Generating diffusions with fractional Brownian motion

Abstract

We study fast / slow systems driven by a fractional Brownian motion $B$ with Hurst parameter $H\in (\frac{1}{3},1]$. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $Y^{\epsilon }$ denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $\epsilon \ll 1$, the solutions of the equation $$d{X}^{\epsilon }={\epsilon }^{\frac{1}{2}-H}F(X^{\epsilon },Y^{\epsilon })dB+F_0(X^{\epsilon },Y^{\epsilon })dt$$ converge to a regular diffusion without having to assume that $F$ averages to $0$, provided that $H<\frac{1}{2}$. For $H>\frac{1}{2}$, 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 homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($H=1$) and the averaging of diffusion processes ($H=\frac{1}{2}$).

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Cited by in corpus (2)

• Weak error estimates for rough volatility models
• Large deviations of slow-fast systems driven by fractional Brownian motion