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Maxima of stochastic processes driven by fractional Brownian motion

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Boris Buchmann  and
Claudia Klüppelberg
Boris Buchmann*
Affiliation:
Australian National University
Claudia Klüppelberg*
Affiliation:
Munich University of Technology
*
∗ Postal address: Centre of Excellence for Mathematics and Statistics of Complex Systems, Mathematical Science Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: [email protected]
∗∗ Postal address: Centre for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany. Email address: [email protected]
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Abstract

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We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

Keywords

Extreme-value theoryfractional Brownian motionfractional Ornstein-Uhlenbeck processfractional stochastic differential equationlong-range dependencepartial maximummaximum domain of attractionstate space transform

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes
Secondary: 60G10: Stationary processes 60H20: Stochastic integral equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 743 - 764
Copyright
Copyright © Applied Probability Trust 2005 

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