Abstract
The Poisson process suitably models the time of successive events and thus has numerous applications in statistics, in economics, it is also fundamental in queueing theory. Economic applications include trading and nowadays particularly high frequency trading. Of outstanding importance are applications in insurance, where arrival times of successive claims are of vital importance. It turns out, however, that real data do not always support the genuine Poisson process. This has lead to variants and augmentations such as time dependent and varying intensities, for example. This paper investigates the fractional Poisson process. We introduce the process and elaborate its main characteristics. The exemplary application considered here is the Carmér–Lundberg theory and the Sparre Andersen model. The fractional regime leads to initial economic stress. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process.
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References
Ahmadi-Javid, A., Pichler, A.: An analytical study of norms and Banach spaces induced by the entropic value-at-risk. Math. Financ. Econ. 11(4), 527–550 (2017)
Aletti, G., Leonenko, N.N., Merzbach, E.: Fractional Poisson processes and martingales. J. Stat. Phys. 170(4), 700–730 (2018)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)
Beghin, L., Orsingher, E.: Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14(61), 1790–1827 (2009)
Beghin, L., Orsingher, E.: Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15(22), 684–709 (2010)
Bingham, N.H.: Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheorie Verw. Geb. 17, 1–22 (1971)
Borodin, A.N., Salminen, P.: Handbook of Brownian motion-facts and formulae. Birkhäuser (2012). https://doi.org/10.1007/978-3-0348-7652-0
Cont, R., Tankov, P.: Financial Modeling with Jump Processes. CRC Press, Boca Raton (2004)
Daley, D.J.: The Hurst index for a long-range dependent renewal processes. Ann. Probab. 27(4), 2035–2041 (1999)
Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. Art. ID 298628, 51 (2011)
Kataria, K.K., Vellaisamy, P.: On densities of the product, quotient and power of independent subordinators. J. Math. Anal. Appl. 462, 1627–1643 (2018)
Kerss, A., Leonenko, N.N., Sikorskii, A.: Fractional Skellam processes with applications to finance. Fract. Calc. Appl. Anal. 17, 532–551 (2014)
Khinchin, A.Y.: Mathematical Methods in the Theory of Queueing. Hafner Publishing Co., New York (1969)
Kumar, A., Vellaisamy, P.: Inverse tempered stable subordinators. Stat. Probab. Lett. 103, 134–141 (2015)
Kusuoka, S.: On law invariant coherent risk measures. In: Advances in Mathematical Economics, vol. 3 Ch. 4, Springer, pp. 83–95 (2001)
Leonenko, N.N., Meerschaert, M.M., Schilling, R.L., Sikorskii, A.: Correlation structure of time-changed Lévy processes. Commun. Appl. Ind. Math. 6(1), e-483,22 (2014)
Leonenko, N.N., Scalas, E., Trinh, M.: Limit Theorems for the Fractional Non-homogeneous Poisson Process. J. Appl. Prob. (in Press) (2019)
Leonenko, N.N., Scalas, E., Trinh, M.: The fractional non-homogeneous Poisson process. Stat. Probab. Lett. 120, 147–156 (2017)
Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64 (2004)
Mainardi, F., Gorenflo, R., Vivoli, A.: Renewal processes of Mittag-Leffler and Wright type. Fract. Calc. Appl. Anal. 8(1), 7–38 (2005)
Meerschaert, M.M., Nane, E., Vellaisamy, P.: The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16(59), 1600–1620 (2011)
Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter, Berlin (2012)
Mikosch, T.: Non-life Insurance Mathematics: An Introduction with the Poisson Process. Springer, Berlin (2009)
Pflug, G.C., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007). https://doi.org/10.1142/9789812708724
Raberto, M., Scalas, E., Mainardi, F.: Waiting times and returns in high-frequency financial data: an empirical study. Phys. A 314, 749–755 (2002)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)
Scalas, E., Gorenflo, R., Luckock, H., Mainardi, F., Mantelli, M., Raberto, M.: Anomalous waiting times in high-frequency financial data. Quant. Finance 4, 695–702 (2004)
Veillette, M., Taqqu, M.S.: Numerical computation of first passage times of increasing Lévy processes. Methodol. Comput. Appl. Probab. 12(4), 695–729 (2010)
Veillette, M., Taqqu, M.S.: Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes. Stat. Probab. Lett. 80(7–8), 697–705 (2010)
Young, V.R.: Premium principles. In: Encyclopedia of Actuarial Science (2006)
Acknowledgements
N. Leonenko was supported in particular by Australian Research Council’s Discovery Projects funding scheme (Project DP160101366) and by Project MTM2015-71839-P of MINECO, Spain (co-funded with FEDER funds).
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Kumar, A., Leonenko, N. & Pichler, A. Fractional risk process in insurance. Math Finan Econ 14, 43–65 (2020). https://doi.org/10.1007/s11579-019-00244-y
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DOI: https://doi.org/10.1007/s11579-019-00244-y