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A discrete-time ruin model with dependence between interclaim arrivals and claim sizes
- Zhenhua Bao^{1}Email author and
- Ye Liu^{1, 2}
https://doi.org/10.1186/s13662-016-0893-4
© Bao and Liu 2016
Received: 26 November 2015
Accepted: 9 June 2016
Published: 11 July 2016
Abstract
We construct a discrete-time ruin model with general premium rate and dependent setting, where the time between two occurrences depends on the previous claim size. The generating function and defective renewal equation satisfied by the Gerber-Shiu expected discounted penalty function are derived by using the roots of a generalized Lundberg’s equation. Explicit expressions for the Gerber-Shiu function are obtained with discrete \(K_{m}\)-family claim sizes and geometric thresholds. Numerical illustration is then examined.
Keywords
- dependence
- Gerber-Shiu discounted penalty function
- defective renewal equation
- Lundberg’s equation
MSC
- 62P05
- 91B30
1 Introduction
In ruin theory, the compound binomial model and the risk model based on a discrete-time renewal process have been extensively analyzed by [1, 2], among many others. Note that, for these mentioned risk models, it is explicitly assumed that claim sizes and claim intervals are independent, which can be restrictive in practical context. Cossette et al. [3] propose a compound Markov binomial model based on the Markov Bernoulli process that introduces dependence between claim occurrences. Woo [4] analyzes a generalized Gerber-Shiu function in a discrete-time renewal risk model with an arbitrary dependence structure. Liu and Bao [5] consider a particular dependence structure among the interclaim time and the subsequent claim size and derive defective renewal equation satisfied by the Gerber-Shiu expected discounted penalty function. We mention that the dependence among claim sizes and interclaim arrivals through bivariate geometric distributions and copula functions have been investigated by Marceau [6], where explicit expressions for the Gerber-Shiu expected penalty function are derived.
As mentioned by Landriault [7], unlike the classical compound Poisson model, in which a unite premium can be assumed without loss of generality, it is clear that such reasoning does not hold for compound binomial model. In that paper, the author studies the evaluation of the generalized expected penalty function in the compound binomial risk model in which the premium rate received per period is c (\(c\in\mathbb{N}^{+}\)). See also Liu and Bao [5] for a discrete-time risk model with general premium rate and time-dependent claim sizes.
The rest of the paper is structured as follows: In Section 2, we analyze the roots of the generalized Lundberg’s equation. The defective renewal equations for the Gerber-Shiu expected discounted penalty function are derived in Section 3. In Section 4, we obtain the explicit expressions for the Gerber-Shiu function when the claim sizes have discrete \(K_{m}\) distributions and the random thresholds follow geometric distributions. A numerical example is also provided.
2 Generalized Lundberg’s equation
In the following lemma, we first apply Rouché’s theorem on a given contour to identify the number of roots of equation (2.12) for \(v\in(0,1)\).
Lemma 1
For \(v\in(0,1)\), there are 2c solutions of generalized Lundberg’s equation (2.12), say \(z_{i}=z_{i}(v)\) for \(i=1,2,\ldots,2c\), inside the unit circle \(\mathcal{C}=\{z:\vert z\vert =1\}\).
Proof
In the case of \(v=1\), the conditions of Rouché’s theorem are no longer satisfied. However, we can determine the roots of (2.12) by applying Theorem 1 of Klimenok [10] as follows.
Lemma 2
For \(v=1\), there are \(2c-1\) roots, say \(z_{1},\dots, z_{2c-1}\), to equation (2.12) inside \(\mathcal{C}\), in addition to the trivial root \(z_{2c}=1\).
Proof
Theorem 3
3 Defective renewal equations
Equations (3.6) and (3.9) eventually lead to the defective renewal equation for the Gerber-Shiu discounted penalty function \(m_{i}(u)\), as presented in the following theorem.
Theorem 4
Proof
4 Applications with geometric thresholds
The thresholds can be viewed as a criterion for classifying claims as large or small. In this section, we suppose that the random thresholds \(\{Q_{i},i=1,2,\dots\}\) follow a geometric distribution with p.f. \((1-p_{3})p_{3}^{n-1}\), \(n\in\mathbb{N}^{+}\), \(0< p_{3}<1\). We derive an explicit expression for the Gerber-Shiu function when the claim sizes belong to the discrete \(K_{m}\)-family and present some special cases.
4.1 The Gerber-Shiu function with \(K_{m}\)-family claim sizes
4.2 The generating function of the time to ruin with geometric distribution
Example
5 Conclusions
We study a discrete-time ruin model with general premium rate and dependence structure, in which the distribution of the time until the next claim depends on the amount of the previous claim. Some analytic techniques are applied to study the Gerber-Shiu expected discounted penalty function. In particular, we show that the Gerber-Shiu function satisfies a defective renewal equation. Explicit expressions for the Gerber-Shiu function are obtained with discrete \(K_{m}\)-family claim sizes and geometric thresholds. The model in this paper can be further extended. For instance, suppose that the premium charged varies depending on the possible change in the distribution of interclaim time. Then the related ruin problems can be solved for this modified model.
Declarations
Acknowledgements
The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions. This research was supported by the Ministry of Education of Humanities and Social Science Project (15YJC910001) and the Program for Liaoning Excellent Talents in University (LR2014031).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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