Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent

Abstract

We study the persistence properties of a fractional Brownian motion whose Hurst exponent is a random variable instead of a fixed constant. For each fixed $H\in (0,1)$, it is well known that the persistence probability of an FBM below a constant barrier decays like $T^{-(1-H)+o(1)}$, as $T$ tends to infinity, cf. Molchan (1999). Our object of interest is the persistence probability of the process resulting from first randomly selecting $H\in (0,1)$ and then considering a fractional Brownian motion with this value of $H$ as a Hurst exponent, a process that is referred to as a fractional Brownian motion with random exponent. We prove that its persistence probability decays as $T^{-(1-H_0)+o(1)}$, as $T$ tends to infinity, where $H_0$ is the essential supremum of the distribution of the random Hurst exponent.

Technical correction to Corollary 4.2, with corresponding changes in Proposition 4.6 and the proof of the upper bound. The main results are unaffected