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In this paper, we consider 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. 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. Finally, the matrix form of systems of integro-differential equations satisfied by the Gerber-Shiu function is presented.

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 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 [

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 [

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.

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

the structure of a semi-Markov dependence type insurance problem as follows. Let

where

In our risk model, we assume that the insurer could borrow money with the amount equal to the deficit at a debit interest force

define N layers

grows to the next higher layer. Meanwhile, the premium will be collected with rate

where

where

Note that the surplus is no longer able to become positive when the negative surplus attains the level

time of the model (2.3) by

where

and the

with

where,

For fix

In this section, we give the integro-differential equations for the moment-generating function

For notational convenience, let

Theorem 3.1. For

and, for

Proof. Fix

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

Similarly, when

By Taylor’s expansion, we have

Substituting (3.7) into (3.6), and then dividing both sides of (3.6) by t and letting

Theorem 3.2. For

Proof.

1) If

2) For

Similarly, we have

where

When

3) For Eq. (3.10), the method is similar to Equation (3.9), so we omit it here.

4) If

5) For

The proof of Theorem 3.2 is complete.

In this section, we get the integro-differential equations for

with

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

Theorem 4.2. For

Proof. This method is similar to Theorem 3.2.

In this section, systems of integro-differential equations for the Gerber-Shiu function are presented. For

Theorem 5.1. For

and, for

with boundary conditions

where

Proof. Fix

By Taylor’s expansion, we have

Substituting (5.7) into (5.6), and then dividing both sides of (5.6) by t and letting

Similarly,when

By Taylor’s expansion, we have

Substituting (5.9) into (5.8), and then dividing both sides of (5.8) by t and letting

Integro-differential Equations (5.1) and (5.2) can be rewritten in matrix form.

Let

and

where T denoting transpose. We have the following theorem.

Theorem 5.2.

with boundary conditions

where

are all

are all d-dimensional vector, in which

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).

BanglingLi,ShixiaMa, (2016) Markov-Dependent Risk Model with Multi-Layer Dividend Strategy and Investment Interest under Absolute Ruin. Journal of Mathematical Finance,06,260-268. doi: 10.4236/jmf.2016.62022