Rough Hurst function estimation

Abstract

The fractional Brownian motion (fBm) is parameterized by the Hurst exponent $H\in(0,1)$, which determines the dependence structure and regularity of sample paths. Empirical findings suggest that the Hurst exponent may be non-constant in time, giving rise to the so-called multifractional Brownian motion (mBm). The It\^o-mBm is an alternative to the classical mBm, and has been shown to admit more intuitive sample path properties in case the Hurst function is rough. In this paper, we show that the It\^o-mBm also allows for a simplified statistical treatment compared to the classical mBm. In particular, estimation of the local Hurst parameter $H(t)$ with H\"older exponent $\eta>0$ achieves rates of convergence which are standard in nonparametric regression, whereas similar results for the classical mBm only hold for the smoother regime $\eta>1$. Furthermore, we derive an estimator of the integrated Hurst exponent $\int_0^t H(s)\, ds$ which achieves a parametric rate of convergence, and use it to construct goodness-of-fit tests.