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Exceedance of power barriers for integrated continuous-time stationary ergodic stable processes

Part of: Stochastic processes Applications

Published online by Cambridge University Press:  01 July 2016

Uğur Tuncay Alparslan
Uğur Tuncay Alparslan*
Affiliation:
University of Nevada, Reno
*
∗ Current address: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016, USA. Email address: [email protected]
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Abstract

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We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable processes generated by ergodic dissipative flows is established. In the second part an example with the integral of a stable process generated by a conservative flow is analyzed. Finally, the difference in the order of magnitude of the exceedance probability in the two cases is related to the dependence structure of the underlying stable process.

Keywords

Continuous-time stationary stable processexceedance probabilitynonlinear barrierslong-range dependenceintegrated stable process

MSC classification

Primary: 60G52: Stable processes
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 874 - 892
Copyright
Copyright © Applied Probability Trust 2009 

References

Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory (Math. Surveys Monogr. 50). American Mathematical Society, Providence, RI.Google Scholar
Albeverio, S., Hilbert, A. and Kolokoltsov, V. (1997). Transience of stochastically perturbed classical Hamiltonian systems and random wave operators. Stoch. Stoch. Reports 60, 41–55.Google Scholar
Alparslan, U. T. and Samorodnitsky, G. (2007). Asymptotic analysis of the ruin with stationary stable steps generated by dissipative flows. Scand. Actuarial J.. 2007, 180–201.CrossRefGoogle Scholar
Alparslan, U. T. and Samorodnitsky, G. (2007). Ruin probability with certain stationary stable claims generated by conservative flows. Adv. Appl. Prob. 39, 360–384.Google Scholar
Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Berman, S. M. (1973/74). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69–94.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University.Google Scholar
Braverman, M. (2004). Tail probabilities of subadditive functionals on stable processes with continuous and discrete time. Stoch. Process. Appl. 112, 157–183.Google Scholar
Cohen, S. and Samorodnitsky, G. (2006). Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Prob. 16, 1432–1461.Google Scholar
Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.Google Scholar
Kac, M. (1962). Probability theory: Its role and its impact. SIAM Rev. 4, 1–11.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
Kolokoltsov, V. N. (1997). A note on the long time asymptotics of the Brownian motion with application to the theory of quantum measurement. Potential Anal. 7, 759–764.Google Scholar
Krengel, U. (1985). Ergodic Theorems (de Gruyter Stud. Math. 6). Walter de Gruyter, Berlin.Google Scholar
Lachal, A. (1990). Sur l'intégrale du mouvement brownien. C. R. Acad. Sci. Paris 311, 461–464.Google Scholar
Lachal, A. (1991). Sur le premier instant de passage de l'intégrale du mouvement brownien. Ann. Inst. H. Poincaré Prob. Statist. 27, 385–405.Google Scholar
Lachal, A. (1992). Sur les excursions de l'intégrale du mouvement brownien. C. R. Acad. Sci. Paris 314, 1053–1056.Google Scholar
Lachal, A. (1993). L'intégrale du mouvement brownien. J. Appl. Prob. 30, 17–27.Google Scholar
Lachal, A. (1994). Dernier instant de passage pour l'intégrale du mouvement brownien. Stoch. Process. Appl. 49, 57–64.Google Scholar
Lachal, A. (1995). Sur les temps de passages successifs de l'intégrale du mouvement brownien. C. R. Acad. Sci. Paris 321, 903–908.Google Scholar
Lachal, A. (1996). Sur la distribution de certaines fonctionnelles de l'intégrale du mouvement brownien avec dérives parabolique et cubique. Commun. Pure Appl. Math. 49, 1299–1338.Google Scholar
McKean, H. P. Jr. (1963). A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2, 227–235.Google Scholar
Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 1814–1851.Google Scholar
Resnick, S. I. (1999). A Probability Path. Birkhäuser, Boston, MA.Google Scholar
Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Tech. J. 23, 282–332.Google Scholar
Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187.Google Scholar
Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Prob. 32, 1438–1468.Google Scholar
Samorodnitsky, G. (2004). Maxima of continuous-time stationary stable processes. Adv. Appl. Prob. 36, 805–823.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar