Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin ()
1. Introduction
Recently ruin theory under regime-switching model is becoming a popular topic. This model is proposed in Reinhard [1] and Asmussen [2]. Asmussen [2] calls it a Markov-modulated risk model in which both the frequency of the claim arrivals and the distribution of the claim amounts are influenced by an external environment process. This model is a generalization of the classical compound Poisson risk model and the primary motivation for this generalization is the enhanced flexibility that it permits for the modeling of the claim arrival process and the claim severity distribution assumed in the classical risk process. The model builds a Markov chain whose states represent different states of an economy into the insurance risk model. The regime-switching of the states of the economy can be attributed to the structural changes in the (macro-) economic conditions, the changes in political regimes, the impact of (macro-) economic news and business cycles, etc. There are many papers published on ruin probabilities and the related problems under the Markov-modulated (or Markov regime-switching) risk model. Ng and Yang [3] give closed form solutions for the joint distribution of the surplus before and after ruin when the initial surplus is zero or when the claim amount distributions are phase-type distributed. Li and Lu [4] study the moments of the present value of the dividend payments and the distribution of the total dividends prior to ruin for the Markov-modulated risk model modified by the introduction of a barrier dividend. Lu and Li [5] and Liu et al. [6] consider a regime-switching risk model with a threshold dividend strategy. Zhu and Yang [7] study a more general Markovian regimeswitching risk model in which the premium, the claim intensity, the claim amount, the dividend payment rate and the dividend threshold level are influenced by an external Markovian environment process. Wei et al. [8] consider the Markov-modulated insurance risk model with tax.
Moreover, in recent years, semi-Markovian risk model has attracted attention in the literature. Albrecher and Boxma [9] study the expected discounted penalty function in a semi-Markovian dependent risk model in which at each instant of a claim, the underlying Markov chain jumps to a new state and the distribution of claim depends on this state. Liu et al. [10,11] consider the expected discounted penalty function under the constant dividend barrier and dividends payments under the threshold strategy in a Markov-dependent risk model, respectively. They consider the structure of a semiMarkovian dependence type as follows. Let 
denote the time between the arrival of the 
th and the ith claims and 
a.s., then
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 [9] and Liu et al. [10,11], in this paper we propose to generalize the semiMarkovian risk model to the absolute ruin risk model. In the new risk model, we assume that the insurer could borrow an amount of money equal to the deficit at a debit interest force 
when the surplus is negative. Meanwhile, the insurer will repay the debts continuously from his/her premium income. When the negative surplus attains the level 
or is below
, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. Moreover, when the surplus exceeds the constant barrier
, dividends are paid continuously so the surplus stays at the level b until a new claim occurs. Some recent references about absolute ruin risk model include Zhou and Zhang [12], Cai [13], Gerber and Yang [14], Yuen et al. [15], Yuan and Hu [16],Wang and Yin [17], Ming et al. [18], Wang et al. [19], Zhang et al. [20], Yu and Huang [21] and references therein.
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 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
. 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.
2. Moment-Generating Function of Du,b
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
(2.1)
and, for
,
(2.2)
with boundary conditions, for
,
(2.3)
(2.4)
where,
.
Proof. Fix
, and
. Considering a small time interval
, such that
. In view of the strong Markov property of the surplus process 
, we have
(2.5)
Conditioning on the event occurring in the interval
, we obtain
(2.6)
Taylor’s expansion gives
(2.7)
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
(2.8)
By Taylor’s expansion
(2.9)
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
(2.10)
Using Taylor’s expansion, we have,
(2.11)
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
,
(2.12)
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
(2.13)
Similarly, we obtain
(2.14)
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.
3. Higher Moment of the Dividend Payments
We now derive a system of integro-differential equations satisfied by
. By the definitions of 
and
, we obtain, for
,
(3.1)
(3.2)
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:
(3.3)
for
, and for
,
(3.4)
Substituting (3.1) into (2.3), similarly, we obtain
(3.5)
Thus, 
is an obvious result since 
.
Substituting (3.1) and (3.2) into (2.4) and (2.12), we obtain, for 
(3.6)
(3.7)
Letting 
in (3.3) and 
in (3.4) and using (3.7), we obtain, for
.
(3.8)
where, “
” denoting decreasing approach.
4. Explicit Expressions for Exponential Claims to 
for a Two-State Model
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 
(4.1)
(4.2)
Applying the operator 
and 
on
(4.1) and (4.2), respectively, and rearranging them, we obtain, for 
(4.3)
(4.4)
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
(4.5)
(4.6)
The corresponding homogeneous differential Equations of (4.5) and (4.6) are
(4.7)
(4.8)
The general solutions of Equations (4.7) and (4.8) are
(4.9)
(4.10)
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
(4.11)
(4.12)
Solving the above two equations, we obtain
then we have, for 
So the general solutions of Equations (4.5) and (4.6) are, for
,
(4.13)
(4.14)
5. The Discounted Penalty Function
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
, 
,
(5.1)
and, for
,
(5.2)
with boundary conditions
(5.3)
(5.4)
(5.5)
where
Proof. For 
and
. Similar to argument as in Section 2, we condition on the events that can occur in the small time interval
.
(5.6)
Since
we then get
(5.7)
Equation (5.7) can be rewritten as
(5.8)
Letting 
in (5.8), we obtain (5.1).
For 
and
, we have
(5.9)
By Taylor’s expansion
(5.10)
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
,
(5.11)
and, for
,
(5.12)
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
,
.
6. Acknowledgements
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).