## On the discounted penalty function in a Markov-dependent risk model.(English)Zbl 1129.91023

Summary: We present a unified approach to the analysis of several popular models in collective risk theory. Based on the analysis of the discounted penalty function in a semi-Markovian risk model by means of Laplace-Stieltjes transforms, we rederive and extend some recent results in the field. In particular, the classical compound Poisson model, Sparre Andersen models with phase-type interclaim times and models with causal dependence of a certain Markovian type between claim sizes and interclaim times are contained as special cases.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60K15 Markov renewal processes, semi-Markov processes 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
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### References:

 [1] Adan, I.; Kulkarni, V., Single-server queue with Markov dependent interarrival and service times, Queueing systems, 45, 113-134, (2003) · Zbl 1036.90029 [2] Albrecher, H., Discussion on “the time value of ruin in a sparre Andersen model” by H. gerber and E. shiu, North American actuarial journal, 9, 2, 71-74, (2005) [3] Albrecher, H.; Boxma, O., A ruin model with dependence between claim sizes and claim intervals, Insurance mathematics and economics, 35, 2, 245-254, (2004) · Zbl 1079.91048 [4] Albrecher, H., Teugels, J., 2004. Exponential behavior in the presence of dependence in risk theory. EURANDOM Research Report 2004-011, TU Eindhoven. · Zbl 1097.62110 [5] Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore [6] Avram, F.; Usabel, M., Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times, ASTIN bulletin, 34, 2, 315-332, (2004) · Zbl 1274.91244 [7] Badescu, A.; Breuer, L.; da Silva Soares, A.; Latouche, G.; Remiche, M.; Stanford, D., Risk processes analyzed as fluid queues, Scandinavian actuarial journal, 2, 127-141, (2005) · Zbl 1092.91037 [8] Cheng, Y.; Tang, Q., Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process, North American actuarial journal, 7, 1-12, (2003) · Zbl 1084.60544 [9] de Smit, J., The queue $$G I / M / s$$ with customers of different types or the queue $$G I / H_m / s$$, Advances in applied probability, 15, 392-419, (1983) · Zbl 0512.60086 [10] Dickson, D., On the distribution of the surplus prior to ruin, Insurance mathematics and economics, 11, 3, 191-207, (1992) · Zbl 0770.62090 [11] Dickson, D., Discussion on “on the time value of ruin” by H. gerber and E. shiu, North American actuarial journal, 2, 1, 74, (1998) [12] Dickson, D.; Drekic, S., The joint distribution of the surplus prior to ruin and the deficit at ruin in some sparre Andersen models, Insurance mathematics and economics, 34, 97-107, (2004) · Zbl 1043.60036 [13] Dickson, D.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insurance mathematics and economics, 29, 333-344, (2001) · Zbl 1074.91549 [14] Doetsch, G., Theorie und anwendung der Laplace-transformation, (1937), Springer Berlin · JFM 63.0368.01 [15] Dufresne, F.; Gerber, H., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance mathematics and economics, 7, 3, 193-199, (1988) · Zbl 0674.62072 [16] Gerber, H.; Shiu, E., On the time value of ruin, North American actuarial journal, 2, 1, 48-72, (1998) [17] Gerber, H.; Shiu, E., The time value of ruin in a sparre Andersen model, North American actuarial journal, 9, 2, 49-69, (2005) · Zbl 1085.62508 [18] Jacobsen, M., Martingales and the distribution of the time to ruin, Stochastic processes and their applications, 107, 1, 29-51, (2003) · Zbl 1075.60554 [19] Janssen, J.; Reinhard, J., Probabilités de ruine pour une classe de modles de risque semi-markoviens, ASTIN bulletin, 15, 2, 123-134, (1985) [20] Li, S.; Garrido, J., On ruin for the Erlang(n) risk process, Insurance mathematics and economics, 34, 3, 391-408, (2004) · Zbl 1188.91089 [21] Li, S., Garrido, J., 2005. On a general class of renewal risk process: analysis of the Gerber-Shiu function. Advances in Applied Probability, in press. · Zbl 1077.60063 [22] Lin, X.; Willmot, G., The moments of the time to ruin, the surplus before ruin and the deficit at ruin, Insurance mathematics and economics, 27, 19-44, (2000) · Zbl 0971.91031 [23] Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Allyn and Bacon Boston · Zbl 0126.02404 [24] Sun, L.; Yang, H., On the joint distributions of surplus immediately before ruin and the deficit at ruin for Erlang(2) risk processes, Insurance mathematics and economics, 34, 121-125, (2004) · Zbl 1054.60017
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