Asymptotic correlation structure of discounted incurred but not reported claims under fractional Poisson arrival process.

*(English)*Zbl 1430.90188Summary: This paper studies the joint moments of a compound discounted renewal process observed at different times with each arrival removed from the system after a random delay. This process can be used to describe the aggregate (discounted) incurred but not reported claims in insurance and also the total number of customers in an infinite server queue. It is shown that the joint moments can be obtained recursively in terms of the renewal density, from which the covariance and correlation structures are derived. In particular, the fractional Poisson process defined via the renewal approach is also considered. Furthermore, the asymptotic behaviour of covariance and correlation coefficient of the aforementioned quantities is analyzed as the time horizon goes to infinity. Special attention is paid to the cases of exponential and Pareto delays. Some numerical examples in relation to our theoretical results are also presented.

##### MSC:

90B22 | Queues and service in operations research |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60G22 | Fractional processes, including fractional Brownian motion |

##### Keywords:

applied probability; fractional Poisson process; incurred but not reported (IBNR) claims; infinite server queues; correlation
PDF
BibTeX
XML
Cite

\textit{E. C. K. Cheung} et al., Eur. J. Oper. Res. 276, No. 2, 582--601 (2019; Zbl 1430.90188)

Full Text:
DOI

##### References:

[1] | Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, Applied mathematics series 55, (1972), Dover Publications |

[2] | Badescu, A. L.; Lin, X. S.; Tang, D., A marked Cox model for the number of IBNR claims: Theory, Insurance: Mathematics and Economics, 69, 29-37, (2016) |

[3] | Badescu, A. L., Lin, X. S., & Tang, D. (2016b). A marked Cox model for the number of IBNR claims: Estimation and application. Preprint available at SSRN at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2747223. |

[4] | Benson, D. A.; Schumer, R.; Meerschaert, M. M., Recurrence of extreme events with power-law interarrival times, Geophysical Research Letters, 34, 16, L16404, (2007) |

[5] | Biard, R.; Saussereau, B., Fractional Poisson process: Long-range dependence and applications in ruin theory, Journal of Applied Probability, 51, 727-740, (2014) |

[6] | Blanchet, J.; Lam, H., A heavy traffic approach to modeling large life insurance portfolios, Insurance: Mathematics and Economics, 53, 237-251, (2013) |

[7] | Blom, J.; De Turck, K.; Mandjes, M., Refined large deviations asymptotics for Markov-modulated infinite-server systems, European Journal of Operational Research, 259, 1036-1044, (2017) |

[8] | Blom, J.; Kella, O.; Mandjes, M.; Thorsdottir, H., Markov-modulated infinite-server queues with general service times, Queueing Systems, 76, 403-424, (2014) |

[9] | Blom, J.; Mandjes, M., A large-deviations analysis of Markov-modulated infinite-server queues, Operations Research Letters, 41, 220-225, (2013) |

[10] | Brown, M.; Ross, S. M., Some results for infinite server queues, Journal of Applied Probability, 6, 604-611, (1969) |

[11] | Cahoy, D. O.; Uchaikin, V. V.; Woyczynski, W. A., Parameter estimation for fractional Poisson processes, Journal of Statistical Planning and Inference, 140, 3106-3120, (2010) |

[12] | Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29, 333-344, (2001) |

[13] | Finkelstein, M.; Cha, J. H., Stochastic modeling for reliability, Springer series in reliability engineering, (2013), Springer-Verlag: Springer-Verlag London |

[14] | Jansen, H. M.; Mandjes, M. R.H.; De Turck, K.; Wittevrongel, S., A large deviations principle for infinite-server queues in a random environment, Queueing Systems, 82, 199-235, (2016) |

[15] | Karlsson, J. E., A stochastic model for time lag in reporting of claims, Journal of Applied Probability, 11, 382-387, (1974) |

[16] | Keilson, J.; Seidmann, A., M/G/∞ with batch arrivals, Operations Research Letters, 7, 219-222, (1988) |

[17] | Keilson, J.; Servi, L. D., Networks of non-homogeneous M(t)/G/∞ systems, Journal of Applied Probability, 31A, 157-168, (1994) |

[18] | Kummer, E. E., De integralibus quibusdam definitis et seriebus infinitis, Journal für die reine und angewandte Mathematik, 17, 228-242, (1837) |

[19] | Landriault, L.; Willmot, G. E.; Xu, D., Analysis of IBNR claims in renewal insurance models, Scandinavian Actuarial Journal, 7, 628-650, (2017) |

[20] | Laskin, N., Fractional Poisson process, Communications in Nonlinear Science and Numerical Simulation, 8, 201-213, (2003) |

[21] | Leonenko, N. N.; Meerschaert, M. M.; Schilling, R. L.; Sikorskii, A., Correlation structure of time-changed Lévy processes, Communications in Applied and Industrial Mathematics, 6, E-483, (2014) |

[22] | Léveillé, G.; Adékambi, F., Covariance of discounted compound renewal sums with a stochastic interest rate, Scandinavian Actuarial Journal, 1, 1-20, (2010) |

[23] | Léveillé, G.; Adékambi, F., Joint moments of discounted compound renewal sums, Scandinavian Actuarial Journal, 1, 40-55, (2012) |

[24] | Léveillé, G.; Hamel, E., A compound renewal model for medical malpractice insurance, European Actuarial Journal, 3, 471-490, (2013) |

[25] | Liu, L.; Kashyap, B. R.K.; Templeton, J. G.C., On the GI^x/G/∞ system, Journal of Applied Probability, 27, 671-683, (1990) |

[26] | Maheshwari, A.; Vellaisamy, P., On the long-range dependence of fractional Poisson and negative binomial processes, Journal of Applied Probability, 53, 989-1000, (2016) |

[27] | Mainardi, F.; Gorenflo, R.; Scalas, E., A fractional generalization of the Poisson processes, Vietnam Journal of Mathematics, 32, 53-64, (2004) |

[28] | Meerschaert, M. M.; Nane, E.; Vellaisamy, P., The fractional Poisson process and the inverse stable subordinator, Electronic Journal of Probability, 16, 1600-1620, (2011) |

[29] | Mikosch, T., Non-life insurance mathematics: An introduction with the Poisson process, (2009), Springer |

[30] | Moiseev, A.; Nazarov, A., Queueing network MAP - (GI/∞)^k with high-rate arrivals, European Journal of Operational Research, 254, 161-168, (2016) |

[31] | Orsingher, O.; Polito, F., On a fractional linear birth-death process, Bernoulli, 17, 114-137, (2011) |

[32] | Pang, G.; Whitt, W., Infinite-server queues with batch arrivals and dependent service times, Probability in the Engineering and Informational Sciences, 26, 197-220, (2012) |

[33] | Repin, O. N.; Saichev, A. I., Fractional Poisson law, Radiophysics and Quantum Electronics, 43, 738-741, (2000) |

[34] | Resnick, S.; Rootzen, H., Self-similar communication models and very heavy tails, Annals of Applied Probability, 10, 753-778, (2000) |

[35] | Ridder, A., Importance sampling algorithms for first passage time probabilities in the infinite server queue, European Journal of Operational Research, 199, 176-186, (2009) |

[36] | Ross, S., Introduction to probability models, (2014), Academic Press |

[37] | Shaked, M.; Shanthikumar, G., Stochastic orders, Springer series in statistics, (2007) |

[38] | Willmot, G. E., A queueing theoretic approach to the analysis of the claims payment process, Transactions of Society of Actuaries, 42, 447-497, (1990) |

[39] | Woo, J. K., On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays, Insurance Mathematics and Economics, 70, 354-363, (2016) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.