Limit theorem for maximum of the storage process with fractional Brownian motion as input

The maximum MT of the storage process Y(t)=sups[greater-or-equal, slanted]t(X(s)-X(t)-c(s-t)) in the interval [0,T] is dealt with, in particular, for growing interval length T. Here X(s) is a fractional Brownian motion with Hurst parameter, 0<H<1. For fixed T the asymptotic behaviour of MT was analysed by Piterbarg (Extremes 4(2) (2001) 147) by determining an approximation for the probability P MT>u for u-->[infinity]. Using this expression the convergence P MT<uT(x) -->G(x) as T-->[infinity] is derived where uT(x)-->[infinity] is a suitable normalization and G(x)=exp(-exp(-x)) the Gumbel distribution. Also the relation to the maximum of the process on a dense grid is analysed.