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

Zhimin Zhang; Eric C. K. Cheung

doi:10.3934/jimo.2017036

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2018, Volume 14, Issue 1: 35-63. Doi: 10.3934/jimo.2017036

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

Zhimin Zhang^1, , and
Eric C. K. Cheung^2,

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
* Corresponding author: Zhimin Zhang

Received: September 2015

Revised: February 2017

Early access: March 2017

Published: January 2018

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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Abstract

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.

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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 $V_{1, \delta}(u;b)$

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

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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

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Table 1. 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	10.900	10.906	10.911
$V_{1, \delta}(5;b^{*})$	39.9857	40.1571	40.2133	40.2412	40.2579	40.2690	40.2769	40.2828
$SD_\delta(5;b^{*})$	15.9415	16.0128	16.0385	16.0478	16.0577	16.0627	16.0647	16.0660
$V_{1, \delta}(10;b^{*})$	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	12.1830	12.1789

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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)

$m$	1	2	3	4	5	6	7	8
$b^{*}$	13.036	13.209	13.270	13.300	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	36.4890	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	44.0650	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	47.5610	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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