Large deviations of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field ZH(τ,s)=BH(s+τ)−BH(s),H∈(0,1), where BH(s) is a standard fractional Brownian motion with Hurst index H∈(0,1). In this paper we derive the exact asymptotic behaviour of the maximum MH(T)=max(τ,s)∈[0,1]×[0,T]ZH(τ,s) for any H∈(0,1/2) complimenting thus the result of Zholud (2008) which establishes the exact tail asymptotic behaviour of M1/2(T).