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Diffusion approximation of a risk model with non-stationary Hawkes arrivals of claims. (English) Zbl 1447.91038

Summary: We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given.

MSC:

91B05 Risk models (general)
60F17 Functional limit theorems; invariance principles
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Bacry, E.; Mastromatteo, I.; Muzy, JF, Hawkes processes in finance, Market Microstruct Liquid, 1, 1-59 (2015)
[2] Bäuerle, N., Approximation of optimal reinsurance and dividend payout policies, Math Finance, 14, 99-113 (2004) · Zbl 1097.91052 · doi:10.1111/j.0960-1627.2004.00183.x
[3] Blundell C, Beck J, Heller KA (2012) Modelling reciprocating relationships with Hawkes processes. In: Advances in neural information processing systems, pp 2600-2608
[4] Billingsley, P., Convergence of probability measures (1999), New York: Wiley-Interscience, New York · Zbl 0172.21201
[5] Crane, R.; Sornette, D., Robust dynamic classes revealed by measuring the response function of a social system, Proc Natl Acad Sci USA, 105, 15649 (2008) · doi:10.1073/pnas.0803685105
[6] Cramér, H., On the mathematical theory of risk. Skandia Jubilee Volume, Stockholm. Reprinted in: MartinLof, A. (Ed.) Cramer, H. (1994). Collected Works (1930), Berlin: Springer, Berlin
[7] Dȩbicki, K.; Hashorva, E.; Ji, L.; Tan, Z., Finite-time ruin probability of aggregate Gaussian processes, Markov Process Related Fields, 20, 435-450 (2014) · Zbl 1320.60095
[8] Embrechts, P.; Schmidli, H., Ruin estimation for a general insurance risk model, Adv Appl Probab, 26, 2, 404-422 (1994) · Zbl 0811.62096 · doi:10.2307/1427443
[9] Gao, Xuefeng; Zhu, Lingjiong, Large deviations and applications for Markovian Hawkes processes with a large initial intensity, Bernoulli, 24, 4, 2875-2905 (2018) · Zbl 1429.60048 · doi:10.3150/17-BEJ948
[10] Gao X, Zhu L (2018b) Functional central limit theorem for stationary Hawkes processes and its application to infinite-server queues. Queue Syst 60:161-206 · Zbl 1418.60024
[11] Grandell, Jan, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977, sup1, 37-52 (1977) · Zbl 0384.60057 · doi:10.1080/03461238.1977.10405071
[12] Gusto, G.; Schbath, S., FADO: a statistical method to detect favored or avoided distances between occurrences of motifs using the Hawkes’ model, Stat Appl Genet Mol Biol, 4, 1 (2005) · Zbl 1095.62126 · doi:10.2202/1544-6115.1119
[13] Harrison, JM, Ruin problems with compounding assets, Stoch Process Appl, 5, 67-79 (1977) · Zbl 0361.60053 · doi:10.1016/0304-4149(77)90051-5
[14] HAWKES, ALAN G., Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58, 1, 83-90 (1971) · Zbl 0219.60029 · doi:10.1093/biomet/58.1.83
[15] Hawkes, Alan G., Point Spectra of Some Mutually Exciting Point Processes, Journal of the Royal Statistical Society: Series B (Methodological), 33, 3, 438-443 (1971) · Zbl 0238.60094
[16] Iglehart, DL, Diffusion approximations in collective risk theory, J Appl Prob, 6, 285-292 (1969) · Zbl 0191.51202 · doi:10.2307/3211999
[17] Johnson, DH, Point process models of single-neuron discharges, J Comput Neurosci, 3, 4, 275-299 (1996) · doi:10.1007/BF00161089
[18] Lundberg, F., Approximerad framställning av sannolikhetsfunktionen. Aterförsäkring av kollektivrisker. Akad Afhandling (1903), Uppsala: Almqvist och Wiksell, Uppsala
[19] Ogata, Y., Statistical models for earthquake occurrences and residual analysis for point processes, J Amer Stat Assoc, 83, 401, 9-27 (1988) · doi:10.1080/01621459.1988.10478560
[20] Pernice, V.; Staude, B.; Carndanobile, S.; Rotter, S., How structure determines correlations in neuronal networks, PLoS Comput Biol, 85, 031916 (2012)
[21] Reynaud-Bouret, P.; Schbath, S., Adaptive estimation for Hawkes processes; application to genome analysis, Ann Statist, 38, 5, 2781-2822 (2010) · Zbl 1200.62135 · doi:10.1214/10-AOS806
[22] Reynaud-Bouret P, Rivoirard V, Tuleau-Malot C (2013) Inference of functional connectivity in neurosciences via Hawkes processes. In: 1st IEEE Global conference on signal and information processing · Zbl 1197.62033
[23] Schmidli, H., Diffusion approximations for a risk process with the possibility of borrowing and investment, Comm Stat Stoch Models, 10, 365-388 (1994) · Zbl 0793.60095 · doi:10.1080/15326349408807300
[24] Stabile, G.; Torrisi, GL, Risk processes with non-stationary Hawkes arrivals, Methodol Comput Appl Prob, 12, 415-429 (2010) · Zbl 1231.91239 · doi:10.1007/s11009-008-9110-6
[25] Whitt W (2002) Stochastic-process limits: an introduction to stochastic-process limits and their application to queues. Springer Science and Business Media · Zbl 0993.60001
[26] Zhu, Lingjiong, Moderate deviations for Hawkes processes, Statistics & Probability Letters, 83, 3, 885-890 (2013) · Zbl 1266.60090 · doi:10.1016/j.spl.2012.12.011
[27] Zhu, Lingjiong, Central Limit Theorem for Nonlinear Hawkes Processes, Journal of Applied Probability, 50, 3, 760-771 (2013) · Zbl 1306.60015 · doi:10.1239/jap/1378401234
[28] Zhu, Lingjiong, Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims, Insurance: Mathematics and Economics, 53, 3, 544-550 (2013) · Zbl 1290.91107
[29] Zhu, L., Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps, J Appl Prob, 51, 699-712 (2014) · Zbl 1307.60033 · doi:10.1239/jap/1409932668
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