Received 1 February 2016; accepted 6 March 2016; published 9 March 2016
The dividend problem has long been an important issue in finance and actuarial sciences. Due to the importance of the dividend problem, the study of the risk model with dividend strategy has received more and more at- tention. Most of the strategies considered are of two kinds: one is the barrier strategy; another is the threshold strategy. For more recent studies about dividend problems, see  - . In these papers, they extend the threshold dividend strategy to the multiple case, and make in-depth study of the model by the probabilistic and differential equation approaches. Under such a dividend strategy, many authors have extensively studied the Gerber-Shiu function for both the classical and the renewal risk model.
In classical insurance theory, we usually say that ruin occurs when the surplus is below zero. But in reality, the insurer could borrow an amount of money equal to the deficits at a debit interest rate to continue his business when the surplus falls below zero. Meanwhile, the insurer will repay the debts from his premium income. If debts are reasonable, the negative surplus may return to a positive level. However, when the negative surplus is below some certain level, the insurer is no longer allowed to run his business and absolute ruin occurs at this situation.
Absolute ruin probability has been frequently considered in recent research works. Dassios and Embrechts considered the absolute ruin, and by a martingale approach they derived the explicit expression for the probability of absolute ruin in the case of exponential individual claim in  . Cai defined Gerber-Shiu function at absolute ruin and derived a system of the integro-differential equations satisfied by the Gerber-Shiu function in  .
Most of the literature in finance is based on the assumption that the inter-arrival time between two successive claims and the claim amounts are independent. However, the independence assumption can be inappropriate and unrealistic in practical contexts. So in recent years, the risk model with dependence structure between inter- arrival times and claim sizes has got more and more attention. For example, see  -  . Yu and Huang  studied the dividend payments prior to absolute ruin in a Markov-dependent risk process. Zhou et al.  proposed a Markov-dependent risk model with multi-layer dividend strategy.
To the best of our knowledge, Markov-dependent risk model with multi-layer dividend strategy and investment interest under absolute ruin has not been investigated. This motivates us to investigate such a risk model in this work. Generally, the authors only extensively consider Gerber-Shiu function in risk models with multi-layer dividend strategy. In this paper, we study not only Gerber-Shiu function, but also the moment-generating func- tion and the nth moment of the discounted dividend payments prior to absolute ruin.
The rest of the paper is organized as follows. In Section 2, the model is described and basic concepts are introduced. In Sections 3, we get integro-differential equations for the moment-generating function of the dis- counted dividend payments prior to absolute ruin and boundary conditions. In Section 4, the integro-differential equations satisfied by higher moment of the discounted dividend payments prior to absolute ruin and boundary conditions are derived. In Section 5, we obtain the systems of integro-differential equations for the Gerber-Shiu function and its matrix form. Section 6 concludes the paper.
2. The Model
In this section, we investigate the Markov-dependent risk model with multi-layer dividend strategy and investment interest under absolute ruin, in which the claim occurrence and the claim amount are regulated by an external
discrete time Markov chain. First, let be an irreducible discrete time Markov chain with finite state space and transition matrix. Similar to Albrecher and Boxma  , we define
the structure of a semi-Markov dependence type insurance problem as follows. Let denote the time be- tween the arrival of the and the ith claims and a.s., then
where is the amount of the nth claim. Thus at each instant of a claim, the Markov chain jumps to a state j and the distribution of the claim depends on the new state j, and has a positive mean. Then, the next interarrival time is exponentially distributed with parameter. Note that given the states and, the quantities and are independent, but there is an autocorrelation among consecutive claim sizes and among consecutive interclaim times as well as cross-correlation between and.
In our risk model, we assume that the insurer could borrow money with the amount equal to the deficit at a debit interest force when the surplus falls below zero or the company is on deficit. And when the surplus becomes positive, the insurer could earn interest at an investment rate. We also assume that the premium rate is a step function, instead of a constant, dependent on the current surplus level. More precisely,
define N layers. When the surplus is in layer k, i.e., premium is collected with rate until a claim causes the surplus to a lower layer or the surplus
grows to the next higher layer. Meanwhile, the premium will be collected with rate when the surplus becomes negative. In reality, we assume. When the surplus is in layer k, dividends are paid continuously at a constant rate. Furthermore, we assume the net profit condition is fulfilled in each layer, that is
where is the stationary distribution of process. We denote the surplus by. Then the dynamics of can be expressed as
where is the number of claims up to time t.
Note that the surplus is no longer able to become positive when the negative surplus attains the level or is below, because the insurer cannot repay all his debts for his business. We denote the absolute ruin
time of the model (2.3) by, which is defined by, and if, for all. Given the initial surplus u, and the force of interest, the present value of all dividends until time of absolute ruin is defined by
where is the cumulative amount of dividends paid out up to time t. In the sequel we will be interested in the moment-generating function
and the th moment function
with, and the expected discounted penalty function, for
where, is the surplus prior to absolute ruin and is the deficit at absolute ruin. The penalty function is an arbitrary nonnegative measurable function defined on.
For fix, throughout this paper we assume that, and are sufficiently smooth functions in u and y in their respective domains.
3. Integro-Differential Equations for
In this section, we give the integro-differential equations for the moment-generating function. Clearly, the moment-generating function behaves differently. For, we define
For notational convenience, let,.
Theorem 3.1. For, ,
and, for, ,
Proof. Fix, and, let be the solution to the equation of, namely, which is the time when the surplus returns to the level zero if no claim occurs to time. Then is the surplus at time if no claim occurs prior to time. We consider a small time interval, where. In view of the strong Markov property of the surplus process, we have
Thus conditioning on the time and the amount of the first claim, we obtain,
By Taylor's expansion, we have
Substituting (3.5) into (3.4), and then dividing both sides of (3.4) by t and letting, we get (3.1).
Similarly, when, we still consider a small time interval, where is sufficiently small so that the surplus process will not reach. Conditioning on the event occurring in the interval, we obtain
By Taylor’s expansion, we have
Substituting (3.7) into (3.6), and then dividing both sides of (3.6) by t and letting, we get (3.2). So the proof is completed.
Theorem 3.2. For, and satisfy
1) If, the absolute ruin is immediate and no dividend is paid, so (3.8) holds.
2) For, letting be the time that the surplus reach 0 for the first time from and using the Markov property of the surplus process, we have
Similarly, we have
where is the time of the first claim.
When, we notice that and both go into zero. Letting in (3.13) and (3.14) and in view of, we derive (3.9).
3) For Eq. (3.10), the method is similar to Equation (3.9), so we omit it here.
4) If, then, so (3.11) holds.
5) For, letting in (3.2), and in (3.2) when, , we can get (3.12).
The proof of Theorem 3.2 is complete.
4. Integro-Differential Equations for
In this section, we get the integro-differential equations for,. First, for, define
Using the representation
we have the following integro-differential equations.
Theorem 4.1. For, ,
and, for, ,
Proof. Substituting (4.1) into (3.1), and then equating the coefficients of, we can get (4.3). Similarly, sub- stituting (4.1) and (4.2) into (3.2), and then equating the coefficients of, we obtain (4.4).
Theorem 4.2. For, and satisfy
Proof. This method is similar to Theorem 3.2.
5. The Gerber-Shiu Function
In this section, systems of integro-differential equations for the Gerber-Shiu function are presented. For, define
Theorem 5.1. For, ,
and, for, ,
with boundary conditions
Proof. Fix, and. Similar to argument as in Section 3, conditioning on the events that can occur in the small time interval, we obtain,
By Taylor’s expansion, we have
Substituting (5.7) into (5.6), and then dividing both sides of (5.6) by t and letting, we get (5.1).
Similarly,when, we still consider a small time interval. We obtain
By Taylor’s expansion, we have
Substituting (5.9) into (5.8), and then dividing both sides of (5.8) by t and letting, we get (5.2). For the boundary conditions (5.3)-(5.5), the method is similar to Theorem 3.2. So the proof is completed.
Integro-differential Equations (5.1) and (5.2) can be rewritten in matrix form.
where T denoting transpose. We have the following theorem.
Theorem 5.2. and satisfy the following integro-differential equations
with boundary conditions
are all matrices, and defined by
are all d-dimensional vector, in which is an column vector.
In this paper, we investigate the Markov-dependent risk model with multi-layer dividend strategy and investment interest under absolute ruin. This complex model is more realistic. We derive systems of integro-differential equations satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the Gerber-Shiu function. Generally, many authors only extensively consider Gerber- Shiu function in risk models with multi-layer dividend strategy. However, due to the importance of the dividend problem, the problems considered by this paper are more important and interesting.
In addition that, we only obtain systems of integro-differential equations. As far as we know, it is not easy to derive the explicit expressions for the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the Gerber-Shiu function. But, maybe we find some numerical method which can solve these equations. We leave it for the further research topic.
We would like to thank the referees for their constructive comments and suggestions which have improved the paper. This work was supported by the the Natural Sciences Foundation of China (grants 11301133 and 11471218).