Analysis of a threshold dividend strategy for a MAP risk model. (English) Zbl 1164.91024

Badescu, Andrei; Drekic, Steve; Landriault, Daviv

Scand. Actuar. J. 2007, No. 4, 227-247 (2007).

The authors consider the risk model described as \[ dR^{b}(t)=\begin{cases} c_1dt-d\left(\sum_{n=1}^{N(t)}U_{n}\right), &\text{for}\;R^{b}(t)<b,\\ c_2dt-d\left(\sum_{n=1}^{N(t)}U_{n}\right), & \text{for}\;R^{b}(t)\geq b, \end{cases} \] where \(\{R^{b}(t),\;t\geq0\}\) represents the surplus process. The detailed structure of the Markovian risk model under consideration in this paper is presented. It is based on the notion that the claim number process \(N(t)\) is governed by a Markovian arrival process, which includes both the classical Poisson and most Sparre Andersen risk models of interest as well as models with correlated inter-claim times. The authors derive the Laplace-Stieltjes transform of the distribution of the time to ruin as well as the Laplace-Stieltjes transform of the joint distribution of the time to ruin, the surplus prior to ruin and the deficit at ruin. For the case, when an insurer pays dividends continuously at rate \(c_1-c_2\) whenever the surplus level is above \(b\), the expected discounted value of total dividend payments made prior to ruin is derived.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

Sparre Andersen risk model; phase-type distribution; Markovian arrival process; Laplace-Stieltjes transform; correlated claims; surplus process; fluid queues

References:

[1]	DOI: 10.1081/STM-120028392 · Zbl 1038.60086 · doi:10.1081/STM-120028392
[2]	DOI: 10.1239/jap/1118777186 · Zbl 1085.60065 · doi:10.1239/jap/1118777186
[3]	DOI: 10.1080/15326340500481788 · Zbl 1350.60095 · doi:10.1080/15326340500481788
[4]	Ahn S., Queueing Systems (2007)
[5]	Albrecher H., North American Actuarial Journal (2007)
[6]	DOI: 10.1016/j.insmatheco.2005.05.004 · Zbl 1117.91377 · doi:10.1016/j.insmatheco.2005.05.004
[7]	DOI: 10.1080/03461230510006946 · Zbl 1092.91036 · doi:10.1080/03461230510006946
[8]	Albrecher H., ASTIN Bulletin (2007)
[9]	Asmussen S., Advances in queueing: theory, methods, and open problems pp 79– (1995)
[10]	DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[11]	DOI: 10.1080/03461230410000565 · Zbl 1092.91037 · doi:10.1080/03461230410000565
[12]	DOI: 10.1080/03461230510009835 · Zbl 1143.91025 · doi:10.1080/03461230510009835
[13]	DOI: 10.2143/AST.34.1.504954 · Zbl 1097.91040 · doi:10.2143/AST.34.1.504954
[14]	DOI: 10.1017/S1748499500000166 · doi:10.1017/S1748499500000166
[15]	DOI: 10.1016/j.insmatheco.2004.11.003 · Zbl 1117.91381 · doi:10.1016/j.insmatheco.2004.11.003
[16]	DOI: 10.1287/opre.14.3.444 · doi:10.1287/opre.14.3.444
[17]	DOI: 10.1137/1.9780898719734 · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[18]	DOI: 10.1016/j.insmatheco.2004.08.004 · Zbl 1122.91345 · doi:10.1016/j.insmatheco.2004.08.004
[19]	DOI: 10.1016/j.insmatheco.2005.08.001 · Zbl 1157.91383 · doi:10.1016/j.insmatheco.2005.08.001
[20]	DOI: 10.1016/j.insmatheco.2003.08.004 · Zbl 1103.91369 · doi:10.1016/j.insmatheco.2003.08.004
[21]	Neuts M., Structured stochastic matrices of M/G/1 type and their applications (1989) · Zbl 0695.60088
[22]	Ramaswami V., Teletraffic engineering in a competitive world – Proceedings of the 16th International Teletraffic Congress pp 1019– (1999)
[23]	DOI: 10.1007/s11009-006-0426-9 · Zbl 1110.60067 · doi:10.1007/s11009-006-0426-9
[24]	Stehfest H., Communications of the ACM 13 pp 45– (1970)