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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: boris.buchmann@maths.anu.edu.au
∗∗ Postal address: Centre for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany. Email address: cklu@ma.tum.de
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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 

References

Berman, S. M. (1971). Maxima and high level excursions of stationary Gaussian processes. Trans. Amer. Math. Soc. 160, 65–185.Google Scholar
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth, Belmont, CA.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Borkovec, M. and Klüppelberg, C. (1998). Extremal behavior of diffusion models in finance. Extremes 1, 47–80. (Correction: 379–380.)Google Scholar
Buchmann, B. and Klüppelberg, C. (2005). Fractional integral equations and state space transforms. To appear in Bernoulli.Google Scholar
Burnecki, K. and Michna, Z. (2002). Simulation of Pickands constants. Prob. Math. Statist. 22, 193–199.Google Scholar
Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Prob. 8, 14 pp.Google Scholar
Davis, R. A. (1982). Maximum and minimum of one-dimensional diffusions. Stoch. Process. Appl. 13, 1–9.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Pickands, J. III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Trans. Math. Monogr. 148). American Mathematical Society, Providence, RI.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, London.Google Scholar