Prediction in a Poisson cluster model
Muneya Matsui and Thomas Mikosch
Source: J. Appl. Probab. Volume 47, Number 2
(2010), 350-366.
Abstract
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.
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Keywords: Poisson cluster model; prediction; conditional expectation; claims reserving; insurance; Lévy process; shot noise
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Permanent link to this document: http://projecteuclid.org/euclid.jap/1276784896
Digital Object Identifier: doi:10.1239/jap/1276784896
Zentralblatt MATH identifier: 05758474
Mathematical Reviews number (MathSciNet): MR2668493
References
Bartlett, M. S. (1963). The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264--296.
Mathematical Reviews (MathSciNet): MR171334
JSTOR: links.jstor.org
Fa\"y", G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems 54, 121--140.
Mathematical Reviews (MathSciNet): MR2268057
Digital Object Identifier: doi:10.1007/s11134-006-9348-z
Zentralblatt MATH: 1119.60075
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.
Mathematical Reviews (MathSciNet): MR1235604
Zentralblatt MATH: 0771.60019
Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709--730.
Mathematical Reviews (MathSciNet): MR809427
Zentralblatt MATH: 0609.60036
Digital Object Identifier: doi:10.2307/1427084
JSTOR: links.jstor.org
Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513--525.
Mathematical Reviews (MathSciNet): MR1013648
Zentralblatt MATH: 0681.60110
Digital Object Identifier: doi:10.2307/1427633
JSTOR: links.jstor.org
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/$\sim$mikosch/Preprint/Prediction/prediction260309.pdf.
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.
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.
Mathematical Reviews (MathSciNet): MR2087964
Zentralblatt MATH: 1075.60020
Digital Object Identifier: doi:10.1016/j.spa.2004.03.015
Klüppelberg, C. and Mikosch, T. (1995). Delay in claim settlement and ruin probability approximations. Scand. Actuarial J. 1995, 154--168.
Mathematical Reviews (MathSciNet): MR1366822
Zentralblatt MATH: 0836.62086
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125--147.
Mathematical Reviews (MathSciNet): MR1354458
Digital Object Identifier: doi:10.2307/3318683
Project Euclid: euclid.bj/1186078364
Zentralblatt MATH: 0842.60030
Klüppelberg, C., Mikosch, T. and Schärf, A. (2003). Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9, 467--496.
Mathematical Reviews (MathSciNet): MR1997493
Digital Object Identifier: doi:10.3150/bj/1065444814
Project Euclid: euclid.bj/1065444814
Zentralblatt MATH: 1044.60013
Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28, 215--243.
Mathematical Reviews (MathSciNet): MR1628422
Digital Object Identifier: doi:10.1023/A:1019190821105
Zentralblatt MATH: 0908.90131
Kurtz, T. G. (1997). Limit theorems for workload input models. In Stochastic Networks. Theory and Applications, eds F. P. Kelly, S. Zachary and I. Ziedins, Oxford University Press, pp. 119--139.
Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287--301.
Mathematical Reviews (MathSciNet): MR741131
Zentralblatt MATH: 0541.60018
Digital Object Identifier: doi:10.2307/3213640
JSTOR: links.jstor.org
Lane, J. A. (1987). The Berry--Esseen bound for the Poisson shot-noise. Adv. Appl. Prob. 19, 512--514.
Mathematical Reviews (MathSciNet): MR889950
Zentralblatt MATH: 0618.60046
Digital Object Identifier: doi:10.2307/1427432
JSTOR: links.jstor.org
Levy, J. B. and Taqqu, M. S. (2000). Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6, 23--44.
Mathematical Reviews (MathSciNet): MR1781180
Digital Object Identifier: doi:10.2307/3318631
Project Euclid: euclid.bj/1082665378
Zentralblatt MATH: 0954.60071
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.
Mathematical Reviews (MathSciNet): MR174090
JSTOR: links.jstor.org
Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bull. 23, 213--225.
Mack, T. (1994). Which stochastic model is underlying the chain ladder method? Insurance Math. Econom. 15, 133--138.
Mathematical Reviews (MathSciNet): MR1333086
McCormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643--656.
Mathematical Reviews (MathSciNet): MR1464600
Zentralblatt MATH: 0886.60041
Digital Object Identifier: doi:10.2307/3215091
JSTOR: links.jstor.org
Mikosch, T. (2009). Non-Life Insurance Mathematics, 2nd edn. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR2503328
Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Operat. Res. 32, 890--918.
Mathematical Reviews (MathSciNet): MR2363203
Zentralblatt MATH: 05279760
Digital Object Identifier: doi:10.1287/moor.1070.0267
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.
Mathematical Reviews (MathSciNet): MR1890056
Digital Object Identifier: doi:10.1214/aoap/1015961155
Project Euclid: euclid.aoap/1015961155
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.
Mathematical Reviews (MathSciNet): MR1777685
Digital Object Identifier: doi:10.2307/3318508
Project Euclid: euclid.bj/1081449595
Zentralblatt MATH: 0963.60032
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.
Mathematical Reviews (MathSciNet): MR1466727
Zentralblatt MATH: 0861.60057
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Mathematical Reviews (MathSciNet): MR1739520
Stegeman, A. (2002). Extremal behavior of heavy-tailed on-periods in a superposition of on/off processes. Adv. Appl. Prob. 34, 179--204.
Mathematical Reviews (MathSciNet): MR1895337
Zentralblatt MATH: 1013.60010
Digital Object Identifier: doi:10.1239/aap/1019160956
Project Euclid: euclid.aap/1019160956
Vere-Jones, D. (1970). Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1--62.
Mathematical Reviews (MathSciNet): MR272087
JSTOR: links.jstor.org
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