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(sgreater than or equal tot)(X(s) - X(t) - c(s - t)) in the interval [0, 7] 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{M-T > u} for u --> infinity. Using this expression the convergence P{M-T < u(T)(x)} --> G(x) as T --> infinity is derived where u(T)(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. (C) 2004 Elsevier B.V. All rights reserved.