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

The maximum M T of the storage process Y ( t ) = sup s ⩾ 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 M T was analysed by Piterbarg (Extremes 4(2) (2001) 147) by determining an approximation for the probability P { M T > u } for u → ∞ . Using this expression the convergence P { M T < u T ( x ) } → G ( x ) as T → ∞ is derived where u T ( x ) → ∞ 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.