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Prediction in a Poisson cluster model

Part of:Special processesStochastic processes

Published online by Cambridge University Press:  14 July 2016

Muneya Matsui and
Thomas Mikosch
Muneya Matsui*
Affiliation:
Keio University
Thomas Mikosch*
Affiliation:
University of Copenhagen
*
∗ Current address: Mathematical Engineering No.4 Laboratory, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
∗∗ Postal address: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. Email address: mikosch@math.ku.dk
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Abstract

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We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over a finite interval, we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure.

Keywords

Poisson cluster modelpredictionconditional expectationclaims reservinginsuranceLévy processshot noise

MSC classification

Secondary:60K30: Applications (congestion, allocation, storage, traffic, etc.)60G25: Prediction theory60G55: Point processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010, pp. 350 - 366
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported by a JSPS Research Fellowship for Young scientists.

Research partly supported by the danish Reasearch Council (FNU) grants 272-06-0442 and 09-072331

References

[1] Bartlett, M. S. (1963). The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264–296.Google Scholar
[2] Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems 54, 121–140.Google Scholar
[3] Giraitis, L., Molchanov, S. A. and Surgailis, D. (1993). Long memory shot noises and limit theorems with application to Burgers' equation. In New Directions in Time Series Analysis, Part II (IMA Vol. Math. Appl. 46), Springer, New York, pp. 153–176.CrossRefGoogle Scholar
[4] Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.Google Scholar
[5] Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513–525.Google Scholar
[6] Jessen, A. H., Mikosch, T. and Samorodnitsky, G. (2009). Prediction of outstanding payments in a Poisson cluster model. Preprint. Available at www.math.ku.dk/∼mikosch/Preprint/Prediction/prediction260309.pdf.Google Scholar
[7] Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.CrossRefGoogle Scholar
[8] Klüppelberg, C. and Kühn, C. (2004). Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance. Stoch. Process. Appl. 113, 333–351.Google Scholar
[9] Klüppelberg, C. and Mikosch, T. (1995). Delay in claim settlement and ruin probability approximations. Scand. Actuarial J. 1995, 154–168.Google Scholar
[10] Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.Google Scholar
[11] Klüppelberg, C., Mikosch, T. and Schärf, A. (2003). Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9, 467–496.Google Scholar
[12] Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28, 215–243.Google Scholar
[13] Kurtz, T. G. (1997). Limit theorems for workload input models. In Stochastic Networks. Theory and Applications, eds Kelly, F. P., Zachary, S. and Ziedins, I., Oxford University Press, pp. 119–139.Google Scholar
[14] Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.Google Scholar
[15] Lane, J. A. (1987). The Berry–Esseen bound for the Poisson shot-noise. Adv. Appl. Prob. 19, 512–514.Google Scholar
[16] Levy, J. B. and Taqqu, M. S. (2000). Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6, 23–44.Google Scholar
[17] Lewis, P. A. W. (1964). A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398–456.Google Scholar
[18] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bull. 23, 213–225.Google Scholar
[19] Mack, T. (1994). Which stochastic model is underlying the chain ladder method? Insurance Math. Econom. 15, 133–138.Google Scholar
[20] McCormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643–656.Google Scholar
[21] Mikosch, T. (2009). Non-Life Insurance Mathematics, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[22] Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Operat. Res. 32, 890–918.Google Scholar
[23] Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.Google Scholar
[24] Pipiras, V. and Taqqu, M. S. (2000). The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6, 607–614.Google Scholar
[25] Samorodnitsky, G. (1996). A class of shot noise models for financial applications. In Athens Conference on Applied Probability and Time Series Analysis (Lecture Notes Statist. 114), Vol. I, Springer, New York, pp. 332–353.Google Scholar
[26] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[27] Stegeman, A. (2002). Extremal behavior of heavy-tailed on-periods in a superposition of on/off processes. Adv. Appl. Prob. 34, 179–204.Google Scholar
[28] Vere-Jones, D. (1970). Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1–62.Google Scholar