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A note on a Lévy insurance risk model under periodic dividend decisions

  • * Corresponding author: Zhimin Zhang

    * Corresponding author: Zhimin Zhang 
Zhimin Zhang is supported by the National Natural Science Foundation of China [11471058,11661074], the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007] and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 16YJC910005). Eric Cheung gratefully acknowledges the support from the Research Grants Council of the Hong Kong Special Administrative Region (Project Number: HKU 17324016). This research is also partially supported by the CAE 2013 research grant from the Society of Actuaries. Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA.
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  • In this paper, we consider a spectrally negative Lévy insurance risk process with a barrier-type dividend strategy. In contrast to the traditional barrier strategy in which dividends are payable to the shareholders immediately when the surplus process reaches a fixed level b (as long as ruin has not yet occurred), it is assumed that the insurer only makes dividend decisions at some discrete time points in the spirit of [1]. Under such a dividend strategy with Erlang inter-dividend-decision times, expressions for the Gerber-Shiu expected discounted penalty function proposed in [24] and the moments of total discounted dividends payable until ruin are derived. The results are expressed in terms of the scale functions of a spectrally negative Lévy process and an embedded spectrally negative Markov additive process. Our analyses rely on the introduction of a potential measure associated with an Erlang random variable. Numerical illustrations are also given.

    Mathematics Subject Classification: Primary: 60G51, 60J75; Secondary: 91B30, 62P05.


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  • Figure 1.  Impact of the parameter $\beta$. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$

    Figure 2.  Impact of claim distributions. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$

    Figure 3.  Expected discounted dividends $V_{1, \delta}(u;b)$ as a function of $b$. (a) Brownian motion model. (b) Compound Poisson model with exponential claims

    Table 1.  Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the Brownian motion model

    $ m$12345678
    $V_{1, \delta}(5;b^{*})$39.985740.157140.213340.241240.257940.269040.276940.2828
    $V_{1, \delta}(10;b^{*})$47.008447.209147.275847.308247.328347.341947.350847.3574
    $V_{1, \delta}(b^{*};b^{*})$47.713348.034548.145248.197948.232948.254348.268848.2802
     | Show Table
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    Table 2.  Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the compound Poisson model (ⅱ)

    $V_{1, \delta}(0;b^{*})$13.164513.211413.226613.234113.238613.241613.243713.2454
    $V_{1, \delta}(5;b^{*})$36.276436.405736.447736.468436.480836.489036.494836.4993
    $V_{1, \delta}(10;b^{*})$43.808343.964544.015144.040244.055144.065044.072144.0775
    $V_{1, \delta}(b^{*};b^{*})$46.974747.314647.430947.487247.522747.547447.561047.5771
     | Show Table
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