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Ruin probability with certain stationary stable claims generated by conservative flows

Part of: Stochastic processes Applications

Published online by Cambridge University Press:  01 July 2016

Uğur Tuncay Alparslan  and
Gennady Samorodnitsky
Uğur Tuncay Alparslan*
Affiliation:
University of Nevada, Reno
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
∗ Postal address: Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA. Email address: [email protected]
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA.
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Abstract

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We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.

Keywords

Ruin probabilitystable processheavy tailergodic theorylong range dependenceconservative flownull-recurrent Markov chainfractional Brownian motion

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 39 , Issue 2 , June 2007 , pp. 360 - 384
Copyright
Copyright © Applied Probability Trust 2007 

References

Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes (Inst. Math. Statist. Lecture Notes Monogr. Ser. 12). Institute of Mathematical Statistics, Hayward, CA.CrossRefGoogle Scholar
Alparslan, U. T. and Samorodnitsky, G. (2006). Asymptotic analysis of exceedance probability with stationary stable steps generated by dissipative flows. To appear in Scand. Actuarial J..Google Scholar
Berman, S. M. (1973). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, 69–94.CrossRefGoogle Scholar
Braverman, M. (2004). Tail probabilities of subadditive functionals on stable processes with continuous and discrete time. Stoch. Process. Appl. 112, 157–183.CrossRefGoogle Scholar
Cohen, S. and Samorodnitsky, G. (2006). Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Prob. 16, 1432–1461.CrossRefGoogle Scholar
Cramér, H. (1926). Review of F. Lundberg's “Försäkringsteknisk riskutjämning”. Skand. Aktuarietidskr.Google Scholar
Cramér, H. (1930). On the mathematical theory of risk. Skandia Jubilee, Stockholm.Google Scholar
Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. (New York) 33). Springer, Berlin.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
Krengel, U. (1985). Ergodic Theorems (De Gruyter Studies Math. 6). De Gruyter, Berlin.CrossRefGoogle Scholar
Lundberg, F. (1903). I approximerad framställning av sannolikhetsfunktionen. ii å terförsäkring av kollektivrisker. , Akad. Afhandling. Almqvist och Wiksell, Uppsala.Google Scholar
Lundberg, F. (1909). Über die theorie der rückversicherung. Trans. VI Int. Congr. Actuaries 1, 877–955.Google Scholar
Lundberg, F. (1926). Försäkringsteknisk Riskutjämning. F. Englunds Boktryckeri AB, Stockholm.Google Scholar
Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 1814–1851.CrossRefGoogle Scholar
Resnick, S. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.Google Scholar
Resnick, S. I. (1999). A Probability Path. Birkhäuser, Boston, MA.Google Scholar
Resnick, S., Samorodnitsky, G. and Xue, F. (1999). How misleading can sample ACFs of stable MAs be? (Very!). Ann. Appl. Prob. 9, 797–817.CrossRefGoogle Scholar
Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187.CrossRefGoogle Scholar
Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2, 365–377.CrossRefGoogle 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 & Hall, New York.Google Scholar
Teugels, J. L. (1970). Regular variation of Markov renewal functions. J. London Math. Soc. (2) 2, 179–190.CrossRefGoogle Scholar