^{*}

^{*}

In this paper, we study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the discounted penalty function, given the initial environment state, are derived. In the two-state risk model, explicit solutions to the integro-differential equations satisfied by the nth moment of the discounted dividend payments prior to absolute ruin are obtained when the claim size distribution is exponentially distributed. Finally, the matrix form of systems of integro-differential equations satisfied by the discounted penalty function are presented.

Recently ruin theory under regime-switching model is becoming a popular topic. This model is proposed in Reinhard [

Moreover, in recent years, semi-Markovian risk model has attracted attention in the literature. Albrecher and Boxma [

where is an irreducible discrete time Markov chain with state space and transition matrix, 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 crosscorrelation between and.

Inspired by Albrecher and Boxma [

The surplus process under the Markovdependent risk model is given by

where is the initial surplus, c the premium rate, the debit interest, the number of claims up to time t, and means the indicator function of an event B. Furthermore, we assume the net profit condition holds, that is

where is the stationary distribution of process.

Let be the cumulative amount of dividends paid out up to time t and the force of interest, then

is the present value of all dividends until time of ruin, where denoted by is the time of absolute ruin.

In the sequel we will be interested in the momentgenerating function

,

and theth 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. Throughout this paper we assume that, and are sufficiently smooth functions in u and y, respectively.

Then, fix, the expected present value of the total dividend payments until ruin in the stationary case is given by

The rest of the paper is organized as follows. In Sections 2, we get integro-differential equations for the moment-generating function and boundary conditions in a Markov-dependent risk model. In section 3, the integro-differential equations satisfied by higher moment of the dividend payments and boundary conditions are derived. Examples for a two-state risk model are illustrated in section 4 when the claim size distribution is exponentially distributed. In the last section, we obtain the systems of integro-differential equations for the discounted penalty function and its matrix form.

In this section, we discuss the integro-differential equations satisfied by the moment-generating function at absolute ruin. We point out that has different paths for and. For, we define

Theorem 2.1 For, , we have

and, for,

with boundary conditions, for,

where,.

Proof. Fix, and. Considering a small time interval, such that. In view of the strong Markov property of the surplus process , we have

Conditioning on the event occurring in the interval, we obtain

Taylor’s expansion gives

Substituting (2.7) into (2.6), dividing both sides by t, and letting, we obtain (2.1).

Similarly, when, we still consider a small time interval, with being sufficiently small so that the surplus will not reach 0 in the time interval. Let be the solution to

Then is the surplus at time if no claim occurs prior to time. We assume. So conditioning on the time and the amount of the first claim, we have

By Taylor’s expansion

Substituting (2.9) into (2.8), dividing both sides by t, and letting, we obtain (2.2).

When the initial surplus is b, we obtain

Using Taylor’s expansion, we have,

Letting in (2.1) and comparing it with (2.11), we obtain (2.3). Where “” denoting increasing approach.

When, absolute ruin is immediate. Thus, no dividend is paid. So we obtain (2.4). Theorem 2.1 is proved.

Theorem 2.2 For,

Proof. 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 obtain

Similarly, we obtain

where is the time of the first claim.

When, we notice that and both go into zero. Letting in (2.13) and (2.14) and in view of

we obtain (2.12). Theorem 2.2 is proved.

We now derive a system of integro-differential equations satisfied by. By the definitions of and, we obtain, for,

where, is defined by

Substituting (3.1) and (3.2) into (2.1) and (2.2), respectively, and comparing the coefficients of yield the following integro-differential equations:

for, and for,

Substituting (3.1) into (2.3), similarly, we obtain

Thus, is an obvious result since .

Substituting (3.1) and (3.2) into (2.4) and (2.12), we obtain, for

Letting in (3.3) and in (3.4) and using (3.7), we obtain, for.

where, “” denoting decreasing approach.

In this section, we consider a two-state Markov-dependent risk model. Then is a two-state Markov chain, which reflects the random environmental effects due to “normal” vs. “abnormal”, or “high season” vs. “low season” conditions. We derive the explicit formulae for when the claim size is exponentially dis tributed,. Set,. In view of Equation (3.3) and the expo nential density function, Equation (3.3) are reduced to, for

Applying the operator and on

(4.1) and (4.2), respectively, and rearranging them, we obtain, for

They are second-order linear non-homogeneous differential equations with constant coefficients. For convenient writing, let

,

,

,

Then Equation (4.3) and Equation (4.4) can be rewriteten as

The corresponding homogeneous differential Equations of (4.5) and (4.6) are

The general solutions of Equations (4.7) and (4.8) are

where, , , are arbitrary constants,

According to the variation of constants method, we assume and are special solutions of Equations (4.5) and (4.6), respectively. Then we have

Solving the above two equations, we obtain

then we have, for

So the general solutions of Equations (4.5) and (4.6) are, for,

In this section, we derive integro-differential equations for the discounted penalty functions. For, define

Note that in the stationary case, we have

Theorem 5.1 For, ,

and, for,

with boundary conditions

where

Proof. For and. Similar to argument as in Section 2, we condition on the events that can occur in the small time interval.

Since

we then get

Equation (5.7) can be rewritten as

Letting in (5.8), we obtain (5.1).

For and, we have

By Taylor’s expansion

Substituting (5.10) into (5.9), dividing both sides by t, and letting, we obtain (5.2). Theorem 5.1 is proved.

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

Let

,.

“T” denoting transpose. Rewritten (5.1) and (5.2) in matrix, then we have Theorem 5.2 The integro-differential equation in ma trix form for and, for,

and, for,

where

are all matrices, and and defined by

are all m-dimensional vector, in which is an column vector. The continuity condition and derivative condition for and is

,

.

The authors are very grateful to the editor and two anonymous referees for their valuable comments and suggestions which led to the present improved version of the manuscript. This research was supported by Humanities and Social Sciences Project of the Ministry Education of China (No. 09YJC910004; No. 10YJC630092) and Shandong Provincial Natural Science Foundation of China (No. ZR2010GL013) and Research Program of Higher Education of Shandong Province (No. J10WF84) and Natural Science Foundation of Shandong Jiaotong University (No. Z201031).