Asymptotic correlation structure of discounted Incurred But Not Reported claims under fractional Poisson arrival process

Abramowitz, 1972, Handbook of mathematical functions with formulas, graphs, and mathematical tables Badescu, 2016, A marked Cox model for the number of IBNR claims: Theory, Insurance: Mathematics and Economics, 69, 29 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. Benson, 2007, Recurrence of extreme events with power-law interarrival times, Geophysical Research Letters, 34, L16404, 10.1029/2007GL030767 Biard, 2014, Fractional Poisson process: Long-range dependence and applications in ruin theory, Journal of Applied Probability, 51, 727, 10.1239/jap/1409932670 Blanchet, 2013, A heavy traffic approach to modeling large life insurance portfolios, Insurance: Mathematics and Economics, 53, 237 Blom, 2017, Refined large deviations asymptotics for Markov-modulated infinite-server systems, European Journal of Operational Research, 259, 1036, 10.1016/j.ejor.2016.10.050 Blom, 2014, Markov-modulated infinite-server queues with general service times, Queueing Systems, 76, 403, 10.1007/s11134-013-9368-4 Blom, 2013, A large-deviations analysis of Markov-modulated infinite-server queues, Operations Research Letters, 41, 220, 10.1016/j.orl.2013.01.009 Brown, 1969, Some results for infinite server queues, Journal of Applied Probability, 6, 604, 10.2307/3212106 Cahoy, 2010, Parameter estimation for fractional Poisson processes, Journal of Statistical Planning and Inference, 140, 3106, 10.1016/j.jspi.2010.04.016 Dickson, 2001, On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29, 333 Finkelstein, 2013, Stochastic modeling for reliability, 10.1007/978-1-4471-5028-2_2 Jansen, 2016, A large deviations principle for infinite-server queues in a random environment, Queueing Systems, 82, 199, 10.1007/s11134-015-9470-x Karlsson, 1974, A stochastic model for time lag in reporting of claims, Journal of Applied Probability, 11, 382, 10.2307/3212761 Keilson, 1988, M/G/∞ with batch arrivals, Operations Research Letters, 7, 219, 10.1016/0167-6377(88)90034-X Keilson, 1994, Networks of non-homogeneous M(t)/G/∞ systems, Journal of Applied Probability, 31A, 157, 10.2307/3214954 Kummer, 1837, De integralibus quibusdam definitis et seriebus infinitis, Journal für die reine und angewandte Mathematik, 17, 228, 10.1515/crll.1837.17.228 Landriault, 2017, Analysis of IBNR claims in renewal insurance models, Scandinavian Actuarial Journal, 7, 628, 10.1080/03461238.2016.1225265 Laskin, 2003, Fractional Poisson process, Communications in Nonlinear Science and Numerical Simulation, 8, 201, 10.1016/S1007-5704(03)00037-6 Leonenko, 2014, Correlation structure of time-changed Lévy processes, Communications in Applied and Industrial Mathematics, 6, E-483, 10.1685/journal.caim.483 Léveillé, 2010, Covariance of discounted compound renewal sums with a stochastic interest rate, Scandinavian Actuarial Journal, 1, 1 Léveillé, 2012, Joint moments of discounted compound renewal sums, Scandinavian Actuarial Journal, 1, 40, 10.1080/03461238.2010.503426 Léveillé, 2013, A compound renewal model for medical malpractice insurance, European Actuarial Journal, 3, 471, 10.1007/s13385-013-0080-x Liu, 1990, On the GIx/G/∞ system, Journal of Applied Probability, 27, 671, 10.2307/3214550 Maheshwari, 2016, On the long-range dependence of fractional Poisson and negative binomial processes, Journal of Applied Probability, 53, 989, 10.1017/jpr.2016.59 Mainardi, 2004, A fractional generalization of the Poisson processes, Vietnam Journal of Mathematics, 32, 53 Meerschaert, 2011, The fractional Poisson process and the inverse stable subordinator, Electronic Journal of Probability, 16, 1600, 10.1214/EJP.v16-920 Mikosch, 2009 Moiseev, 2016, Queueing network MAP - (GI/∞)k with high-rate arrivals, European Journal of Operational Research, 254, 161, 10.1016/j.ejor.2016.04.011 Orsingher, 2011, On a fractional linear birth-death process, Bernoulli, 17, 114, 10.3150/10-BEJ263 Pang, 2012, Infinite-server queues with batch arrivals and dependent service times, Probability in the Engineering and Informational Sciences, 26, 197, 10.1017/S0269964811000337 Repin, 2000, Fractional Poisson law, Radiophysics and Quantum Electronics, 43, 738, 10.1023/A:1004890226863 Resnick, 2000, Self-similar communication models and very heavy tails, Annals of Applied Probability, 10, 753, 10.1214/aoap/1019487509 Ridder, 2009, Importance sampling algorithms for first passage time probabilities in the infinite server queue, European Journal of Operational Research, 199, 176, 10.1016/j.ejor.2008.10.028 Ross, 2014 Shaked, 2007, Stochastic orders Willmot, 1990, A queueing theoretic approach to the analysis of the claims payment process, Transactions of Society of Actuaries, 42, 447 Woo, 2016, On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays, Insurance Mathematics and Economics, 70, 354, 10.1016/j.insmatheco.2016.07.004