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Risk processes analyzed as fluid queues. (English) Zbl 1092.91037
This paper deals with the risk model based on the Markovian arrival process which makes it fairly general. On the one hand, this model includes both the classical and most Sparre Andersen models, so long as claim amounts are modelled as phase-type random variables. On the other hand, Markovian arrival processes allow for correlated arrival processes, and can be used to model situations where environmental factors change radically. The proposed approach is based on the explicit recognition of the similarity between the evolution of the risk process \(R(t)\) and that of a fluid queue defined as follows. Take a Markov process \(\{\phi(x)\}_{x\geq0}\) on a finite state space \(S\) and associate a real number \(r_{i}\) to each state \(i\in S\): \(r_{i}\) is the rate at which some fluid increases or decreases when the Markov process is in the state \(i\). Define the piecewise linear function \(F(x)=u+\int_{0}^{x}r_{\phi(v)}\,dv\), where \(u\) is the fluid level at the time zero. The fluid queue is defined as \(\Phi(x)=F(x)-\min(0,\widetilde F(x))\), where \(\widetilde F(x)=\min_{0\leq v\leq x}F(v)\). The Laplace-Stieltjes transform of the time of ruin is derived.

MSC:
91B30 Risk theory, insurance (MSC2010)
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References:
[1] Seal H, Survival Probabilities (1978)
[2] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[3] DOI: 10.1016/S0167-6687(03)00117-3 · Zbl 1074.91026 · doi:10.1016/S0167-6687(03)00117-3
[4] Avram F and Usábel M 2003b Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times Technical Report, Université de Pau · Zbl 1274.91244
[5] DOI: 10.1137/1.9780898719734 · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[6] DOI: 10.1214/aoap/1050689597 · Zbl 1030.60067 · doi:10.1214/aoap/1050689597
[7] DOI: 10.2307/3214845 · Zbl 0778.60035 · doi:10.2307/3214845
[8] Asmussen S, Scandinavian Journal Statistics 23 pp 419– (1996)
[9] DOI: 10.1023/A:1020981005544 · Zbl 1088.62511 · doi:10.1023/A:1020981005544
[10] DOI: 10.1016/0167-9473(95)00025-9 · Zbl 0875.62405 · doi:10.1016/0167-9473(95)00025-9
[11] Barlow MT, Séminaire de Probabilités XIV, Volume 784 of Lecture Notes in Math pp pp. 324–331– (1980)
[12] London RR, Séminaire de Probabilités XVI, Volume 920 of Lecture Notes in Math pp pp. 68–90– (1982)
[13] DOI: 10.1080/15326349508807330 · Zbl 0817.60086 · doi:10.1080/15326349508807330
[14] Asmussen S, Advances in queueing: Theory, methods, and open problems pp pp. 79–102– (1995)
[15] Ramaswami V, Teletraffic engineering in a competitive world (Proceedings of the 16th International Teletraffic Congress) pp pp. 1019–1030– (1999)
[16] DOI: 10.2143/AST.32.2.1029 · Zbl 1081.60028 · doi:10.2143/AST.32.2.1029
[17] da Silva Soares A, Proceedings of the 4th International Conference on Matrix-Analytic Methods pp pp. 89–106– (2002) · Zbl 1162.90407
[18] DOI: 10.1214/aoap/1177005065 · Zbl 0806.60052 · doi:10.1214/aoap/1177005065
[19] Faddeev DK, Computational methods of linear algebra (1963)
[20] DOI: 10.1080/03461230110106471 · Zbl 1142.62088 · doi:10.1080/03461230110106471
[21] Asmussen S, Applied Probability and Queues (1987)
[22] Neuts MF, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach (1981)
[23] DOI: 10.2307/3214773 · Zbl 0789.60055 · doi:10.2307/3214773
[24] DOI: 10.1016/S0024-3795(01)00426-8 · Zbl 0994.65047 · doi:10.1016/S0024-3795(01)00426-8
[25] DOI: 10.1137/S0895479800381872 · Zbl 1005.65014 · doi:10.1137/S0895479800381872
[26] DOI: 10.1007/s002110050303 · Zbl 0889.65145 · doi:10.1007/s002110050303
[27] Meini B, Advances in Performance Analysis 1 pp 215– (1998) · Zbl 0934.60086
[28] Bean N O’Reilly M and Taylor PG 2003 Hitting probabilities and hitting times for stochastic fluid flows Submitted for publication
[29] DOI: 10.1016/S0167-6687(99)00029-3 · Zbl 1028.91561 · doi:10.1016/S0167-6687(99)00029-3
[30] Asmussen S, Scandinavian Actuarial Journal pp 31– (1984)
[31] Thorin O, ASTIN Bulletin 7 pp 137– (1973) · doi:10.1017/S0515036100005808
[32] Stanford DA Avram F Badescu A Breuer L da Silva Soares A and Latouche G 2003 Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models. Universite Libr de Bruxelles, Technical Report Di511 Submitted for publication · Zbl 1123.62078
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