Abstract
A periodic dividend problem is studied in this paper. We assume that dividend payments are made at a sequence of Poisson arrival times, and ruin is continuously monitored. First of all, three integro-differential equations for the expected discounted dividends are obtained. Then, we investigate the explicit expressions for the expected discounted dividends, and the optimal dividend barrier is given for exponential claims. A similar study on a generalized Gerber–Shiu function involving the absolute time is also performed. To demonstrate the existing results, we give some numerical examples.
1. Introduction
Suppose the dynamics of the surplus process of an insurance company at time t is defined as the solution towhere is the premium charged in the unit time and is the debit interest. is a compound Poisson process with intensity representing the total claim amounts until time is the claim size. Suppose that the claim sizes are independent of each other and have common density function We also assume that and are mutually independent.
Model (1) means that the insurer can make loans at the rate once the process goes below zero. When the insurer can be prohibited to run its business, i.e., the insurer is absolutely ruined. The absolute ruin risk models have been investigated extensively, see Gerber and Yang [1], Cai [2], Yuen et al. [3], Wang and Yin [4], and Cai and Yang [5], among others.
In recent years, the research problems in risk theory are more and more closely related to real life. The risk model with debit interest is a good example. The insurer cannot monitor the surplus continuously, and the dividend can be paid at some certain times. Because of its importance in real life, the topic of periodic dividends has become very popular in risk theory during the last twenty years. Considering dividends can only be made at some discrete times in practice, Albrecher et al. [6] put forward the periodic barrier dividends in this type of risk model. They assumed that both barrier dividends and ruin can only be observed at some randomized times. Considering the insurer monitors bankruptcy more closely than dividends, Avanzi et al. [7] investigated periodic dividend barrier strategy in the dual model where the solvency is monitored continuously. For those risk models with periodic dividends and bankruptcy, the reader can refer to Zhang and Cheung [8, 9], Avanzi et al. [10], Dong et al. [11], and Peng et al. [12], among others. Different to the papers mentioned above, the periodic dividend strategy in Peng et al. [12] is threshold strategy.
Now, we study surplus process (1) under periodic dividend strategy. We assume the ruin is continuously monitored as usual. Let be the sequence of dividend observation times with and be an exponentially random variable with In addition, we assume that are independent of and . Furthermore, we assume that no dividends are made at time 0. The modified surplus process is given by with being the constant level of the dividend barrier. Denote by the time of absolute ruin for with the convention that
Letbe the expected discounted dividend before time and The generalized Gerber–Shiu function is also our concern:where denotes the indicator function of event is a bounded measurable function of , is the surplus prior to time T, and is the deficit at time When is the Gerber–Shiu function proposed by Gerber and Shiu [13]. When describes the joint distribution of the absolute time T and
When is the probability generating function of . The topic on the number of claims has been studied by many scholars in various forms. Some scholars focus on some generalized Gerber–Shiu functions, see Li and Lu [14] and Wang et al. [15]; some only considered the Laplace transform, see Egidio dos Reis [16]; and some studied the density function, see Dickson [17] and Czarna et al. [18].
We arrange the paper as follows. In Section 2, we first derive a system of integro-differential equations for . For exponential claims, explicit results for are given. The generalized Gerber–Shiu function is discussed in Section 3. In Section 4, some numerical results are shown to illustrate the given results.
2. Results for
2.1. Integro-differential Equations for
The integro-differential equation is a conventional method; we start this section with the expression for .
Theorem 1. For we havefor we haveand for we have
In addition, we have the following relations:
Proof. A standard method is used here, see also Yuen et al. [3] and Albrecher et al. [6]. For a small interval , all the possible events are taken into account. Then, we havefor and for ,where
Taking derivative on s in (13) and letting we arrive at (6) and (7). By a similar method, (5) can be obtained from (14).
Continuity condition (8) can be obtained from (13), and (10) can be obtained by comparing (13) with (14). (9) can be obtained by using (6) and (7), while (11) is derived by (5) and (6). Let in (5), and we have boundary condition (12).
Remark 1. Obviously, (6) and (7) are the same as (2.1) and (2.2) in Yuen et al. [3], respectively.
2.2. Explicit Expressions of
In this section, the density function of claim sizes is supposed to be Applying the operator to (5), then we havefor
By the transforms and , (15) is reduced to a confluent hypergeometric equation:
It follows from Abramowitz and Stegun [19] that admits the following expression:where and are two constants, is the first kind of confluent hypergeometric function, and is the second kind of confluent hypergeometric function. Hence, the solution of (15) can be expressed as
Due to boundary condition (13), one immediately deduces
Using a similar procedure to (15), we deducefor andfor Solving (19) and (20) by the knowledge of the ordinary differential equation, one deduceswhere are roots of the equation
Clearly, is bounded. Hence, By (18), (21), and (22) and (8)–(11), we get the following equations:where
Substituting (22) into (7) and equating the coefficients of x lead toand then equating the constant term yields
Solving (24)–(30), we getwhere
Remark 2. For it is easy to check that Letting in (31) and (32) and noticing , we havewhich are in accordance to (3.4) in Yuen et al. [3].
When the initial surplus we can investigate the optimal dividend barrier which can maximize before absolute ruin. By (31) and (32), we identify that their numerators have nothing to do with the barrier b, and their denominators are We also find that their numerators are bigger than 0. So, the optimal barrier solves i.e.,When , which is the same as (3.5) in Yuen et al. [3].
Remark 3. When the claim sizes are general distributions, e.g., Erlang(n) or mixture of exponential distribution, the explicit expression for cannot be provided for Then, some numerical methods will be helpful, e.g., Yu et al. [20] and Zhang et al. [21].
3. Results for
Using a similar method for Theorem 1, we can obtain the following results for the generalized Gerber–Shiu function
Theorem 2. The generalized Gerber–Shiu function admits the following expressions:with the following conditions:
3.1. Explicit Expressions of
We also assume that the claim sizes have the density function similar to Section 2.2, and we have
Then, solving the above system of equations leads towhereand are solutions of the equation
Since is a bounded function, then By conditions (40)–(43), we have
Inserting (45) into (36) and comparing the coefficients of constants yield
Solving (48)–(53), one obtainswhere
4. Numerical Examples
In this last section, we give some numerical examples to illustrate the results. In Figures 1–4, we assume that the claim size follows exponential distribution with mean 2, the premium rate the Poisson intensity the discounted factor , and the debit interest Figure 1 depicts the profiles of for We find that is not a monotone function of When the curves of and are given in Figures 2 and 3 for several different β, respectively. It is shown that is a decreasing function of β, while is a increasing function of The bigger β means the more frequent observation, which can lead to the earlier ruin. By Figure 3, we also find that is more and more close to as β becomes more and more large. The profiles of are given in Figure 4 for some different initial surplus By Figure 4, we know that the optimal barrier is which has nothing to do with the initial surplus.
The numerical results for are given in Table 1 for various observation intensity β and debit interest The optimal dividend barrier is an increasing function of which is also an increasing function of α. More frequent observation will give rise to more dividend payments in the initial stage and earlier absolute ruin. In order to avoid the earlier absolute ruin, a bigger dividend barrier will be needed to decrease the dividend payments at the beginning.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11701319 and 11571198) and Fuyang Municipal Government-Fuyang Normal College Horizontal Cooperation Projects in 2017 (no. XDHXTD201709).