\; \end{aligned}$$ d X ε = ε 1 2 - H F ( X ε , Y ε ) d B + F 0 ( X ε , Y ε ) d t converge to a regular diffusion without having to assume that F averages to 0, provided that $$H< \frac{1}{2}$$ H < 1 2 . For $$H > \frac{1}{2}$$ H > 1 2 , a similar result holds, but this time it does require F to average to 0. We also prove that the n-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the time homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($$H=1$$ H = 1 ) and the averaging of diffusion processes ($$H= \frac{1}{2}$$ H = 1 2 ).