Abstract
A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.
1. Introduction
The compound binomial model is a discrete time analogue of compound Poisson model. In the compound binomial model, the counting process is a binomial process. From the compound binomial model proposed by Gerber [1], a series of papers and books have studied this model (see Gerber [1]; Shiu [2]; Cossette [3]; Wu [4]; Peng et al. [5] and references therein).
As a class of important stochastic point processes, the batch Markovian arrival process (BMAP), proposed by Lucantoni [6], is dense in the class of stationary point processes. BMAP is used to model the stochastic processes in finance, computer, reliability, communication, and inventory conveniently. Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP, introduced by Neuts [7], includes phase-type (PH) renewal processes and nonrenewal processes such as the Markov modulated Poisson process (MMPP). Like Ahn et al. [8], Eric et al. [9], Artalejo et al. [10], Dong and Liu [11] and many authors have studied the compound Poisson process with MAP.
Inspired by Ahn et al. [8], Badescu et al. [12], Eric et al. [9], Artalejo et al. [10], and Dong and Liu [11], we discuss the compound Binomial model with BMAP. In this model, the counting process is a BMAP, which is a reasonable assumption. For example, an insurance company, which accepts the car insurance policies, might need to deal with several traffic accidents a day. Moreover, in different circumstances, the probability of traffic accident and the claim sizes are of big differences. So it may be more reasonable that the premium rate of car insurance is different in different environments. Therefore, we assume that the premium rate, probability of the claim occurring, and the claim amount are all influenced by the phase process of BMAP. Also, we study the compound binomial model with BMAP and threshold dividend. This study has certain guiding significance in insurance company and shareholders.
This paper is structured as follows: the specific definition of a compound binomial model with BMAP is introduced in Section 2. In Section 3, we discuss the ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm is proposed. In Section 4, we also discuss research on the compound binomial model with BMAP and threshold dividend. The recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the explicit expression of the total discounted dividend values are obtained and proved. Finally, we present some numerical examples to illustrate in Section 5.
2. Model
Let be a probability space with filtration containing all objects defined in this paper. Assume that satisfies the usual conditions, i.e., is right-continuous and -complete. At first, we will introduce the compound binomial model and the batch Markovian arrival process.
2.1. Compound Binomial Model
In the compound binomial model, denotes the surplus process of an insurer and is given bywhere the initial surplus is a nonnegative integer, is the aggregate claim up to time , which is described byand . In any time period, the probability with only a claim occurrence is , and the probability with no claim occurrence is . We denote by the event where a claim occurs in the time period , and we denote by the event where no claim occurs in the time period . The occurrences of claims in different time periods are independent events. denotes the claim amount that probably occurs at time , and are mutually independent, identically distributed (i.i.d.), positive integer-valued random variables, which have a common discrete distribution . Denote with . And the claim amounts are independent of .
2.2. Batch Markovian Arrival Process
Definition 1. Let . Given a series of matrixes and which satisfied the following conditions:(1) are substochastic matrixes(2)The matrix is nonsingular(3) is a stochastic matrix, and Then, is called the numerical characteristic of discrete-time batch Markovian arrival process.
Proposition 1. Assume that be the numerical characteristic of discrete-time batch Markovian arrival process. Then,(1) is a conservative matrix, and each state is sojourned.(2)Let and Then, is a conservative matrix, and each state is sojourned.(3) and are both regular.
Definition 2. Given the numerical characteristics of batch Markovian arrival process , let be a stochastic process with transition probability matrix and be a two-dimensional discrete-time batch Markovian process. Let be the initial probability distribution vector of , which satisfied . We call as a discrete-time batch Markovian arrival process (DTBMAP); for short, we can denote it as DTBMAP . is called the counting process and is called the phase process.
The BMAP is one of the most flexible stochastic processes and is defined as a specific Markov chain (MC). More precisely, the BMAP consists of two different processes with discrete state space. One process represents the dynamics of internal state called phase process, and the other process corresponds to the number of events, i.e., the counting process like a binomial process. The phase process is usually modeled by a MC, and the counting process is modulated by the phase process. In fact, Markov-modulated Bernoulli process and discrete-time platoon arrival process, which are specific and subclasses of BMAP, have been utilized to evaluate the information communication systems based on the queueing analysis and finance. BMAP enables one to capture the realistic assumptions as much as possible and provide solutions that practitioners can implement.
2.3. Modified Model
The model we considered in this paper can be described bywhere is the counting process of DTBMAP with state space . and , representing the size of the th premium and claim, respectively, are both dependent on the state of the phase process of DTBMAP . That is, given , is i.i.d. variable with the common binomial distribution and is i.i.d. positive and integer-valued stochastic series with the mean and the common distribution . Denote with . And assume , , and are independent.
We should note the following: (1) gives the probability of no state changes without claim arrivals; (2) gives the probability of state changes to state without claim arrivals; (3) gives the probability of no state changes with claims arrival; (4) gives the probability of state changes to state with claims arrival. Furthermore, every insurer would want to make a profit. That is, the expected claim size over a single period is strictly inferior to the premium size, i.e.,where is the safety factor.
Before introducing the main results, we should point out that the DTBMAP is very general. On the one hand, it may represent a renewal process where the interclaim times follow binomial distributions and negative binomial distribution or even discrete-time phase-type distributions. On the other hand, it allows for situations where numbers of claim time and claim size random variables are dependent.
Remark 1. When , the compound binomial model with DTBMAP degenerates to the compound binomial model. When , then it is the Markov-modulated compound binomial model, a degenerate case of the compound binomial model with DTBMAP .
3. Ruin Probability
3.1. Introduction
We define the time of ruin as
If ruin never occurs, . Also, let us define the conditional finite-time ruin probability asand conditional finite-time non-ruin probability as
Denote
Obviously, we can see that the unconditional finite-time ruin and nonruin probability, and , can be derived from the conditional ones with following formulas, respectively:
For convenience, we also define the infinite-time ones by simply letting in our previous conditional or unconditional finite-time ruin or nonruin probabilities. Thus, if we obtain the conditional finite-time ruin probability, all of ruin probabilities of this model are solved.
3.2. Main Result
For convenience, in the next article, we denotewhere is a function of .
Theorem 1. In the compound binomial model with DTBMAP , the conditional finite-time nonruin probabilities satisfy the following recursive formula:where . Then,where represents the th convolution of . And
Proof. We can separate some possible cases by conditioning on the r.v.‘s . There are possible cases as follows:(1)No state changes and no claim arrivals(2)State changes to state and no claim arrivals(3)No state changes and claims arrival(4)State changes to state and claims arrivalThen, the following formula can be easily derived by using the formula of full probability and the Markov property. For all , we haveThus, (12) is derived when we rewrite (15) into the matrix form.
In order to proof the following theorem, some definitions are required to be introduced. Let be the elapsed time by the phase process in state over the first periods where if is true and if is false. We also denote the amount of decrease of the surplus process over the first periods when the phase process is in state , i.e., . Denote as the amount of decrease of the surplus process over the first periods. Furthermore, and are denoted as the th premium amount and claim size when the phase process is in state , respectively.
Proposition 2. The infinite-time ruin probability tends to 0 as initial surplus tends to .
Proof. Obviously, we can see .
Taking the limit as of yieldsSince is irreducible and ergodic, it follows thatwhere is the stationary distribution of .
We can easily see that for given , and are both i.i.d. and is distributed as . Because the phase process is irreducible, :where . Therefore, is a random walk for and from the strong law of large numbers, we can find :By combining (16) to (19), we can obtain thatEquation (20) and the safety loading condition imply that and thus ensure that is finite. Consequently,
Theorem 2. In the compound binomial model with DTBMAP , the numerical algorithm proposed to obtain the conditional finite-time nonruin probabilities is as follows: Fix for and . Find for and by solving the following system of equations with unknown parameters:
Proof. First, by conditioning, respectively, on the random variables , four cases probably occurred. And from the stationarity of the surplus process, we can findfor and . Given that for and , we must solve the system of equations- unknown parameters given by equation (23) for and .
4. Compound Binomial Model with BMAP and Dividend
In this section, we will embed a threshold dividend strategy in the compound binomial model with BMAP . First, we will introduce the specific description of this model.
4.1. Description
Based on the compound binomial model with BMAP , we can define the compound binomial model with BMAP and threshold dividend strategy. The surplus process of an insurer is given bywhere are entirely the same as the description in model (4). is the dividend threshold, i.e., if the surplus of an insurer is greater than , the exceed part will pay out as dividend to the shareholders and if the surplus of an insurer is smaller than , nothing is needed to do. And we should point out the assumption that the dividend is paid out after the premium is received and claims are paid out.
Similarly, we define the ruin time of this model as
If ruin never occurs, .
4.2. Gerber–Shiu Function
The Gerber–Shiu function, also called expected discounted penalty function, was first introduced by Gerber–Shiu [1]. Many papers and books have studied it.
Definition 3. The Gerber–Shiu function is defined bywhere is the discount factor, is a binary function, and represents the surplus before ruin. represents the deficit at ruin, and is the indicator function of an event taking value 1 whenever the event occurs and 0 when it does not.
The Gerber–Shiu function plays an important role in risk theory. When , the Gerber–Shiu function changes to the ruin probability. When , it changes to the discounted surplus before ruin time. When , it changes to the discounted deficit at ruin. Studying on the Gerber–Shiu function can understand this model more deeply and enable to properly handle the operations of an insurance company.
To solve the problem, we denote some auxiliary functions. Denote the conditional Gerber–Shiu function asand denoteWe can easily see thatThus, we can solve the Gerber–Shiu function by solving . Next, we will derive the solution of .
Theorem 3. In the compound binomial model with DTBMAP and dividend threshold , the conditional Gerber–Shiu functions satisfy the following recursive formula.
For ,whereand for ,where
Proof. Similarly, by conditioning on , we can derive that for ,But for ,Rewrite the abovementioned equations into matrix, and we can derive the theorem.
Remark 2. When we want to calculate , we can perform the following: Choose ; then, we can derive an equation system containing equations with unknown numbers. The solutions of the equation system are just . , can be calculated by (32).
4.3. Discounted Dividend Value
For the risk model with dividend, we are also interested in the expected discounted dividend value of all dividends up to the ruin time in general. Especially in shareholders’ standpoint, the dividend value is the aim and the only focus thing. So, we will discuss research on the expected discounted dividend value of all dividends up to the ruin time. Let denote the discounted factor, denote the expected discounted dividend value of the first dividend under the conditions and , denote the expected discounted dividend value of all dividends up to the ruin time under the conditions and , and denote the expected discounted dividend value of all dividends up to the ruin time under the conditions .
Theorem 4. In the compound binomial model with DTBMAP and dividend threshold , we have(1) for and .(2)For ,(3), for and , satisfied the following recursive formula:where
Theorem 5. In the compound binomial model with DTBMAP and dividend threshold , we have(1) for and .(2), , and satisfied the following expression:whereFor ,and is a series of matrixes:(1)(2), ,(3),(4), ,
Proof. Similar to the method of the previous theorem, we can obtain a series of equations. Then, we can rewrite the equation system into matrix as follows:Hence, to prove this theorem, the most important thing is to prove that is nonsingular.
It can be easily seen that since is a probability distribution function, and ,In each row, we can write .(1)When , we can see Because is a stochastic matrix, we can derive that So by combining (48) with (49) we can derive Inequality (50) can be written into That is,(2)When , by using the same method, we can also obtainThat is,Inequality (52) and inequality (54) lead to that the absolute value of diagonal (the th element in the th row) is greater than the sum of others in this row. For being a arbitrary, is a (row) strictly diagonally dominant matrix. Hence, is nonsingular, which leads to the result.
5. Numerical illustration
Example 1. Let . The matrixes are given as follows:When , we let . And assume , , , and . Then, and . From it, we can see that is a “good” state, but is a “bad” state. Our example is structured in order to differentiate the “good “state and “bad” state, so we can see the difference between and . When we let , we can obtain the expected dividend value of all dividends up to the ruin time by using Theorem 5. All results are shown in Figure 1.
In order to reflect the effect of each factor, we let ; ; and , respectively, for comparison. The results of the situations ; ; and are shown in Figures 2–4, respectively.
Combining all results, we can analyze the following:(1) is gradually increased with the increase of initial surplus (2)For a given , is gradually increased with the increase of initial surplus (3)When the initial surplus is more and more big, the difference between and is more and more small(4)For is a “good” state and is a “bad” state, is larger than when is equal(5) is larger than , but smaller than when initial surplus is equal(6)The safety factor is larger and the expected dividend value of all dividends up to the ruin time is larger when initial surplus is equal
Data Availability
The data used to support the findings of this study are currently under embargo while the research findings are commercialized. Requests for data, 6 months after publication of this article, will be considered by the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by theNational Natural Science Foundation (No. 11701175), the Hunan Natural Science Foundation (No. 2019JJ50024), and the Hunan Philosophy and Social Science Foundation (Nos. 17YBQ020 and 17YBA291).