Recently, there was a more and more interest in the issue of risk models with dividend strategies. Obviously, dividend strategies can reflect the surplus cash flows more realistically in a insurance portfolio. The theories developed are very valuable in the devising and managing of products with dividends. In this paper, we discuss the dividend-penalty identity problem in a Markov risk model, which governed by a Markov arrival claim process and allows for claim sizes to be correlated with inter-claim times. The purpose of this paper is to show how the dividend penalty identity can be obtained by general solution of the integro-differential equation.

Suppose $\{Z_{n}; n\geq 0 \}$ is an irreducible discrete time Markov chain with state space $E=\{1,\cdot\cdot\cdot,m \}$ and transition matrix

$\mathbf{P} =(p_{i,j})_{i,j=1}^{m}.$

Let $u$ be the initial capital, $c$ the premium rate, $X{_j}$ the size of the $j$th claim and $N(t)$ is the number of claims up to time $t$. $F(x)$ is the distribution of the claim $X{_j}$. The surplus process $\{R(t);t\geq 0 \}$ is defined as

$\begin{array}{ll} R(t)=u+ct-\sum\limits_{j=1}^{N(t)}X_{j}. \end{array}$

(1)

Let $W_{i}$ denote the duration between the arrivals of the $(i-1)$th and the $i$th claim and $W_0=X_0=0$ a.s., then

$\begin{array}{ll} &P(W_{n+1}\leq x,X_{n+1}\leq y,Z_{n+1}=j|\ Z_{n}=i,(W_r ,X_r ,Z_r),0\leq r\leq n)\\= &P(W_1\leq x,X_1\leq y,Z_1=j|\ Z_0=i)=(1-e^{-\lambda_ix})p_{ij}F_{j}(y). \end{array}$

(2)

The model is enriched by the payment of dividends to the share-holders of the company, and the surplus is modified accordingly. When the surplus exceeds a constant barrier $b\geq u$, dividends are paid continuously so that the surplus stays at the level $b$ until next claim occurs. Let $\{R_{b}(t);t\geq 0\}$ be the surplus process with initial surplus $u$ under the constant barrier above.

Define $T_b=$inf$\{t\geq 0:R_b(t)<0\}$ to be the time of ruin. Let $\delta >0$ be the force interest and $w(x,y)$, for $x,y\geq 0$, be non-negative valued of penalty function. Let $\mathbf{p} _{i}={\bf p}(\cdot | Z_{0}=i) $, define

$\begin{array}{ll} \phi_{i}(u;b)=\mathbf{E} _{i}[e^{-\delta T_{b}}w(R(T_{b}-),|R(T_{b})|)I(T_{b}<\infty)|R(0)=u] \end{array}$

(3)

to be the discounted penalty function (Gerber-Shiu function) at $T_b$ given that the initial surplus is $u$, given that the initial environment is $i$. Denote that when $b=\infty$,

$\phi_{i}(u;b)=\phi_{i}(u),$

that is the Gerber-Shiu function without dividend barrier.

The Gerber-Shiu discounted penalty function under the constant dividend barrier is associated with the discounted penalty function for the process without dividend strategy. Define $T=\inf\{t\geq 0: R(t)<0\}$ to be the time of ruin of the surplus process (1), and for $\delta\geq 0$,

$\phi_{i}(u)=\mathbf{E} _{i}[e^{-\delta T}\omega(R(T-),|R(T)|)I(T<\infty)|R(0)=u],\ \ u\geq 0,\ i\in E $

to be the Gerber -Shiu discounted penalty function, given the initial surplus $u$ and the initial state $i$. We denote by $c$ the constant premium rate for the surplus process (1) without dividend strategy. Function $\phi_{i}(u)$ investigated in Albrecher and Boxma (2005), which satisfies the following integro-differential equation, for $i\in E$,

$\begin{eqnarray*} &&c\phi_{i}^{'}(u)\\ &=&(\lambda_{i}+\delta)\phi_{i}(u)-\lambda_{i}\sum\limits_{j=1}^{m}p_{ij}\left(\displaystyle\int \limits_{0}^{u}\phi_{j}(u-y)dF_{j}(y) +\displaystyle\int\limits_u^{\infty}\omega(u,y-u)dF_{j}(y)\right). \end{eqnarray*}$

Let

$\vec{\Phi}(u)=(\phi_1(u),\cdot\cdot\cdot,\phi_m(u))^{\top },$

then an integro-differential equation in matrix form for $\vec{\Phi}(u)$ is given by

$ \vec{\Phi}^{'}(u)={\bf H}_{c}\vec{\Phi}(u)+\displaystyle\int\limits_0^{u}{\bf G}_{c}(x)\vec{\Phi}(u-x)dx+\vec{h}(u), 0<u<\infty, $

(4)

where

$\mathbf{H} _{c}={\rm diag}((\lambda_1+\delta)/c,\cdot\cdot\cdot,(\lambda_m+\delta/)c)$

and

$ \mathbf{G} _{c}(x)=-\left( \begin{array}{ccc} {\lambda_1\over c}&\ &\ \\ \ &\ddots&\ \\ \ &\ &{\lambda_m\over c} \end{array} \right) \mathbf{P} \left( \begin{array}{ccc} f_1(x)&\ &\ \\ \ &\ddots&\ \\ \ &\ &f_m(x) \end{array} \right) $

are $m\times m$ matrices, and $\vec{h}(u)$ is an m-dimensional vector which is given by

$ \vec{h}(u)=\displaystyle\int\limits_u^{\infty}{\bf G}_{c}(x)\omega(u,x-u)\vec{\mathbf{1} }dx, $

(5)

where $\vec{\mathbf{1} }=(1,\cdot\cdot\cdot,1)^{\top} $ is an $m$-dimensional column vector. The corresponding homogenous integro-differential equation of $(4)$ is

$ \vec{\Phi}^{'}(u)={\bf H}_{c}\vec{\Phi}(u)+\displaystyle\int\limits_0^{u}{\bf G}_{c}(x)\vec{\Phi}(u-x)dx. $

(6)

By Theorem 2.3.1 in Burton [2], we give the analytical expression for $\vec{\Phi}(u)$ in the following lemma.

Lemma 1 Let $\mathbf{v} (u)=(v_{i,j}(u))_{i,j=1}^{m}$ be the $m\times m$ matrix whose columns are particular solutions to (6) with $\mathbf{v} (0)=\mathbf{I} $, where $\mathbf{I}$ is the $m\times m$ identity matrix. The solutions to equation (4) is

$ \vec{\Phi}(u)=\mathbf{v} (u)\vec{\Phi}(0)+\int\limits_0^{u}{\bf v}(u-x)\vec{h}(x)dx, 0\leq u <\infty, $

where $\mathbf{v} (u)$, $\vec {\Phi}(0)$ was given by (3.6) and (4.4) in [7].

As for $\phi_{i}(u;b)$, by similar approach as in Liu et al. (2010), we can get the Gerber-Shiu function (3) under the constant dividend strategy, satisfying the following integro-differential equation

$\begin{eqnarray*} &&c\phi_{i}^{'}(u;b)\\ &=&(\lambda_i+\delta)\phi_i(u;b)-\lambda_i\sum\limits_{j=1}^{m}p_{ij} (\int\limits_{0}^{u}\phi_{j}(u-x;b)dF_j(x)+\int\limits_{u}^{\infty}w(u,x-u)dF_j(x)) \end{eqnarray*}$

with boundary condition $\phi_i^{'}(b;b)=0$.

Let

$\vec{\mathbf{\Phi} }^{'}(u;b)=(\Phi_1^{'}(u;b),\cdot\cdot\cdot,\phi_m^{'}(u;b))^{\top },$

$dF_i(x)=f_i(x)dx$, then an integro-differential equation in matrix form for $\vec{\Phi}(u;b)$ is given by

$ \vec{\mathbf{\Phi} }^{'}(u;b)={\bf H}_{c}\vec{\Phi}(u;b)+\int\limits_0^{u}\mathbf{G} _{c}(x)\vec{{\bf \Phi}}(u-x;b)dx+\vec{h}(u), 0\leq u\leq b $

(7)

with boundary condition

$ \vec{\mathbf{\Phi} }^{'}(b;b)=\vec{\mathbf{0}} , $

and $\vec{\mathbf{1} }=(1,\cdot\cdot\cdot,1)^{\top} $, $\vec{{\bf 0}}=(0,\cdot\cdot\cdot,0)^{\top} $ are $m\times 1$ vectors.

Again we apply Theorem 2.3.1 in Burton [2] to obtain the analytical expression for $\vec{\Phi}(u;b)$ as follows

$ \vec{\Phi}(u;b)=\mathbf{v} (u)\vec{\Phi}(0;b)+\int\limits_0^{u}{\bf v}(u-x)\vec{h}(x)dx, 0\leq u <b. $

Now restricting $\vec{\Phi}(u;b)$ in (4) to $0\leq u<b$, we have

$ \vec{\Phi}(u;b)-\mathbf{v} (u)\vec{\Phi}(0;b)=\vec{\Phi}(u)-{\bf v}(u)\vec{\Phi}(0), $

then $\vec{\Phi}(u;b)$ in (7) can be rewritten as

$ \vec{\Phi}(u;b)=\mathbf{v} (u)\ [\vec{\Phi}(0;b)-\vec{\Phi}(0)]+\vec{\Phi}(u)={\bf v}(u)\vec{k}(b)+\vec{\Phi}(u), 0\leq u<b, $

(8)

where $\vec{k}(b)=\vec{\Phi}(0;b)-\vec{\Phi}(0)$.

This formula (8) is the so-called dividend-penalty identity for a general class of the Markov risk model. Note that in (8), the expected discounted penalty function $\vec{\Phi}(u;b)$ for the modified surplus processes with dividend strategy can be expressed as the summation of the expected discounted penalty function $\vec{\Phi}(u)$ for the corresponding process without dividend strategy applied and a vector which is the product of ${\bf v}(u)$, a matrix function of $u$, and $\vec{k}(b)$, a vector function of $b.$

When $m=1$, the model reduces to the classical compound Poisson risk model, the expected discounted penalty function $\vec{\Phi}(u;b)$ in (8) simplifies to

$\phi(u;b)=\phi(u)+v(u)k(b), 0\leq u<b,$

which is equation (5.1) in Lin et al. (2003). Here $\phi(u)$ ($m_{\infty}(u)$ in their paper) is the expected discounted penalty function under the classical risk process with premium rate c, and the function $v(u)$ satisfies reduced integro-differential equation (7) and the constant $k(b)$ is determined in their paper. We extend the results in [1] and show the this identity can be obtained by the general solution of the integro-differential equation.