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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 128796, 8 pages
http://dx.doi.org/10.1155/2013/128796
Research Article

Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

1School of Science, Shandong Jiaotong University, Jinan 250023, China
2School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China

Received 11 January 2013; Accepted 5 March 2013

Academic Editor: Hua Su

Copyright © 2013 Yujuan Huang and Wenguang Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplace transformation of the time when the surplus reaches a given level for the first time is discussed, and the expectation and its variance are obtained. Finally, we give the numerical examples.

1. Introduction

In insurance mathematics, the classical risk model has been the center of focus for decades [13]. The surplus in the classical model at time can be expressed as where is the initial capital, is the constant rate of premium, and is a Poisson process, with Poisson rate denoting the number of claims up to time . The individual claim sizes , independent of , are independent and identically distributed nonnegative random variables with common distribution function with mean , variance , and moment generating function .

But in the Poisson process, the expectation and variance are equal. This is obviously not consistent with actual situation. So recently there is a huge amount of literature devoted to the generalization of the classical model in different ways. Lu and Li [4] consider a Markov-modulated risk model in which the claim interarrivals, claim sizes, and premiums are influenced by an external Markovian environment process. Tan and Yang [5] discuss the compound binomial risk model with an interest on the surplus under a constant dividend barrier and periodically paying dividends. Vellaisamy and Upadhye [6] study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. Cossette et al. [7] present a compound Markov binomial model, which is an extension of the compound binomial model. The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences. Recursive formulas are provided for the computation of the ruin probabilities over finite- and infinite-time horizons. A Lundberg exponential bound is derived for the ruin probability, and numerical examples are also provided. Yang and Zhang [8] investigate a Sparre Andersen risk model in which the inter-claim times are generalized Erlang(n) distributed. Czarna and Palmowski [9] focus on a general spectrally negative Levy insurance risk process. For this class of processes, they analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time .

In this paper, we will consider a double Poisson-Geometric risk model with diffusion in which the arrival of policies is a Poisson-Geometric process and the claims process follows the compound Poisson-Geometric process. For more details and new developments on the Poisson-Geometric risk model, the interested readers can refer to [1013].

The rest of the paper is organized as follows. In Section 2, the risk model is introduced. In Section 3, we obtain the adjustment coefficient equation and the formula of ruin probability. Then we present the effect of the related parameters on the adjustment coefficient. In Section 4, using the martingale method, the time when the surplus reaches a level firstly is considered, and the expectation and its variance are obtained. Numerical illustrations are also given.

2. The Risk Model

Definition 1 (see [10]). A distribution is said to be Poisson-Geometric distributed, denoted by , if its generating function is where , . Note that if , then the Poisson-Geometric distribution degenerates into Poisson distribution.

Definition 2 (see [10]). Let and , then is said to be a Poisson-Geometric process with parameters , if it satisfies (1); (2) has stationary and independent increments; (3), is a Poisson-Geometric distributed with parameters , , and
The corresponding moment generating function of is .
Then the double Poisson-Geometric risk model with interference is defined as where is the number of premium up to time and follows a Poisson-Geometric distribution with parameters and ; is the number of claims up to time and follows a Poisson-Geometric distribution with parameters and . is the standard Brownian motion and is a constant, representing the diffusion volatility parameters. Throughout this paper, we assume that , , , and are mutually independent.
In order to ensure the insurance company's stable operation, we assume which implies
Let Then is the relative security loading factor.
For the risk model (3), the time to ruin, denoted by , is defined as And define the ruin probability with an initial surplus by , namely,

3. The Ruin Probability

Define the profits process by ; that is, Obviously we have

Let Then

Lemma 3. The profits process has the following properties:(1);(2) has stationary and independent increments.

Theorem 4. For the profits process , there is a function such that

Proof. Consider Let Then we obtain (13).

Theorem 5. Equation has a unique positive solution , and (16) is said to be an adjustment coefficient equation of the risk model (3) and is said to be an adjustment coefficient.

Proof. From (15), we have , and since which imply
It is easy to see that the moment generating function is an increasing function. Due to , there exists an such that ; that is, when . So when , and is a convex function with . Then it can be shown that has a unique positive solution on .

Example 6. Suppose , , , , and , , . By (16), we obtain the adjustment coefficient . Moreover, we give the effect of related parameters on adjustment coefficient ; see Figures 1, 2, 3, 4, 5, 6, and 7.
For the profits process , let .

128796.fig.001
Figure 1: The impact of on .
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Figure 2: The impact of on .
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Figure 3: The impact of on .
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Figure 4: The impact of on .
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Figure 5: The impact of on .
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Figure 6: The impact of on .
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Figure 7: The impact of on .

Theorem 7. is a martingale, where .

Proof. Consider

Theorem 8. If and satisfy the equation , then the surplus is a martingale.

Proof. Consider

Lemma 9. The ruin time is the stopping time of .

Theorem 10. For for all , the ultimate ruin probability satisfies where .

Proof. For a fixed time , is a bounded stopping time; using the theorem of martingale and stopping time, we have which implies by expectation on both sides of (23), and letting , we can obtain (21).

Theorem 11. The probability of the risk model (3) is

Proof. is a ruin time and for a fixed time , is a bounded stopping time. Using the theorem of martingale and stopping time, we have Let , we have
If is an indicator function of the event , we get
Since by the law of large numbers, when , (a.s.). By dominated convergence theorem, we have Then when in (26), we can obtain (24).

Corollary 12. Consider

Example 13. Suppose , , and . By (30), we give the effect of adjustment coefficient on the upper bound of the ruin probability; see Figure 8.

128796.fig.008
Figure 8: The impact of on ruin probability .

4. The Time to Reach a Given Level

Let Then is the time when the surplus reaches a given level firstly.

Theorem 14. The Laplace transform of is where and satisfy

Proof. For the surplus process , using the theorem of martingale and stopping time, we see that is a stopping rime of . Let . By Theorem 8, the surplus process is a martingale; hence, we have implying that Since , so we get

Theorem 15. The expectation and variance of satisfy

Proof. Let . Using Theorem 11, we have . Then Let . We have

Example 16. Suppose , , , , , , , and . By (37), we give the effect of related parameters on and ; see Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21.

128796.fig.009
Figure 9: The impact of on .
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Figure 10: The impact of on .
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Figure 11: The impact of on .
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Figure 12: The impact of on .
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Figure 13: The impact of on .
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Figure 14: The impact of on .
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Figure 15: The impact of on .
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Figure 16: The impact of on .
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Figure 17: The impact of on .
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Figure 18: The impact of on .
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Figure 19: The impact of on .
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Figure 20: The impact of on .
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Figure 21: The impact of on .

Acknowledgments

Y. Huang thanks the three anonymous referees for the thoughtful comments and suggestions that greatly improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11171187, Grant no. 10921101), National Basic Research Program of China (973 Program, Grant no. 2007CB814906), Natural Science Foundation of Shandong Province (Grant no. ZR2012AQ013, Grant no. ZR2010GL013), Humanities and Social Sciences Project of the Ministry Education of China (Grant no. 10YJC630092, Grant no. 09YJC910004), and 2013 Major Project Cultivation Plan of Shandong Jiaotong University.

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