Simulation of Pickands constants

• Autorzy: Burnecki K. , Michna Z.
• Języki publikacji: EN

Abstrakty

EN

Pickands constants appear in the asymptotic formulas for extremes of Gaussian processes. The explicit formula of Pickands constants does not exist. Moreover, in the literature there is no numerical approximation. In this paper we compute numerically Pickands constants by the use of change of measure technique. To this end we apply two different algorithms to simulate fractional Brownian motion. Finally, we compare the approximations with a theoretical hypothesis and a recently obtained lower bound on the constants. The results justify the hypothesis.

Słowa kluczowe

EN

Pickands constant; fractional Brownian motion; change of measure; Cholesky factorization; fGp algorithm

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2002
• Tom: Vol. 22, Fasc. 1
• Strony: 193--199
• Opis fizyczny: Biblogr. 15 poz., wykr.

Twórcy

autor

Burnecki K.

• Institute of Mathematics, Wroclaw University of Technology, 50-370 Wrocław, Poland

autor

Michna Z.

• Department of Mathematics, Wrocław University of Economics, 53-345 Wrocław, Poland

Bibliografia

• [1] R. J. Adler, An Introduction to Continuity, Extreme, and Related Topics for General Gaussian Processes, Institute of Mathematical Statistics, Hayward, 1990.
• [2] S. Asmussen, Applied Probability and Queues, Wiley, New York 1987.
• [3] S. Asmussen, Stochastic simulation with a view towards stochastic processes, Lecture Notes No 2, Centre for Mathematical Physics and Stochastics, University of Aarhus, Denmark, 1998.
• [4] D. C. Caccia, D. Percival, M. J. Cannon, G. Raymond and J. B. Bassingthwaighte, Analyzing exact fractal time series : evaluating dispersional analysis and rescaled range methods, Physica A 246 (1997), pp. 609-632.
• [5] R. B. Davies and D. S. Harte, Tests for Hurst effect, Biometrika 74 (1987), pp. 95-101.
• [6] K. Dębicki, Ruin probability for Gaussian integrated processes, Stochastic Process. Appl. 98 (1) (2002), pp. 151-174.
• [7] K. Dębicki, Z. Michna and T. Rolski, Bounds and simulation of generalized Pickands constants. Tech. report, University of Wrocław, Wrocław 2000.
• [8] J. Hüsler and V. Piterbarg, Extremes of a certain class of Gaussian processes, Stochastic Process. Appl. 83 (1999), pp. 338-357.
• [9] A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes, Marcel Dekker Inc., New York 1992.
• [10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1991.
• [11] Z. Michna, On tail probabilities and first passage times for fractional Brownian motion, Math. Meth. Oper. Res. 49 (1999), pp. 335-354.
• [12] O. Narayan, Exact asymptotic queue length distribution for fractional Brownian traffic, Advances in Performance Analysis 1 (1) (1998), pp. 39-63.
• [13] J. Pickands III, Asymptotic properties of the maximum in a stationary Gaussian process, Trans. Amer. Math. Soc. 145 (1969), pp. 75-86.
• [14] V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Transi. Math. Monographs 148, AMS, Providence 1996.
• [15] Q. Shao, Bounds and estimators of a basic constant in extreme value theory of Gaussian processes, Statistica Sinica 6 (1996), pp. 245-257.