On a perturbed compound Poisson model with varying premium rates

Zhimin Zhang; Yang Yang; Chaolin Liu

doi:10.3934/jimo.2016043

Article Contents

2017, Volume 13, Issue 2: 721-736. Doi: 10.3934/jimo.2016043

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On a perturbed compound Poisson model with varying premium rates

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.
Department of Statistics, Nanjing Audit University, Nanjing 211815, China

* Corresponding author: Chaolin Liu
* Corresponding author: Chaolin Liu

Received: October 31, 2015

Revised: December 31, 2015

Published: August 2017

Z.M. Zhang was supported by the National Natural Science Foundation of China [11471058,11101451,11301303] and the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007]. The research of Y. Yang was supported by National Natural Science Foundation of China (No. 71471090), the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 14YJCZH182), China Postdoctoral Science Foundation (No. 2014T70449,2012M520964), Natural Science Foundation of Jiangsu Province of China (No. BK20131339), the Major Research Plan of Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJA110001), Qing Lan Project, PAPD, Program of Excellent Science and Technology Innovation Team of the Jiangsu Higher Education Institu-tions of China, Project of Construction for Superior Subjects of Statistics of Jiangsu Higher Education Institutions, Project of the Key Lab of Financial Engineering of Jiangsu Province. The research of C.L. Liu was supported by the Fundamental Research Funds for the Central Universities(No. 106112015CDJXY100006).

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Abstract

In this paper, we consider a perturbed compound Poisson model with varying premium rates. The surplus process is observed at a sequence of review times. The effective premium rate is adjusted according to the surplus increment between the inter-review times. We study the Gerber-Shiu functions by Laplace transform method. When the claim size density is a combination of exponentials, the explicit expressions for the Laplace transforms of ruin time are derived.

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Mathematics Subject Classification: Primary: 91B30; Secondary: 62P05.

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Figure 1. Ruin probabilities for Erlang(2) inter-review times. (a) $f_X(x)=3e^{-1.5x}-3 e^{-3x}$; (b) $f_X(x)=e^{-x}$; (c) $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$

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Table 1. Exact values of ruin probabilities when $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$

$u$	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$
$\phi_1(u)$	0.64215	0.33207	0.15448	0.06979	0.03119	0.01388	0.00617	0.00274	0.00121	0.00054
$\phi_2(u)$	0.41629	0.19545	0.08830	0.03946	0.01757	0.00781	0.00347	0.00154	0.00068	0.00030

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Table 2. Exact values of ruin probabilities when $f_X(x)=e^{-x}$

$u$	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$
$\phi_1(u)$	0.70195	0.42231	0.23766	0.13018	0.07048	0.03797	0.02041	0.01096	0.00588	0.00316
$\phi_2(u)$	0.48671	0.27859	0.15357	0.08339	0.04500	0.02421	0.01301	0.00699	0.00375	0.00201

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Table 3. Exact values of ruin probabilities when $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$

$u$	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$
$\phi_1(u)$	0.75332	0.55726	0.40147	0.28352	0.19789	0.13718	0.09472	0.06524	0.04488	0.03084
$\phi_2(u)$	0.59185	0.43080	0.30693	0.21544	0.14989	0.10374	0.07158	0.04929	0.03390	0.02330

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