Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T03:31:12.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

Part of: Stochastic processes Mathematical economics Special processes Other special functions

Published online by Cambridge University Press:  30 January 2018

Romain Biard  and
Bruno Saussereau
Romain Biard*
Affiliation:
Laboratoire de Mathématiques de Besançon
Bruno Saussereau*
Affiliation:
Laboratoire de Mathématiques de Besançon
*
∗ Postal address: Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France.
∗ Postal address: Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

Keywords

Fractional Poisson processrenewal processlong-range dependenceruin probability

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G55: Point processes 91B30: Risk theory, insurance
Secondary: 60K05: Renewal theory 33E12: Mittag-Leffler functions and generalizations
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 727 - 740
Copyright
© Applied Probability Trust 

References

Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions ({Encyclopedia Math. Appl. 71}). Cambridge University Press.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14), 2nd edn. World Scientific, Hackensack, NJ.Google Scholar
Beghin, L. and Macci, C. (2012). Alternative forms of compound fractional Poisson processes. Abstr. Appl. Anal. 2012, 747503.Google Scholar
Beghin, L. and Macci, C. (2013). Large deviations for fractional Poisson processes. Statist. Prob. Lett. 83, 1193–1202.CrossRefGoogle Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.Google Scholar
Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684–709.Google Scholar
Benson, D. A., Schumer, R. and Meerschaert, M. M. (2007). Recurrence of extreme events with power-law interarrival times. Geophys. Res. Lett. 34, L16404.Google Scholar
Borovkov, K. A. and Dickson, D. C. M. (2008). On the ruin time distribution for a Sparre Andersen process with exponential claim sizes. Insurance Math. Econom. 42, 1104–1108.Google Scholar
Cahoy, D. O., Uchaikin, V. V. and Woyczynski, W. A. (2010). Parameter estimation for fractional Poisson processes. J. Statist. Planning Infer. 140, 3106–3120.Google Scholar
Daley, D. J. (1999). The Hurst index of long-range dependent renewal processes. Ann. Prob. 27, 2035–2041.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Process. Appl. 70, 265–282.Google Scholar
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.Google Scholar
Jumarie, G. (2001). Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fractals 12, 2577–2587.Google Scholar
Lagerås, A. N. (2005). A renewal-process-type expression for the moments of inverse subordinators. J. Appl. Prob. 42, 1134–1144.Google Scholar
Laskin, N. (2003). Fractional Poisson process. Chaotic transport and complexity in classical and quantum dynamics. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213.Google Scholar
Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64.Google Scholar
Malinovskii, V. K. (1998). Non-Poissonian claims' arrivals and calculation of the probability of ruin. Insurance Math. Econom. 22, 123–138.Google Scholar
Mazza, C. and Rullière, D. (2004). A link between wave governed random motions and ruin processes. Insurance Math. Econom. 35, 205–222.Google Scholar
Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2011). The fractional Poisson process and the inverse stable subordinator. Electron. J. Prob. 16, 1600–1620.Google Scholar
Musson, R. M. W., Tsapanos, T. and Nakas, C. T. (2002). A power-law function for earthquake interarrival time and magnitude. Bull. Seismol. Soc. Amer. 92, 1783–1794.Google Scholar
Repin, O. N. and Saichev, A. I. (2000). Fractional Poisson law. Radiophys. Quantum Electron. 43, 738–741.Google Scholar
Saussereau, B. (2012). Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion. Bernoulli 18, 1–23.Google Scholar
Scalas, E. (2012). A class of CTRWs: compound fractional Poisson processes. In Fractional Dynamics, World Scientific, Hackensack, NJ, pp. 353–374.Google Scholar
Uchaikin, V. V., Cahoy, D. O. and Sibatov, R. T. (2008). Fractional processes: from Poisson to branching one. Internat. J. Bifur. Chaos Appl. Sci. Eng. 18, 2717–2725.Google Scholar
Wang, R. and Liu, H. (2002). On the ruin probability under a class of risk processes. Astin Bull. 32, 81–90.Google Scholar