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January  2018, 14(1): 35-63. doi: 10.3934/jimo.2017036

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

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

* Corresponding author: Zhimin Zhang

Received  September 2015 Revised  February 2017 Published  January 2018 Early access  April 2017

Fund Project: 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.

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.

Citation: Zhimin Zhang, Eric C. K. Cheung. A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial & Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036
##### References:

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##### References:
Impact of the parameter $\beta$. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$
Impact of claim distributions. (a) Laplace transform of ruin time $\phi_\delta(u;b)$. (b) Expected discounted dividends $V_{1, \delta}(u;b)$
Expected discounted dividends $V_{1, \delta}(u;b)$ as a function of $b$. (a) Brownian motion model. (b) Compound Poisson model with exponential claims
Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the Brownian motion model
 $m$ 1 2 3 4 5 6 7 8 $b^{*}$ 10.698 10.816 10.858 10.879 10.892 10.9 10.906 10.911 $V_{1, \delta}(5;b^{*})$ 39.9857 40.1571 40.2133 40.2412 40.2579 40.269 40.2769 40.2828 $SD_\delta(5;b^{*})$ 15.9415 16.0128 16.0385 16.0478 16.0577 16.0627 16.0647 16.066 $V_{1, \delta}(10;b^{*})$ 47.0084 47.2091 47.2758 47.3082 47.3283 47.3419 47.3508 47.3574 $SD_\delta(10;b^{*})$ 12.1893 12.2434 12.2672 12.2727 12.2828 12.2921 12.2919 12.2908 $V_{1, \delta}(b^{*};b^{*})$ 47.7133 48.0345 48.1452 48.1979 48.2329 48.2543 48.2688 48.2802 $SD_\delta(b^{*};b^{*})$ 12.1021 12.1415 12.1638 12.1622 12.1763 12.1869 12.183 12.1789
 $m$ 1 2 3 4 5 6 7 8 $b^{*}$ 10.698 10.816 10.858 10.879 10.892 10.9 10.906 10.911 $V_{1, \delta}(5;b^{*})$ 39.9857 40.1571 40.2133 40.2412 40.2579 40.269 40.2769 40.2828 $SD_\delta(5;b^{*})$ 15.9415 16.0128 16.0385 16.0478 16.0577 16.0627 16.0647 16.066 $V_{1, \delta}(10;b^{*})$ 47.0084 47.2091 47.2758 47.3082 47.3283 47.3419 47.3508 47.3574 $SD_\delta(10;b^{*})$ 12.1893 12.2434 12.2672 12.2727 12.2828 12.2921 12.2919 12.2908 $V_{1, \delta}(b^{*};b^{*})$ 47.7133 48.0345 48.1452 48.1979 48.2329 48.2543 48.2688 48.2802 $SD_\delta(b^{*};b^{*})$ 12.1021 12.1415 12.1638 12.1622 12.1763 12.1869 12.183 12.1789
Exact values of $b^*$, $V_{1, \delta}(u;b^*)$ and $SD_\delta(u;b^*)$ in the compound Poisson model (ⅱ)
 $m$ 1 2 3 4 5 6 7 8 $b^{*}$ 13.036 13.209 13.27 13.3 13.319 13.332 13.339 13.349 $V_{1, \delta}(0;b^{*})$ 13.1645 13.2114 13.2266 13.2341 13.2386 13.2416 13.2437 13.2454 $SD_\delta(0;b^{*})$ 19.6093 19.6802 19.7034 19.7146 19.7215 19.7263 19.7293 19.7314 $V_{1, \delta}(5;b^{*})$ 36.2764 36.4057 36.4477 36.4684 36.4808 36.489 36.4948 36.4993 $SD_\delta(5;b^{*})$ 17.2879 17.3529 17.3748 17.3846 17.3911 17.3961 17.3986 17.3996 $V_{1, \delta}(10;b^{*})$ 43.8083 43.9645 44.0151 44.0402 44.0551 44.065 44.0721 44.0775 $SD_\delta(10;b^{*})$ 13.5952 13.6492 13.6681 13.6758 13.6813 13.6864 13.6879 13.6877 $V_{1, \delta}(b^{*};b^{*})$ 46.9747 47.3146 47.4309 47.4872 47.5227 47.5474 47.561 47.5771 $SD_\delta(b^{*};b^{*})$ 13.0272 13.0611 13.0776 13.0796 13.0908 13.0968 13.0962 13.0907
 $m$ 1 2 3 4 5 6 7 8 $b^{*}$ 13.036 13.209 13.27 13.3 13.319 13.332 13.339 13.349 $V_{1, \delta}(0;b^{*})$ 13.1645 13.2114 13.2266 13.2341 13.2386 13.2416 13.2437 13.2454 $SD_\delta(0;b^{*})$ 19.6093 19.6802 19.7034 19.7146 19.7215 19.7263 19.7293 19.7314 $V_{1, \delta}(5;b^{*})$ 36.2764 36.4057 36.4477 36.4684 36.4808 36.489 36.4948 36.4993 $SD_\delta(5;b^{*})$ 17.2879 17.3529 17.3748 17.3846 17.3911 17.3961 17.3986 17.3996 $V_{1, \delta}(10;b^{*})$ 43.8083 43.9645 44.0151 44.0402 44.0551 44.065 44.0721 44.0775 $SD_\delta(10;b^{*})$ 13.5952 13.6492 13.6681 13.6758 13.6813 13.6864 13.6879 13.6877 $V_{1, \delta}(b^{*};b^{*})$ 46.9747 47.3146 47.4309 47.4872 47.5227 47.5474 47.561 47.5771 $SD_\delta(b^{*};b^{*})$ 13.0272 13.0611 13.0776 13.0796 13.0908 13.0968 13.0962 13.0907
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