# Compound Binomial Model with Batch Markovian Arrival Process

Authors: Fang Jin; Chengxun Wu; Hui Ou
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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 Section2. 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

LetΩ,ℱ,ℙ be a probability space with filtration ℱt containing all objects defined in this paper. Assume that ℱt satisfies the usual conditions, i.e., ℱt is right-continuous and ℙ-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,Cn,n=0,1,2,… denotes the surplus process of an insurer and is given by(1)Cn=u+t−Sn,n=0,1,2,…,where the initial surplus u is a nonnegative integer, Sn is the aggregate claim up to time n, which is described by(2)Sn=X1ξ1+X2ξ2+⋯+Xnξn,and S0=0. In any time period, the probability with only a claim occurrence is θ,0<θ≤1, and the probability with no claim occurrence is λ=1−θ. We denote by ξn=1 the event where a claim occurs in the time period n−1,n, and we denote by ξn=0 the event where no claim occurs in the time period n−1,n. The occurrences of claims in different time periods are independent events. X=Xn,t=1,2,… denotes the claim amount that probably occurs at time t, and X1,X2,X3,… are mutually independent, identically distributed (i.i.d.), positive integer-valued random variables, which have a common discrete distribution PX=k=fk,k=1,2,…. Denote Fk=PX≤k=∑j=1kfj with F0=0. And the claim amounts X=Xn,n=1,2,… are independent of ξ=ξn,n=1,2,….


### 2.2. Batch Markovian Arrival Process

Definition 1.
LetE=e1,e2,…,emm≥1. Given a series of m×m matrixes D0=dij0 and Dk=dijkk∈ℕ+ which satisfied the following conditions:(1)
Dkk∈ℕ+ are substochastic matrixes(2)
The matrixI−D0 is nonsingular(3)
P=∑k=0∞Dk=qij is a stochastic matrix, and qij=∑k=0∞dijk
Then,D0,Dkk∈ℕ+ is called the numerical characteristic of discrete-time batch Markovian arrival process.Proposition 1.
Assume thatD0,Dkk∈ℕ+ be the numerical characteristic of discrete-time batch Markovian arrival process. Then,(1)
P=∑k=0∞Dk is a m×m conservative matrix, and each state is sojourned.(2)
LetE∗=n,j,n∈ℕ,ej∈E and(3)P∗=D0D1D2D3⋯0D0D1D2⋯00D0D1⋯000D0⋯⋮⋮⋮⋮⋱.Then,P∗ is a E∗×E∗ conservative matrix, and each state is sojourned.(3)
P and P∗ are both regular.Definition 2.
Given the numerical characteristics of batch Markovian arrival processD0,Dkk∈ℕ+, let X∗=X∗t,t∈ℕ be a stochastic process with transition probability matrix P∗ and X∗t=Nt,Jt be a two-dimensional discrete-time batch Markovian process. Let α=α1,α2,…,αmT be the initial probability distribution vector of X∗, which satisfied ∑j=1mαj=1. We call X∗ as a discrete-time batch Markovian arrival process (DTBMAP); for short, we can denote it as DTBMAP αT,D0,Dkk∈ℕ+. N=Nt,t∈ℕ is called the counting process and J=Jt,t∈ℕ 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 by(4)Un=u+∑k=1nck−∑k=1NnXk,n=0,1,2,…,where Nn is the counting process of DTBMAP αT,D0,Dkk∈ℕ+ with state space E∗=n,j,n∈ℕ,j∈E=e1,e2,e3,…,em. ck and Xk, representing the size of the kth premium and claim, respectively, are both dependent on the state of the phase process Jn of DTBMAP αT,D0,Dkk∈ℕ+. That is, given Jk=ei∈E, ck is i.i.d. variable with the common binomial distribution Bpipi∈0,1,qi=1−pi and Xk is i.i.d. positive and integer-valued stochastic series with the mean μi and the common distribution PXk=x=fix,x=1,2,…. Denote Fik=PXk≥x=∑j=1kfij with Fi0=0. And assume Jn, Nn,Xn, and cn are independent.We should note the following: (1)dii0 gives the probability of no state changes without claim arrivals; (2) dij0i≠j gives the probability of state i changes to state j without claim arrivals; (3) diikk∈ℕ+ gives the probability of no state changes with k claims arrival; (4) dijki≠j,k∈ℕ+ gives the probability of state i changes to state j with k 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.,(5)∑ei∈E∑ej∈E∑k=0∞αidijkpj=1+λ∑ei∈E∑ej∈E∑k=1∞αidijkkμj,where λ>0 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.
WhenE=e1,D0=p,D1=q=1−p,andDk=0k≥2,k∈ℕ+, the compound binomial model with DTBMAP αT,D0,Dkk∈ℕ+ degenerates to the compound binomial model. When E=e1,e2,…,emm≥2,D0=diagpm×m,D1=diag1−pm×m,andDk=0m×mk≥2,k∈ℕ+, then it is the Markov-modulated compound binomial model, a degenerate case of the compound binomial model with DTBMAP αT,D0,Dkk∈ℕ+.


## 2.1. Compound Binomial Model

In the compound binomial model,Cn,n=0,1,2,… denotes the surplus process of an insurer and is given by(1)Cn=u+t−Sn,n=0,1,2,…,where the initial surplus u is a nonnegative integer, Sn is the aggregate claim up to time n, which is described by(2)Sn=X1ξ1+X2ξ2+⋯+Xnξn,and S0=0. In any time period, the probability with only a claim occurrence is θ,0<θ≤1, and the probability with no claim occurrence is λ=1−θ. We denote by ξn=1 the event where a claim occurs in the time period n−1,n, and we denote by ξn=0 the event where no claim occurs in the time period n−1,n. The occurrences of claims in different time periods are independent events. X=Xn,t=1,2,… denotes the claim amount that probably occurs at time t, and X1,X2,X3,… are mutually independent, identically distributed (i.i.d.), positive integer-valued random variables, which have a common discrete distribution PX=k=fk,k=1,2,…. Denote Fk=PX≤k=∑j=1kfj with F0=0. And the claim amounts X=Xn,n=1,2,… are independent of ξ=ξn,n=1,2,….


## 2.2. Batch Markovian Arrival Process

Definition 1.
LetE=e1,e2,…,emm≥1. Given a series of m×m matrixes D0=dij0 and Dk=dijkk∈ℕ+ which satisfied the following conditions:(1)
Dkk∈ℕ+ are substochastic matrixes(2)
The matrixI−D0 is nonsingular(3)
P=∑k=0∞Dk=qij is a stochastic matrix, and qij=∑k=0∞dijk
Then,D0,Dkk∈ℕ+ is called the numerical characteristic of discrete-time batch Markovian arrival process.Proposition 1.
Assume thatD0,Dkk∈ℕ+ be the numerical characteristic of discrete-time batch Markovian arrival process. Then,(1)
P=∑k=0∞Dk is a m×m conservative matrix, and each state is sojourned.(2)
LetE∗=n,j,n∈ℕ,ej∈E and(3)P∗=D0D1D2D3⋯0D0D1D2⋯00D0D1⋯000D0⋯⋮⋮⋮⋮⋱.Then,P∗ is a E∗×E∗ conservative matrix, and each state is sojourned.(3)
P and P∗ are both regular.Definition 2.
Given the numerical characteristics of batch Markovian arrival processD0,Dkk∈ℕ+, let X∗=X∗t,t∈ℕ be a stochastic process with transition probability matrix P∗ and X∗t=Nt,Jt be a two-dimensional discrete-time batch Markovian process. Let α=α1,α2,…,αmT be the initial probability distribution vector of X∗, which satisfied ∑j=1mαj=1. We call X∗ as a discrete-time batch Markovian arrival process (DTBMAP); for short, we can denote it as DTBMAP αT,D0,Dkk∈ℕ+. N=Nt,t∈ℕ is called the counting process and J=Jt,t∈ℕ 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 by(4)Un=u+∑k=1nck−∑k=1NnXk,n=0,1,2,…,where Nn is the counting process of DTBMAP αT,D0,Dkk∈ℕ+ with state space E∗=n,j,n∈ℕ,j∈E=e1,e2,e3,…,em. ck and Xk, representing the size of the kth premium and claim, respectively, are both dependent on the state of the phase process Jn of DTBMAP αT,D0,Dkk∈ℕ+. That is, given Jk=ei∈E, ck is i.i.d. variable with the common binomial distribution Bpipi∈0,1,qi=1−pi and Xk is i.i.d. positive and integer-valued stochastic series with the mean μi and the common distribution PXk=x=fix,x=1,2,…. Denote Fik=PXk≥x=∑j=1kfij with Fi0=0. And assume Jn, Nn,Xn, and cn are independent.We should note the following: (1)dii0 gives the probability of no state changes without claim arrivals; (2) dij0i≠j gives the probability of state i changes to state j without claim arrivals; (3) diikk∈ℕ+ gives the probability of no state changes with k claims arrival; (4) dijki≠j,k∈ℕ+ gives the probability of state i changes to state j with k 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.,(5)∑ei∈E∑ej∈E∑k=0∞αidijkpj=1+λ∑ei∈E∑ej∈E∑k=1∞αidijkkμj,where λ>0 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.
WhenE=e1,D0=p,D1=q=1−p,andDk=0k≥2,k∈ℕ+, the compound binomial model with DTBMAP αT,D0,Dkk∈ℕ+ degenerates to the compound binomial model. When E=e1,e2,…,emm≥2,D0=diagpm×m,D1=diag1−pm×m,andDk=0m×mk≥2,k∈ℕ+, then it is the Markov-modulated compound binomial model, a degenerate case of the compound binomial model with DTBMAP αT,D0,Dkk∈ℕ+.


## 3. Ruin Probability

### 3.1. Introduction

We define the time of ruin as(6)τ=infk∈ℕ,Uk<0.If ruin never occurs,τ=∞. Also, let us define the conditional finite-time ruin probability as(7)ψu,nei=Pτ≤nJ0=eiei∈Eand conditional finite-time non-ruin probability as(8)φu,nei=1−ψu,neiei∈E.Denote(9)φ⟶u,n=φu,ne1φu,ne2φu,ne3⋯φu,nem1×mT.Obviously, we can see that the unconditional finite-time ruin and nonruin probability,ψu,n and φu,n, can be derived from the conditional ones with following formulas, respectively:(10)ψu,n=∑ei∈Eαiψu,nei,φu,n=∑ei∈Eαiφu,nei=αTφ⟶u,n.For convenience, we also define the infinite-time ones by simply lettingn⟶∞ 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 denote(11)hjx=h1jxh2jxh3jx⋯hmjx1×mT,where hijxi=1,2,…,m is a function of x.Theorem 1.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+, the conditional finite-time nonruin probabilities satisfy the following recursive formula:(12)φ⟶u,n+1=D0×P×φ⟶u+1,n+D0×I−P×φ⟶u,n+∑k=1∞Dk×P×h1u.n+∑k=1∞Dk×I−P×h2u.n,where P=diagp1,p2,…,pmandI=diag1,1,…,1. Then,(13)hi1u.n=∑y=0ufi∗ky+1φu−y,nei,hi2u.n=∑y=1uf1∗kyφu−y,ne1,where g∗ky represents the kth convolution of gy. And(14)∀ei∈E,φ0,1ei=1−∑ej∈Edij0−∑ej∈Edij1pjfj1,∀u∈ℕ,∀ei∈E,φu,0ei=1.Proof.
We can separate some possible cases by conditioning on the r.v.‘sJ1,N1,c1,andX1. There are possible cases as follows:(1)
No state changes and no claim arrivals(2)
Statei changes to state ji≠j and no claim arrivals(3)
No state changes andkk≥1 claims arrival(4)
Statei changes to state ji≠j and kk≥1 claims arrival
Then, the following formula can be easily derived by using the formula of full probability and the Markov property. For allei∈E, we have(15)φu,n+1ei=∑ej∈Edij0pjφu+1,nej+qjφu,nej+∑ej∈E∑k=1∞dijkpj∑y=1u+1fj∗kyφu+1−y,nej+qj∑y=1ufj∗kyφu−y,nej.
Thus, (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. LetVin=∑l=1NnIJl=ei be the elapsed time by the phase process Jn in state ei over the first k periods where IA=1 if A is true and IA=0 if A is false. We also denote Win the amount of decrease of the surplus process over the first n periods when the phase process Jn is in state ei, i.e., Win=∑l=1NnXl−clIJl=ei. Denote Wn as the amount of decrease of the surplus process over the first n periods. Furthermore, Xin and cin are denoted as the nth premium amount and claim size when the phase process Jn is in state ei, respectively.Proposition 2.
The infinite-time ruin probability tends to 0 as initial surplusu tends to ∞.Proof.
Obviously, we can seeWn=∑ei∈EWin.
Taking the limit asn⟶∞ of Wn/n yields(16)limn⟶∞Wnn=limn⟶∞∑ei∈EWinn=∑ei∈Elimn⟶∞WinVin×Vinn.
SinceJn is irreducible and ergodic, it follows that(17)limn⟶∞Vinn=ξi,where ξ=ξ1,ξ2,…,ξm is the stationary distribution of Jn.
We can easily see that for givenJn=ei, Xn and cn are both i.i.d. and Win is distributed as ∑l=1VinXil−cil. Because the phase process Jn is irreducible, limn⟶∞Vin=∞:(18)limn⟶∞WinVin=limn⟶∞∑l=1VinXil−cilVin=limn⟶∞∑l=1nXil−ciln=limn⟶∞Linn.where Lin=∑l=1nXil−cil. Therefore, Lin is a random walk for ei∈E and from the strong law of large numbers, we can find ∀ei∈E:(19)limn⟶∞Linn=EXil−cil=μi−pi.
By combining (16) to (19), we can obtain that(20)limn⟶∞Wnn=∑ei∈Eαiμi−pi.
Equation (20) and the safety loading condition imply that limn⟶∞Wn=−∞ and thus ensure that maxn∈ℕWn is finite. Consequently,(21)limn⟶∞ψu=limn⟶∞Pmaxn∈ℕWn>u=0.Theorem 2.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ , the numerical algorithm proposed to obtain the conditional finite-time nonruin probabilities is as follows:Fixφuei=1 for u=n,n+1,n+2,… and ei∈E.Findφuei for u=0,1,2,…,n and ei∈E by solving the following system of m×n equations with m×n unknown parameters:(22)φuei=∑ej∈Edij0pjφu+1ej+qjφuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyφu+1−yej+∑y=1uqjfjyφu−yej.Proof.
First, by conditioning, respectively, on the random variablesJ1,N1,c1,andX1, four cases probably occurred. And from the stationarity of the surplus process, we can find(23)φuei=∑ej∈Edij0pjφu+1ej+qjφuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyφu+1−yej+∑y=1uqjfjyφu−yej,for ei∈E and u∈ℕ. Given that φuei=1 for u=n,n+1,n+2,… and ei∈E, we must solve the system of m×n equations-m×n unknown parameters given by equation (23) for u=0,1,2,…,n and ei∈E.


## 3.1. Introduction

We define the time of ruin as(6)τ=infk∈ℕ,Uk<0.If ruin never occurs,τ=∞. Also, let us define the conditional finite-time ruin probability as(7)ψu,nei=Pτ≤nJ0=eiei∈Eand conditional finite-time non-ruin probability as(8)φu,nei=1−ψu,neiei∈E.Denote(9)φ⟶u,n=φu,ne1φu,ne2φu,ne3⋯φu,nem1×mT.Obviously, we can see that the unconditional finite-time ruin and nonruin probability,ψu,n and φu,n, can be derived from the conditional ones with following formulas, respectively:(10)ψu,n=∑ei∈Eαiψu,nei,φu,n=∑ei∈Eαiφu,nei=αTφ⟶u,n.For convenience, we also define the infinite-time ones by simply lettingn⟶∞ 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 denote(11)hjx=h1jxh2jxh3jx⋯hmjx1×mT,where hijxi=1,2,…,m is a function of x.Theorem 1.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+, the conditional finite-time nonruin probabilities satisfy the following recursive formula:(12)φ⟶u,n+1=D0×P×φ⟶u+1,n+D0×I−P×φ⟶u,n+∑k=1∞Dk×P×h1u.n+∑k=1∞Dk×I−P×h2u.n,where P=diagp1,p2,…,pmandI=diag1,1,…,1. Then,(13)hi1u.n=∑y=0ufi∗ky+1φu−y,nei,hi2u.n=∑y=1uf1∗kyφu−y,ne1,where g∗ky represents the kth convolution of gy. And(14)∀ei∈E,φ0,1ei=1−∑ej∈Edij0−∑ej∈Edij1pjfj1,∀u∈ℕ,∀ei∈E,φu,0ei=1.Proof.
We can separate some possible cases by conditioning on the r.v.‘sJ1,N1,c1,andX1. There are possible cases as follows:(1)
No state changes and no claim arrivals(2)
Statei changes to state ji≠j and no claim arrivals(3)
No state changes andkk≥1 claims arrival(4)
Statei changes to state ji≠j and kk≥1 claims arrival
Then, the following formula can be easily derived by using the formula of full probability and the Markov property. For allei∈E, we have(15)φu,n+1ei=∑ej∈Edij0pjφu+1,nej+qjφu,nej+∑ej∈E∑k=1∞dijkpj∑y=1u+1fj∗kyφu+1−y,nej+qj∑y=1ufj∗kyφu−y,nej.
Thus, (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. LetVin=∑l=1NnIJl=ei be the elapsed time by the phase process Jn in state ei over the first k periods where IA=1 if A is true and IA=0 if A is false. We also denote Win the amount of decrease of the surplus process over the first n periods when the phase process Jn is in state ei, i.e., Win=∑l=1NnXl−clIJl=ei. Denote Wn as the amount of decrease of the surplus process over the first n periods. Furthermore, Xin and cin are denoted as the nth premium amount and claim size when the phase process Jn is in state ei, respectively.Proposition 2.
The infinite-time ruin probability tends to 0 as initial surplusu tends to ∞.Proof.
Obviously, we can seeWn=∑ei∈EWin.
Taking the limit asn⟶∞ of Wn/n yields(16)limn⟶∞Wnn=limn⟶∞∑ei∈EWinn=∑ei∈Elimn⟶∞WinVin×Vinn.
SinceJn is irreducible and ergodic, it follows that(17)limn⟶∞Vinn=ξi,where ξ=ξ1,ξ2,…,ξm is the stationary distribution of Jn.
We can easily see that for givenJn=ei, Xn and cn are both i.i.d. and Win is distributed as ∑l=1VinXil−cil. Because the phase process Jn is irreducible, limn⟶∞Vin=∞:(18)limn⟶∞WinVin=limn⟶∞∑l=1VinXil−cilVin=limn⟶∞∑l=1nXil−ciln=limn⟶∞Linn.where Lin=∑l=1nXil−cil. Therefore, Lin is a random walk for ei∈E and from the strong law of large numbers, we can find ∀ei∈E:(19)limn⟶∞Linn=EXil−cil=μi−pi.
By combining (16) to (19), we can obtain that(20)limn⟶∞Wnn=∑ei∈Eαiμi−pi.
Equation (20) and the safety loading condition imply that limn⟶∞Wn=−∞ and thus ensure that maxn∈ℕWn is finite. Consequently,(21)limn⟶∞ψu=limn⟶∞Pmaxn∈ℕWn>u=0.Theorem 2.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ , the numerical algorithm proposed to obtain the conditional finite-time nonruin probabilities is as follows:Fixφuei=1 for u=n,n+1,n+2,… and ei∈E.Findφuei for u=0,1,2,…,n and ei∈E by solving the following system of m×n equations with m×n unknown parameters:(22)φuei=∑ej∈Edij0pjφu+1ej+qjφuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyφu+1−yej+∑y=1uqjfjyφu−yej.Proof.
First, by conditioning, respectively, on the random variablesJ1,N1,c1,andX1, four cases probably occurred. And from the stationarity of the surplus process, we can find(23)φuei=∑ej∈Edij0pjφu+1ej+qjφuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyφu+1−yej+∑y=1uqjfjyφu−yej,for ei∈E and u∈ℕ. Given that φuei=1 for u=n,n+1,n+2,… and ei∈E, we must solve the system of m×n equations-m×n unknown parameters given by equation (23) for u=0,1,2,…,n and ei∈E.


## 4. Compound Binomial Model with BMAP and Dividend

In this section, we will embed a threshold dividend strategy in the compound binomial model with BMAPαT,D0,Dkk∈ℕ+. First, we will introduce the specific description of this model.


### 4.1. Description

Based on the compound binomial model with BMAPαT,D0,Dkk∈ℕ+, we can define the compound binomial model with BMAP αT,D0,Dkk∈ℕ+ and threshold dividend strategy. The surplus process of an insurer is given by(24)Vn=cn−∑k=Nn−1NnXk,b,n∈ℕ+;V0=u.where Nn,cn,​andXn are entirely the same as the description in model (4). b>0 is the dividend threshold, i.e., if the surplus of an insurer is greater than b, the exceed part will pay out as dividend to the shareholders and if the surplus of an insurer is smaller than b, 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(25)τb=infk∈ℕ,Vk<0.If ruin never occurs,τb=∞.


### 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 by(26)mu=EvτbwUτb−1,UτbIτb<∞U0=u,where v∈0,1 is the discount factor, w⋅,⋅:ℕ+×ℕ+⟶ℕ is a binary function, and Uτb−1 represents the surplus before ruin. Uτb represents the deficit at ruin, and IA is the indicator function of an event A taking value 1 whenever the event A occurs and 0 when it does not.
The Gerber–Shiu function plays an important role in risk theory. Whenw⋅,⋅≡1andv=1, the Gerber–Shiu function changes to the ruin probability. When wx,y=x, it changes to the discounted surplus before ruin time. When wx,y=y, 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 as(27)muei=EvτbwUτb−1,UτbIτb<∞U0=u,J0=ei,and denote(28)mu=mue1mue2mue3⋯muem1×mT.
We can easily see that(29)mu=∑ei∈Eαimuei=αTmu.
Thus, we can solve the Gerber–Shiu functionmu by solving muei. Next, we will derive the solution of muei.Theorem 3.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, the conditional Gerber–Shiu functions satisfy the following recursive formula.
Foru=0,2,…,b−1,(30)mu=vD0×P×mu+1+vD0×I−P×mu+v∑k=1∞Dk×P×h3u+v∑k=1∞Dk×I−P×h4u,where(31)hi3u=∑y=1ufi∗ky+1mu−yei+∑y=1∞fi∗ky+1+uwu,y,hi4u=∑y=0uf1∗kymu−ye1+∑y=1∞f1∗ky+uwu,y,and for u=b,b+1,…,(32)mu=vD0×mb+v∑k=1∞Dk×P×h5u+v∑k=1∞Dk×I−P×h6u,where(33)hi5u=∑y=0u−bfi∗ky+1mbei+∑y=u−b+1ufi∗ky+1mu−yei+∑y=1∞fi∗ky+1+uwu,y,hi6u=∑y=1u−bfi∗kymbei+∑y=u−b+1ufi∗kymu−yei+∑y=1∞fi∗ky+uwu,y.Proof.
Similarly, by conditioning onJ1,N1,c1,andX1, we can derive that for u=b,b+1,…,(34)muei=∑ej∈Edij0pjmb|ej+qjmb|ej+∑ej∈E∑k=1∞dijk∑y=1∞pjfj∗kymbejIu+1−y>b+mu+1−yejI0≤u+1−y≤b+wu,y−u−1Iu+1−y<0+∑y=1∞qjfj∗kymbejIu−y>b+mu−yejI0≤u−y≤b+wu,y−uIu−y<0=∑ej∈Edij0mbej+∑ej∈E∑k=1∞dijkpj∑y=0u−bfj∗ky+1mbej+∑y=u−b+1ufj∗ky+1mu−yej+∑y=1∞fj∗ky+1+uwu,y+∑ej∈E∑k=1∞dijkqj∑y=1u−bfj∗kymbej+∑y=u−b+1ufj∗kymu−yej+∑y=1∞fj∗ky+uwu,y.
But foru=0,2,…,b−1,(35)muei=∑ej∈Edij0pjmu+1ej+qjmuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjymu+1−yej+∑y=1uqjfjymu−yej.
Rewrite the abovementioned equations into matrix, and we can derive the theorem.Remark 2.
When we want to calculatemuei, we can perform the following:Chooseu=0,1,2,…,b; then, we can derive an equation system containing mb+1 equations with mb+1 unknown numbers. The solutions of the equation system are just muei,ei∈E,u=0,1,2,…,b.muei,ei∈E,u≥b+1, 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. Letγ0<γ≤1 denote the discounted factor, B1uei denote the expected discounted dividend value of the first dividend under the conditions V0=u and J0=ei, B1uei denote the expected discounted dividend value of all dividends up to the ruin time under the conditions V0=u and J0=ei, and Bu denote the expected discounted dividend value of all dividends up to the ruin time under the conditions V0=u.Theorem 4.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, we have(1)
B1uei=γu−b for ei∈E and u>b.(2)
Forei∈E,(36)B1bei=γ∑ej∈Edij0pj+qjB1bej+γ∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyB1u+1−yej+∑y=1uqjfjyB1u−yej.(3)
D1uei, for ei∈E and u<b, satisfied the following recursive formula:(37)B1u=γD0×P×B1u+1+γD0×I−P×B1u+γ∑k=1∞Dk×P×h7u+γ∑k=1∞Dk×I−P×h8u,where(38)B1u=B1ue1B1ue2B1ue3⋯B1uem1×mT,hi7u=∑y=1uf1∗ky+1B1u−ye1,hi8u=∑y=0uf1∗kyB1u−ye1.Theorem 5.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, we have(1)
Buei=γu−b+γBbei for ei∈E and u>b.(2)
Buei, ei∈E, and u≤b satisfied the following expression:(39)ℬ=A−1V,(40)B0B1B2⋮Bb=A00A01A02⋯A0bA10A11A12⋯A1bA20A21A22⋯A2b⋮⋮⋮⋱⋮Ab0Ab1Ab2⋯Abb−1v10v11v12⋮v1b,where(41)Bi=Bie1Bie2Bie3⋯Biem1×mT.For i=0,1,2,…,b−1,(42)v1i=000⋯01×mT,v1b=−γ∑ej∈Ed1j0pj−γ∑ej∈Ed2j0pj−γ∑ej∈Ed3j0pj⋯−γ∑ej∈Edmj0pj1×mT,and Asls=0,1,…,b,l=0,1,…,b is a series of m×m matrixes:(1)
s+2≤l,Asl=0m×m(2)
s+1=l, Asl=aijm×m,(43)aij=γdij0pj.(3)
s=l,(44)s≠b,Asl=bijm×m,bij=γqidii0+pidii1fi1−1,i=j;γqjdij0+pjdij1fj1,i≠j.s=b,Abb=b˜ijm×m,b˜ij=γdii0+pidii1fi1−1,i=j;γdij0+pjdij1fj1,i≠j.(4)
s>l, Asl=cijm×m,(45)cij=γ∑k=1∞dijkpjfj∗ks+1−l+qjfj∗ks−l.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:(46)Aℬ=V,
Hence, to prove this theorem, the most important thing is to prove thatA is nonsingular.
It can be easily seen that sincefjx is a probability distribution function, ∀k=1,2,…,m and ∀ej∈E,(47)∑x=1∞fj∗kx=1.
In eachr row, we can write r=sn+l.(1)
Whens≠b, we can see(48)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej∈Eγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj=∑ej∈E∑s>l∑k=1∞drjkpjfj∗ks+1−l+qjfj∗ks−l+qjdrj0+pjdrj1fj1+drj0pj=γ∑ej∈E∑s>l∑k=1∞drjkpjfj∗ks+1−l+qjfj∗ks−l+drj0+pjdrj1fj1≤γ∑ej∈E∑k=1∞∑s>ldrjkpjfj∗ks+1−l+pjfj∗k1+qjfj∗ks−l+drj0=γ∑ej∈E∑k=1∞∑s>ldrjkpjfj∗ks+1−l+fj∗k1+qjfj∗ks−l+drj0≤γ∑ej∈E∑k=1∞∑s>ldrjkpj+qj+drj0=γ∑ej∈E∑k=0∞drjk.BecauseP=∑k=0∞Dk=qij is a stochastic matrix, we can derive that(49)∑ej∈E∑k=0∞drjk=1,∀er∈E.So by combining (48) with (49) we can derive(50)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej∈Eγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ.Inequality (50) can be written into(51)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej≠erγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ−γqrdrr0+prdrr1fr1≤1−γqrdrr0+prdrr1fr1=γqrdrr0+prdrr1fr1−1.That is,(52)∑ej∈Earj+∑ej≠erbrj+∑ej∈Ecrj≤brr.(2)
Whens=b, by using the same method, we can also obtain(53)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej≠erγdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ−γdrr0+prdrr1fr1≤1−γdrr0+prdrr1fr1=γdrr0+prdrr1fr1−1.That is,(54)∑ej∈Earj+∑ej≠erb˜rj+∑ej∈Ecrj≤b˜rr.
Inequality (52) and inequality (54) lead to that the absolute value of diagonal (the rth element in the rth row) is greater than the sum of others in this row. For r being a arbitrary, A is a (row) strictly diagonally dominant matrix. Hence, A is nonsingular, which leads to the result.


## 4.1. Description

Based on the compound binomial model with BMAPαT,D0,Dkk∈ℕ+, we can define the compound binomial model with BMAP αT,D0,Dkk∈ℕ+ and threshold dividend strategy. The surplus process of an insurer is given by(24)Vn=cn−∑k=Nn−1NnXk,b,n∈ℕ+;V0=u.where Nn,cn,​andXn are entirely the same as the description in model (4). b>0 is the dividend threshold, i.e., if the surplus of an insurer is greater than b, the exceed part will pay out as dividend to the shareholders and if the surplus of an insurer is smaller than b, 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(25)τb=infk∈ℕ,Vk<0.If ruin never occurs,τb=∞.


## 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 by(26)mu=EvτbwUτb−1,UτbIτb<∞U0=u,where v∈0,1 is the discount factor, w⋅,⋅:ℕ+×ℕ+⟶ℕ is a binary function, and Uτb−1 represents the surplus before ruin. Uτb represents the deficit at ruin, and IA is the indicator function of an event A taking value 1 whenever the event A occurs and 0 when it does not.
The Gerber–Shiu function plays an important role in risk theory. Whenw⋅,⋅≡1andv=1, the Gerber–Shiu function changes to the ruin probability. When wx,y=x, it changes to the discounted surplus before ruin time. When wx,y=y, 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 as(27)muei=EvτbwUτb−1,UτbIτb<∞U0=u,J0=ei,and denote(28)mu=mue1mue2mue3⋯muem1×mT.
We can easily see that(29)mu=∑ei∈Eαimuei=αTmu.
Thus, we can solve the Gerber–Shiu functionmu by solving muei. Next, we will derive the solution of muei.Theorem 3.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, the conditional Gerber–Shiu functions satisfy the following recursive formula.
Foru=0,2,…,b−1,(30)mu=vD0×P×mu+1+vD0×I−P×mu+v∑k=1∞Dk×P×h3u+v∑k=1∞Dk×I−P×h4u,where(31)hi3u=∑y=1ufi∗ky+1mu−yei+∑y=1∞fi∗ky+1+uwu,y,hi4u=∑y=0uf1∗kymu−ye1+∑y=1∞f1∗ky+uwu,y,and for u=b,b+1,…,(32)mu=vD0×mb+v∑k=1∞Dk×P×h5u+v∑k=1∞Dk×I−P×h6u,where(33)hi5u=∑y=0u−bfi∗ky+1mbei+∑y=u−b+1ufi∗ky+1mu−yei+∑y=1∞fi∗ky+1+uwu,y,hi6u=∑y=1u−bfi∗kymbei+∑y=u−b+1ufi∗kymu−yei+∑y=1∞fi∗ky+uwu,y.Proof.
Similarly, by conditioning onJ1,N1,c1,andX1, we can derive that for u=b,b+1,…,(34)muei=∑ej∈Edij0pjmb|ej+qjmb|ej+∑ej∈E∑k=1∞dijk∑y=1∞pjfj∗kymbejIu+1−y>b+mu+1−yejI0≤u+1−y≤b+wu,y−u−1Iu+1−y<0+∑y=1∞qjfj∗kymbejIu−y>b+mu−yejI0≤u−y≤b+wu,y−uIu−y<0=∑ej∈Edij0mbej+∑ej∈E∑k=1∞dijkpj∑y=0u−bfj∗ky+1mbej+∑y=u−b+1ufj∗ky+1mu−yej+∑y=1∞fj∗ky+1+uwu,y+∑ej∈E∑k=1∞dijkqj∑y=1u−bfj∗kymbej+∑y=u−b+1ufj∗kymu−yej+∑y=1∞fj∗ky+uwu,y.
But foru=0,2,…,b−1,(35)muei=∑ej∈Edij0pjmu+1ej+qjmuej+∑ej∈E∑k=1∞dijk∑y=1u+1pjfjymu+1−yej+∑y=1uqjfjymu−yej.
Rewrite the abovementioned equations into matrix, and we can derive the theorem.Remark 2.
When we want to calculatemuei, we can perform the following:Chooseu=0,1,2,…,b; then, we can derive an equation system containing mb+1 equations with mb+1 unknown numbers. The solutions of the equation system are just muei,ei∈E,u=0,1,2,…,b.muei,ei∈E,u≥b+1, 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. Letγ0<γ≤1 denote the discounted factor, B1uei denote the expected discounted dividend value of the first dividend under the conditions V0=u and J0=ei, B1uei denote the expected discounted dividend value of all dividends up to the ruin time under the conditions V0=u and J0=ei, and Bu denote the expected discounted dividend value of all dividends up to the ruin time under the conditions V0=u.Theorem 4.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, we have(1)
B1uei=γu−b for ei∈E and u>b.(2)
Forei∈E,(36)B1bei=γ∑ej∈Edij0pj+qjB1bej+γ∑ej∈E∑k=1∞dijk∑y=1u+1pjfjyB1u+1−yej+∑y=1uqjfjyB1u−yej.(3)
D1uei, for ei∈E and u<b, satisfied the following recursive formula:(37)B1u=γD0×P×B1u+1+γD0×I−P×B1u+γ∑k=1∞Dk×P×h7u+γ∑k=1∞Dk×I−P×h8u,where(38)B1u=B1ue1B1ue2B1ue3⋯B1uem1×mT,hi7u=∑y=1uf1∗ky+1B1u−ye1,hi8u=∑y=0uf1∗kyB1u−ye1.Theorem 5.
In the compound binomial model with DTBMAPαT,D0,Dkk∈ℕ+ and dividend threshold b, we have(1)
Buei=γu−b+γBbei for ei∈E and u>b.(2)
Buei, ei∈E, and u≤b satisfied the following expression:(39)ℬ=A−1V,(40)B0B1B2⋮Bb=A00A01A02⋯A0bA10A11A12⋯A1bA20A21A22⋯A2b⋮⋮⋮⋱⋮Ab0Ab1Ab2⋯Abb−1v10v11v12⋮v1b,where(41)Bi=Bie1Bie2Bie3⋯Biem1×mT.For i=0,1,2,…,b−1,(42)v1i=000⋯01×mT,v1b=−γ∑ej∈Ed1j0pj−γ∑ej∈Ed2j0pj−γ∑ej∈Ed3j0pj⋯−γ∑ej∈Edmj0pj1×mT,and Asls=0,1,…,b,l=0,1,…,b is a series of m×m matrixes:(1)
s+2≤l,Asl=0m×m(2)
s+1=l, Asl=aijm×m,(43)aij=γdij0pj.(3)
s=l,(44)s≠b,Asl=bijm×m,bij=γqidii0+pidii1fi1−1,i=j;γqjdij0+pjdij1fj1,i≠j.s=b,Abb=b˜ijm×m,b˜ij=γdii0+pidii1fi1−1,i=j;γdij0+pjdij1fj1,i≠j.(4)
s>l, Asl=cijm×m,(45)cij=γ∑k=1∞dijkpjfj∗ks+1−l+qjfj∗ks−l.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:(46)Aℬ=V,
Hence, to prove this theorem, the most important thing is to prove thatA is nonsingular.
It can be easily seen that sincefjx is a probability distribution function, ∀k=1,2,…,m and ∀ej∈E,(47)∑x=1∞fj∗kx=1.
In eachr row, we can write r=sn+l.(1)
Whens≠b, we can see(48)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej∈Eγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj=∑ej∈E∑s>l∑k=1∞drjkpjfj∗ks+1−l+qjfj∗ks−l+qjdrj0+pjdrj1fj1+drj0pj=γ∑ej∈E∑s>l∑k=1∞drjkpjfj∗ks+1−l+qjfj∗ks−l+drj0+pjdrj1fj1≤γ∑ej∈E∑k=1∞∑s>ldrjkpjfj∗ks+1−l+pjfj∗k1+qjfj∗ks−l+drj0=γ∑ej∈E∑k=1∞∑s>ldrjkpjfj∗ks+1−l+fj∗k1+qjfj∗ks−l+drj0≤γ∑ej∈E∑k=1∞∑s>ldrjkpj+qj+drj0=γ∑ej∈E∑k=0∞drjk.BecauseP=∑k=0∞Dk=qij is a stochastic matrix, we can derive that(49)∑ej∈E∑k=0∞drjk=1,∀er∈E.So by combining (48) with (49) we can derive(50)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej∈Eγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ.Inequality (50) can be written into(51)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej≠erγqjdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ−γqrdrr0+prdrr1fr1≤1−γqrdrr0+prdrr1fr1=γqrdrr0+prdrr1fr1−1.That is,(52)∑ej∈Earj+∑ej≠erbrj+∑ej∈Ecrj≤brr.(2)
Whens=b, by using the same method, we can also obtain(53)∑s>l∑ej∈E∑k=1∞γdrjkpjfj∗ks+1−l+qjfj∗ks−l+∑ej≠erγdrj0+pjdrj1fj1+∑ej∈Eγdrj0pj≤γ−γdrr0+prdrr1fr1≤1−γdrr0+prdrr1fr1=γdrr0+prdrr1fr1−1.That is,(54)∑ej∈Earj+∑ej≠erb˜rj+∑ej∈Ecrj≤b˜rr.
Inequality (52) and inequality (54) lead to that the absolute value of diagonal (the rth element in the rth row) is greater than the sum of others in this row. For r being a arbitrary, A is a (row) strictly diagonally dominant matrix. Hence, A is nonsingular, which leads to the result.


## 5. Numerical illustration

Example 1.
Letγ=1,b=5,E=e1,e2,andα=5/6,1/6T. The matrixes are given as follows:(55)D0=3/41/161/21/18,D1=1/161/162/91/36,D2=1/321/321/61/36.
Whenk≥3, we let Dk=02×2. And assume f11=3/4, f12=1/4, f21=2/3, and f22=1/3. Then, f1∗22=9/16,f1∗23=3/8,f1∗24=1/16,f2∗22=4/9,f2∗23=4/9,f2∗24=1/9,μ1=5/4, and μ2=4/3. From it, we can see that e1 is a “good” state, but e2 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 Bue1 and Bue2. When we let p1=2/3andp2=3/4, 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 letp1=3/4,p2=3/4; p1=3/4,p2=4/5; and p1=4/5,p2=3/4, respectively, for comparison. The results of the situations p1=3/4,p2=3/4; p1=3/4,p2=4/5; and p1=4/5,p2=3/4 are shown in Figures 2–4, respectively.
Combining all results, we can analyze the following:(1)
Bu is gradually increased with the increase of initial surplus u(2)
For a givenei, Buei is gradually increased with the increase of initial surplus u(3)
When the initial surplusu is more and more big, the difference between Bue1 and Bue2 is more and more small(4)
Fore1 is a “good” state and e2 is a “bad” state, Bue1 is larger than Bue2 when u is equal(5)
Bu is larger than Bue2, but smaller than Bue1 when initial surplus u 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 surplusu is equalFigure 1
Value of all dividendsp1=2/3andp2=3/4.Figure 2
Value of all dividendsp1=3/4andp2=3/4.Figure 3
Value of all dividendsp1=3/4andp2=4/5.Figure 4
Value of all dividendsp1=4/5andp2=3/4.



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*Source: 1932704-2020-11-28.xml*