Large Deviations of Shepp Statistics for Fractional Brownian Motion

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 $H\in (0,1\mathrm{/}2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.