Mathematical Problems in Engineering

Volume 2015, Article ID 134246, 8 pages

http://dx.doi.org/10.1155/2015/134246

## A Dependent Insurance Risk Model with Surrender and Investment under the Thinning Process

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

Received 26 August 2015; Accepted 17 September 2015

Academic Editor: Xinguang Zhang

Copyright © 2015 Wenguang Yu and Yujuan Huang. 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

A dependent insurance risk model with surrender and investment under the thinning process is discussed, where the arrival of the policies follows a compound Poisson-Geometric process, and the occurrences of the claim and surrender happen as the -thinning process and the -thinning process of the arrival process, respectively. By the martingale theory, the properties of the surplus process, adjustment coefficient equation, the upper bound of ruin probability, and explicit expression of ruin probability are obtained. Moreover, we also get the Laplace transformation, the expectation, and the variance of the time when the surplus reaches a given level for the first time. Finally, various trends of the upper bound of ruin probability and the expectation and the variance of the time when the surplus reaches a given level for the first time are simulated analytically along with changing the investment size, investment interest rates, claim rate, and surrender rate.

#### 1. Introduction

In the classical ruin theory, compound Poisson risk model, is the main research object [1, 2], where is the initial reserve, is the premium rate, and is a Poisson process with intensity , representing the number of claims up to time . The individual claim sizes , independent of , are i.i.d. positive random variables with distribution function and density function with mean . In the model, the premium income process is a linear function of time; it does not matter to claim. But in actual life, the arrival of policy of insurance company is usually associated with occurrence of claim; for example, the more the number of policies sold, the more the number of claims happened. Therefore, many studies in literature discuss the dependent relationship among the premium income, interclaim arrivals, and the claim size. See, for example, Liu et al. [3] considering a Markov-dependent risk model with a constant dividend barrier. Shi et al. [4] explore methods that allow for the correlation among frequency and severity components for microlevel insurance data. Jiang et al. [5] investigate some uniform asymptotic estimates for finite-time ruin probabilities when the claim size vector and its interarrival time are subject to certain general dependence structure. Zhang and Yang [6], Shi et al. [7], and Zou et al. [8] consider a compound Poisson risk model and a dependence structure of the claim size and interclaim time modeled by a Farlie-Gumbel-Morgenstern copula.

The above papers always assume the claim number follows a Poisson distribution, but in fact the claim number does not fully comply with the rule of Poisson distribution and its variance is often greater than the mean. Except the natural environment, an important reason for this phenomenon is that insurance companies have adopted risk aversion mechanism, such as franchise system and no-claim discount system [9]. This makes the policy holder weighs the interests which may not claim for compensation in the event of an accident; it will cause the claim number to be less than the number of accidents. In addition, on the one hand, the insurance company will have huge funds and various kinds of reserves in the operation process, which formed the huge amount of available funds. On the other hand, in order to protect the interests of the insured, the insurance company must use the fund rationally and effectively. In fact, the insurance industry is very active in the financial markets. In the financial markets of western developed countries, the total amount of funds provided by the insurance industry is close to commercial banks. So considering the risk model with investment income has greater practical value and realistic significance [10–12].

In view of the above problems, this paper will promote the premium income process of insurance companies to follow the compound Poisson-Geometric process [13–15], while the counting processes of claim and surrender are the -thinning process and the -thinning process of premium income process and further consideration of the investment interest rate. For the new improved model, we study the properties of surplus process, adjustment coefficient equation, ruin probability, and the expectation and variance of the first time to reach a given level. Finally, numerical analysis is also given.

The contents of this paper are organized as follows: Section 2 introduces the risk model. In Section 3, we give the main results of the paper. Finally, we provide the numerical examples in Section 4.

#### 2. The Risk Model

*Definition 1. *Let and be a probability space; ; then, the surplus process with initial surplus is defined as follows:where represents the initial capital and represents the investment capital, which is based on the size of initial capital, premium income per unit of time, and the predicted claim sizes. represents the investment income per unit of time. is a Poisson-Geometric process with parameters and denoting the number of premiums up to time ; namely, . is a sequence of i.i.d. random variables representing the amount of the th premium and . is the -thinning process of denoting the number of claims up to time ; namely, . The individual claims sizes are a sequence of i.i.d. random variables and . is the -thinning process of denoting the number of surrenders up to time ; namely, and and . The sequence of i.i.d. random variables represents the amount of the th payment of insurance policy and , , and . is a standard Brownian motion denoting the uncertain benefits and payments of insurance companies. is a constant, representing the diffusion volatility parameter. In addition, we suppose that , , , , and are mutually independent. From the theory of point process, and are also mutually independent.

Let be profits process. In order to ensure the insurance company’s steady business, we assume , and the relative security loading factor is defined as follows:

#### 3. Main Results

Lemma 2. *The profits process has the following properties:*(i)* has stationary and independent increments.*(ii)*.*

*Lemma 3. For the profits process , when , one has the following:*

*Lemma 4. For the profits process , suppose for some ; then, there is a function such that*

*Proof . *ConsiderLetwhere is the moment generating function of . Similarly, we can define and .

*The following discussions are adjustment coefficient and the adjustment coefficient equation. Since the ruin probability as a number of indicators can evaluate insurance company solvency, it attracts attention. The research goal is to obtain specific expression of ruin probability. However, it is very difficult to directly obtain the expression of this function, but Lundberg found an indirect expression way by introducing a parameter which can play the intermediary role, namely, Lundberg coefficient or adjustment coefficient. Its principle is that the ruin probability is expressed as a function of adjustment coefficient and then seeks the calculation for adjustment coefficient. Thus, the adjustment coefficient plays a very important role in the study of ruin probability.*

*Lemma 5. Equation is said to be an adjustment coefficient equation of the risk model (2), and it has a unique positive solution , which is called an adjustment coefficient (see Figure 1).*