Compound Binomial Model with Batch Markovian Arrival Process

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.


Introduction
e 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. erefore, 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.
is study has certain guiding significance in insurance company and shareholders.
is 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 finitetime 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. e 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.

Model
Let (Ω, F, P) be a probability space with filtration F t containing all objects defined in this paper. Assume that F t satisfies the usual conditions, i.e., F t is right-continuous and P-complete. At first, we will introduce the compound binomial model and the batch Markovian arrival process.

Compound Binomial Model.
In the compound binomial model, C(n), n � 0, 1, 2, . . . { } denotes the surplus process of an insurer and is given by where the initial surplus u is a nonnegative integer, S(n) is the aggregate claim up to time n, which is described by and S(0) � 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 ξ n � 1 the event where a claim occurs in the time period (n − 1, n], and we denote by ξ n � 0 the event where no claim occurs in the time period (n − 1, n]. e occurrences of claims in different time periods are independent events. X � X n , t � 1, 2, . . . denotes the claim amount that probably occurs at time t, and X 1 , X 2 , X 3 , . . . are mutually independent, identically distributed (i.i.d.), positive integer-valued random variables, which have a common discrete distribution P( And the claim amounts X � X n , n � 1, 2, . . . are independent of ξ � ξ n , n � 1, 2, . . . .

Batch Markovian Arrival Process
which satisfied the following conditions: is a stochastic matrix, and q ij � ∞ k�0 d k ij en, (D 0 , D k (k ∈ N + )) is called the numerical characteristic of discrete-time batch Markovian arrival process.

Proposition 1.
Assume that (D 0 , D k (k ∈ N + )) be the numerical characteristic of discrete-time batch Markovian arrival process. en, (1) P � ∞ k�0 D k is a m × m conservative matrix, and each state is sojourned. (2) Let E * � (n, j), n ∈ N, e j ∈ E and en, 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 process (D 0 , D k (k ∈ N + )), let X * � X * (t), t ∈ N { } be a stochastic process with transition probability matrix P * and X * (t) � (N(t), J(t)) be a twodimensional discrete-time batch Markovian process. Let α � (α 1 , α 2 , . . . , α m ) T be the initial probability distribution vector of X * , which satisfied m j�1 α j � 1. We call X * as a discrete-time batch Markovian arrival process (DTBMAP); for short, we can denote it as DTBMAP e 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. e 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.

Modified Model.
e model we considered in this paper can be described by (4) where N(n) is the counting process of DTBMAP (α T , D 0 , D k (k ∈ N + )) with state space E * � (n, j), n ∈ N, j ∈ E � e 1 , e 2 , e 3 , . . . , e m }. c(k) and X(k), representing the size of the kth premium and claim, respectively, are both dependent on the state of the phase process J(n) at is, given 2 Mathematical Problems in Engineering positive and integer-valued stochastic series with the mean μ i and the common distribution We should note the following: (1) d 0 ii gives the probability of no state changes without claim arrivals; (2) d 0 ij (i ≠ j) gives the probability of state i changes to state j without claim arrivals; (3) d k ii (k ∈ N + ) gives the probability of no state changes with k claims arrival; (4) d k ij (i ≠ 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.
at is, the expected claim size over a single period is strictly inferior to the premium size, i.e., 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.
, then it is the Markov-modulated compound binomial model, a degenerate case of the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )).

Introduction. We define the time of ruin as
If ruin never occurs, τ � ∞. Also, let us define the conditional finite-time ruin probability as and conditional finite-time non-ruin probability as Denote Obviously, we can see that the unconditional finite-time ruin and nonruin probability, ψ(u, n) and φ(u, n), can be derived from the conditional ones with following formulas, respectively: For convenience, we also define the infinite-time ones by simply letting n ⟶ ∞ in our previous conditional or unconditional finite-time ruin or nonruin probabilities.
us, if we obtain the conditional finite-time ruin probability, all of ruin probabilities of this model are solved.

Main
Result. For convenience, in the next article, we denote where

Theorem 1.
In the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )), the conditional finite-time nonruin probabilities satisfy the following recursive formula: where P � diag(p 1 , p 2 , . . . , p m ) and I � diag(1, 1, . . . , 1). en, where g * (k) (y) represents the kth convolution of g(y). And Proof. We can separate some possible cases by conditioning on the r.v.'s J(1), N(1), c (1), and X (1). ere are possible cases as follows: (1) No state changes and no claim arrivals (2) State i changes to state j(i ≠ j) and no claim arrivals (3) No state changes and k(k ≥ 1) claims arrival (4) State i changes to state j(i ≠ j) and k(k ≥ 1) claims arrival Mathematical Problems in Engineering en, the following formula can be easily derived by using the formula of full probability and the Markov property. For all e i ∈ E, we have us, (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 V i (n) � N(n) l�1 I J(l)�e i { } be the elapsed time by the phase process J(n) in state e i over the first k periods where I A � 1 if A is true and I A � 0 if A is false. We also denote W i (n) the amount of decrease of the surplus process over the first n periods when the phase process J(n) is in state e i , i.e., W i (n) � Denote W(n) as the amount of decrease of the surplus process over the first n periods. Furthermore, X i (n) and c i (n) are denoted as the nth premium amount and claim size when the phase process J(n) is in state e i , respectively.

Proposition 2.
e infinite-time ruin probability tends to 0 as initial surplus u tends to ∞.
Proof. Obviously, we can see W(n) � e i ∈E W i (n).
Taking the limit as n ⟶ ∞ of W(n)/n yields Since J(n) is irreducible and ergodic, it follows that where ξ � (ξ 1 , ξ 2 , . . . , ξ m ) is the stationary distribution of J(n). We can easily see that for given J(n) � e i , X(n) and c(n) are both i.i.d. and W i (n) is distributed as where L i (n) � n l�1 (X i (l) − c i (l)). erefore, L i (n) is a random walk for e i ∈ E and from the strong law of large numbers, we can find ∀e i ∈ E: By combining (16) to (19), we can obtain that Equation (20) and the safety loading condition imply that lim n⟶∞ W(n) � −∞ and thus ensure that max n∈N W(n) is finite. Consequently, Theorem 2. In the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )) , the numerical algorithm proposed to obtain the conditional finite-time nonruin probabilities is as follows: Fix φ(u | e i ) � 1 for u � n, n + 1, n + 2, . . . and e i ∈ E. Find φ(u | e i ) for u � 0, 1, 2, . . . , n and e i ∈ E by solving the following system of m × n equations with m × n unknown parameters: Proof. First, by conditioning, respectively, on the random variables J(1), N(1), c (1), and X(1), four cases probably occurred. And from the stationarity of the surplus process, we can find for e i ∈ E and u ∈ N. Given that φ(u | e i ) � 1 for u � n, n + 1, n + 2, . . . and e i ∈ 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 e i ∈ E.

Compound Binomial Model with BMAP and Dividend
In this section, we will embed a threshold dividend strategy in the compound binomial model with BMAP (α T , D 0 , D k (k ∈ N + )). First, we will introduce the specific description of this model.

Description.
Based on the compound binomial model with BMAP (α T , D 0 , D k (k ∈ N + )), we can define the compound binomial model with BMAP (α T , D 0 , D k (k ∈ N + )) and threshold dividend strategy. e surplus process of an insurer is given by where N(n), c(n), and X(n) 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 If ruin never occurs, τ b � ∞.

Gerber-Shiu Function.
e 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.
e Gerber-Shiu function is defined by where v ∈ (0, 1] is the discount factor, w(·, ·): N + × N + ⟶ N is a binary function, and U(τ b − 1) represents the surplus before ruin. |U(τ b )| represents the deficit at ruin, and I(A) is the indicator function of an event A taking value 1 whenever the event A occurs and 0 when it does not. e Gerber-Shiu function plays an important role in risk theory. When w(·, ·) ≡ 1 and v � 1, the Gerber-Shiu function changes to the ruin probability. When w(x, y) � x, it changes to the discounted surplus before ruin time. When w(x, 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 and denote m(u) � m u | e 1 m u | e 2 m u | e 3 · · · m u | e m T 1×m .
We can easily see that us, we can solve the Gerber-Shiu function m(u) by solving m(u | e i ). Next, we will derive the solution of m(u | e i ).

Theorem 3.
In the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )) and dividend threshold b, the conditional Gerber-Shiu functions satisfy the following recursive formula.

Theorem 4.
In the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )) and dividend threshold b, we have , for e i ∈ E and u < b, satisfied the following recursive formula: where Theorem 5. In the compound binomial model with DTBMAP (α T , D 0 , D k (k ∈ N + )) and dividend threshold b, we have (2) B(u | e i ), e i ∈ E, and u ≤ b satisfied the following expression: where and A sl (s � 0, 1, . . . , b, l � 0, 1, . . . , b) is a series of m × m matrixes: Mathematical Problems in Engineering 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.
Combining all results, we can analyze the following: (1) B(u) is gradually increased with the increase of initial surplus u (2) For a given e i , B(u | e i ) is gradually increased with the increase of initial surplus u (3) When the initial surplus u is more and more big, the difference between B(u | e 1 ) and B(u | e 2 ) is more and more small (4) For e 1 is a "good" state and e 2 is a "bad" state, B(u | e 1 ) is larger than B(u | e 2 ) when u is equal (5) B(u) is larger than B(u | e 2 ), but smaller than B(u | e 1 ) when initial surplus u is equal (6) e safety factor is larger and the expected dividend value of all dividends up to the ruin time is larger when initial surplus u is equal

Mathematical Problems in Engineering
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Conflicts of Interest
e authors declare that they have no conflicts of interest.