Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent

TL;DR

This paper investigates the persistence probability decay of fractional Brownian motion with a randomly chosen Hurst exponent, showing it decays at a rate determined by the essential supremum of the exponent's distribution.

Contribution

It establishes the asymptotic decay rate of persistence probability for fractional Brownian motion with a random Hurst exponent, extending known results for fixed Hurst parameters.

Findings

Persistence probability decays as T^{-(1-H_0)+o(1)}

Decay rate depends on the essential supremum of the Hurst distribution

Results generalize fixed Hurst exponent case to random exponents

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.