zbMATH — the first resource for mathematics

On simple ruin expressions in dependent Sparre Andersen risk models. (English) Zbl 1286.91063
Summary: In this note we provide a simple alternative probabilistic derivation of an explicit formula of I. K. M. Kwan and H. Yang [“Ruin probability in a threshold insurance risk model”, Belg. Actuar. Bull. 7, 41–49 (2007), http://www.belgianactuarialbulletin.be/browse.php?issue=77-8] for the probability of ruin in a risk model with a certain dependence between general claim interoccurrence times and subsequent claim sizes of conditionally exponential type. The approach puts the type of formula in a general context, illustrating the potential for similar simple ruin probability expressions in more general risk models with dependence.
Reviewer: Reviewer (Berlin)

91B30 Risk theory, insurance (MSC2010)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
Full Text: DOI Euclid
[1] Adan, I. J. B. F. and Kulkarni, V. G. (2003). Single-server queue with Markov-dependent inter-arrival and service times. Queueing Systems 45, 113-134. · Zbl 1036.90029 · doi:10.1023/A:1026093622185
[2] Ahn, S. and Badescu, A. L. (2007). On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance Math. Econom. 41, 234-249. · Zbl 1193.60103 · doi:10.1016/j.insmatheco.2006.10.017
[3] Albrecher, H. and Boxma, O. J. (2004). A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econom. 35, 245-254. · Zbl 1079.91048 · doi:10.1016/j.insmatheco.2003.09.009
[4] Albrecher, H. and Boxma, O. J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom. 37, 650-672. · Zbl 1129.91023 · doi:10.1016/j.insmatheco.2005.06.007
[5] Albrecher, H. and Teugels, J. L. (2006). Exponential behavior in the presence of dependence in risk theory. J. Appl. Prob. 43, 257-273. · Zbl 1097.62110 · doi:10.1239/jap/1143936258 · euclid:jap/1143936258
[6] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[7] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities , 2nd edn. World Scientific, Hackensack, NJ. · Zbl 1247.91080
[8] Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006). On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuarial J. 2006, 265-285. · Zbl 1145.91030 · doi:10.1080/03461230600992266
[9] Breuer, L. (2008). First passage times for Markov additive processes with positive jumps of phase type. J. Appl. Prob. 45, 779-799. · Zbl 1156.60059 · doi:10.1239/jap/1222441829
[10] Cheung, E. C. K., Landriault, D. and Badescu, A. L. (2011). On a generalization of the risk model with Markovian claim arrivals. Stoch. Models 27, 407-430. · Zbl 1237.91124 · doi:10.1080/15326349.2011.593403
[11] Cheung, E. C. K., Landriault, D., Willmot, G. E. and Woo, J.-K. (2010). Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models. Insurance Math. Econom. 46, 117-126. · Zbl 1231.91157 · doi:10.1016/j.insmatheco.2009.05.009
[12] D’Auria, B., Ivanovs, J., Kella, O. and Mandjes, M. (2010). First passage of a Markov additive process and generalized Jordan chains. J. Appl. Prob. 47, 1048-1057. · Zbl 1213.60144 · doi:10.1239/jap/1294170518
[13] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis . Cambridge University Press. · Zbl 0704.15002
[14] Ivanovs, J. (2013). A note on killing with applications in risk theory. Insurance Math. Econom. 52, 29-34. · Zbl 1291.91114
[15] Kwan, I. K. M. and Yang, H. (2007). Ruin probability in a threshold insurance risk model. Belg. Actuarial Bull. 7, 41-49.
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.