Abstract
We consider risk processes with non-stationary Hawkes claims arrivals, and we study the asymptotic behavior of infinite and finite horizon ruin probabilities under light-tailed conditions on the claims. Moreover, we provide asymptotically efficient simulation laws for ruin probabilities and we give numerical illustrations of the theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albrecher H, Asmussen S (2006) Ruin probabilities and aggregate claims distributions for shot noise Cox processes. Scand Actuar J 2:86–110
Asmussen S (2000) Ruin probabilities. World Scientific, Singapore
Bordenave C, Torrisi GL (2007) Large deviations of Poisson cluster processes. Stoch Models 23:593–625
Brémaud P (1981) Point processes and queues. Springer, New York
Brémaud P (2000) An insensitivity property of Lundberg’s estimate for delayed claims. J Appl Probab 37:914–917
Brémaud P, Massoulié L (1996) Stability of nonlinear Hawkes processes. Ann Probab 24:1563–1588
Brémaud P, Nappo G, Torrisi GL (2002) Rate of convergence to equilibrium of marked Hawkes processes. J Appl Probab 39:123–136
Chavez-Demoulin V, Davison AC, Mc Neil AJ (2005) Estimating value-at-risk: a point process approach. Quantitative Finance 5:227–234
Daley DJ, Vere-Jones D (2003) An introduction to the theory of point processes, 2nd edn. Springer, New York
Dembo A, Zeitouni O (1998) Large deviations techniques and applications, 2nd edn. Springer, New York
Duffield NG, O’Connell N (1995) Large deviations and overflow probabilities for the general single-server queue, with applications. Math Proc Camb Philos Soc 118:363–374
Ganesh A, O’Connell N, Wischik D (2004) Big queues. In: Lecture Notes in Mathematics, vol 1838. Springer, Berlin
Ganesh A, Torrisi GL (2006) A class of risk processes with delayed claims: ruin probabilities estimates under heavy-tailed conditions. J Appl Probab 43:916–926
Hawkes AG (1971a) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58:83–90
Hawkes AG (1971b) Point spectra of some mutually exciting point processes. J R Stat Soc Ser B 33:438–443
Hawkes AG, Oakes D (1974) A cluster representation of a self-exciting process. J Appl Probab 11:493–503
Jacod J, Mémin J (1975) Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Dép math. Inform. Univ. de Rennes
Jagers P (1975) Branching processes with biological applications. Wiley, London
Kerstan J (1964) Teilprozesse Poissonscher Prozesse. In: Transactions of the third Prague conference on information theory, statistical decision functions, random processes. Czech. Academy of Science, Prague, pp 377–403
Klüppelberg C, Mikosch T (1995) Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1:125–147
Lehtonen T, Nyrhinen H (1992) Simulating level-crossing probabilities by importance sampling. Adv Appl Probab 24:858–874
Lipster RS, Shiryayev AN (1978) Statistics of random processes II: applications. Springer, New York
Massoulié L (1998) Stability results for a general class of interacting point processes dynamics, and applications. Stoch Process their Appl 75:1–30
Rolski T, Schmidli V, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, Chichester
Siegmund D (1976) Importance sampling in the Monte Carlo study of sequential tests. Ann Stat 4:673–684
Torrisi GL (2002) A class of interacting marked point processes: rate of convergence to equilibrium. J Appl Probab 39:137–161
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stabile, G., Torrisi, G.L. Risk Processes with Non-stationary Hawkes Claims Arrivals. Methodol Comput Appl Probab 12, 415–429 (2010). https://doi.org/10.1007/s11009-008-9110-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-008-9110-6