Abstract
In this paper, we consider a classical risk model refracted at given level. We give an explicit expression for the joint density of the ruin time and the cumulative number of claims counted up to ruin time. The proof is based on solving some integro-differential equations and employing the Lagrange’s Expansion Theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Badescu, A., Drekic, S., Landriault, D.: Analysis of a threshold dividend strategy for a MAP risk model. Scand. Actuar. J. 4, 227–247 (2007)
De Finetti, B.: Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, pp. 433–443 (1957)
Dickson, D.C.M.: The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model. Insur. Math. Econ. 50(3), 334–337 (2012)
Dickson, D.C.M., Hipp, C.: On the time to ruin for Erlang (2) risk processes. Insur. Math. Econ. 29(3), 333–344 (2001)
Dickson, D.C.M., Willmot, G.E.: The density of the time to ruin in the classical Poisson risk model. Astin Bull. 35(1), 45–60 (2005)
Frostig, E., Pitts, S.M., Politis, K.: The time to ruin and the number of claims until ruin for phase-type claims. Insur. Math. Econ. 51(1), 19–25 (2012)
Gao, H., Yin, C.: The perturbed Sparre Andersen model with a threshold dividend strategy. J. Comput. Appl. Math. 220(1–2), 394–408 (2008)
Gerber, H.U., Shiu, E.S.W.: The time value of ruin in a Sparre Andersen model. North Am. Actuar. J. 9(2), 49–69 (2005)
Kyprianou, A.E., Loeffen, R.L.: Refracted Lévy processes. Ann. Inst. H. Poincaré Probab. Stat. 46(1), 24–44 (2010)
Lagrange, J.L.: Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Chez Haude et Spener, Libraires de la Cour & de l’Académie royale (1770)
Landriault, D., Shi, T., Willmot, G.E.: Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions. Insur. Math. Econ. 49(3), 371–379 (2011)
Li, S., Garrido, J.: On a class of renewal risk models with a constant dividend barrier. Insur. Math. Econ. 35(3), 691–701 (2004)
Li, S., Garrido, J.: On ruin for the Erlang (n) risk process. Insur. Math. Econ. 34(3), 391–408 (2004)
Li, S., Lu, Y.: The distribution of total dividend payments in a Sparre Andersen model. Statist. Probab. Lett. 79(9), 1246–1251 (2009)
Li, S., Lu, Y.: On the time and the number of claims when the surplus drops below a certain level. Scand. Actuar. J. 5, 420–445 (2016)
Lin, X.S., Pavlova, K.P.: The compound Poisson risk model with a threshold dividend strategy. Insur. Math. Econ. 38(1), 57–80 (2006)
Lin, X.S., Willmot, G.E., Drekic, S.: The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insur. Math. Econ. 33(3), 551–566 (2003)
Lu, Y., Li, S.: The Markovian regime-switching risk model with a threshold dividend strategy. Insur. Math. Econ. 44(2), 296–303 (2009)
Willmot, G.E., Dickson, D.C.M.: The Gerber-Shiu discounted penalty function in the stationary renewal risk model. Insur. Math. Econ. 32(3), 403–411 (2003)
Zhao, C., Zhang, C.: Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang (n) claims. J. Comput. Appl. Math. 244, 102–114 (2013)
Zhou, X.: On a classical risk model with a constant dividend barrier. North Am. Actuar. J. 9(4), 95–108 (2005)
Zhu, J., Yang, H.: Ruin theory for a Markov regime-switching model under a threshold dividend strategy. Insur. Math. Econ. 42(1), 311–318 (2008)
Acknowledgements
Chunming Zhao is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2682017CX065) and by the FP7 Grant PIRSES-GA-2012-318984. Zbigniew Palmowski is supported by the National Science Centre under the grant 2013/09/B/HS4/01496. Zbigniew Palmowski thanks the organizers of the wonderful program Mathematics of Risk for all work done to make this event happened. In particular, he is grateful to Kostya Borovkov (University of Melbourne) and Kais Hamza (Monash University) for all the help and nice discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Li, Y., Palmowski, Z., Zhao, C., Zhang, C. (2019). Number of Claims and Ruin Time for a Refracted Risk Process. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_49
Download citation
DOI: https://doi.org/10.1007/978-3-030-04161-8_49
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04160-1
Online ISBN: 978-3-030-04161-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)