Large Deviations of Shepp Statistics for Fractional Brownian Motion

TL;DR

This paper derives the exact asymptotic behavior of the maximum of fractional Brownian motion increments for Hurst indices less than 1/2, extending previous results for standard Brownian motion.

Contribution

It provides the first precise asymptotic analysis of Shepp statistics for fractional Brownian motion with H<1/2, complementing earlier Brownian motion results.

Findings

Exact asymptotics for the maximum of fractional Brownian increments.

Extension of Shepp statistics analysis to H<1/2.

Complements previous Brownian motion results.

Abstract

Define the incremental fractional Brownian field $B_{H}(s+\tau)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum $\max_{(\tau,s)\in[0,1]\times[0,T]} (B_{H}(s+\tau)-B_{H}(s)) $ for any $H\in (0,1/2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.