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

MSC:
91B30 Risk theory, insurance (MSC2010)
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