The Dividend Problems for a Discrete Markov Risk Model with Stochastic Return

DENG Ying-chun,YUE Sheng-jie, XIAO He-lu, ZHAO Chang-bao

(Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China,College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China)

The Dividend Problems for a Discrete Markov Risk Model with Stochastic Return

DENG Ying-chun*,YUE Sheng-jie, XIAO He-lu, ZHAO Chang-bao

(Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China,College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China)

A discrete Markov risk model with stochastic return is considered, in which both the claim occurrence and the claim amount are regulated by a discrete time Markov process. A set of linear equations for the expected present value of all dividends paid out until ruin occurs are derived when insurer uses a constant dividend strategy. Finally, explicit formula for the expected total discounted dividends until ruin is given.

Markov risk model; stochastic return; dividend barrier; expected discounted dividends

1 Introduction

We consider a discrete time risk process based on the compound binomial model. The surplus process is described as follows.

LetYkn(n=1,2,…) denote the return on the investment by an insurer in the time period (n-1,n] when the surplus iskat timen-1. And assume that for allk(k=0,1,2,…), {Ykn,n=1,2,…} is a sequence of mutually independent, identically distributed, integer-valued random variables, and independent of {Sn,n=1,2,…}. Besides, fork≠landn≠m,Yknis independent ofYlm. LetU(n) denote the surplus at timen(n=0,1,…) when no dividend is considered. Then

U(n)=U(n-1)+c+YU(n-1),n-Xnξn,

(1)

whereU(0)=u.

We introduce a constant dividend barrier into the model (1). Assume that any surplus of the insurer above the levelb(a positive integer) is immediately paid out to the shareholders so that the surplus is brought back to the levelb. When the surplus is below, nothing is done. Once the surplus is negative, the insurer is ruined and the process stops. LetV(n) denote the surplus at timen. Then

V(n)=min{V(n-1)+c+YV(n-1),n-Xnξn,b},

(2)

whereV(0)=u. We are interested in the expected present value of the accumulated dividends up to the time of ruin in the risk model (2).

In this paper, we consider the modified compound binomial risk model, in which the claim occurrence, the claim amount are both regulated by an underlying Markov environment process. Denote the environment process by {Jn:n=0,1,2,…}, which is a discrete time Markov process. Assume that {Jn} is homogeneous and irreducible. Denote the state space byE={1,2,…,m}, one-step transition probability matrix byA(aij), whereaij=P(Jn+1=j|Jn=i). Letπ=(π1,π2,…,πm) be its stationary distribution.

We assume that the claim occurrence sequence {ξn,n≥1}, the claim amount sequence {Xn,n≥1} are both governed by the Markov process {Jn,n≥0}. WhenJn-1=i, the claim occurrence r.v.ξnat timenis Bernoulli distributed as follows:

P(ξn=1|Jn-1=i)=pi,0

P(ξn=0|Jn-1=i)=qi=1-pi,

however,Xnhas a p.f. dependent on the current environment state

P(Xn=k|Jn-1=i)=f(i)(k),k=1,2,3,…

It is reasonable that the incomes from investments are brought in. The incomeYkn(k≥0,n≥1) is certainly related to the capitalk, and whenk=0 the income should be 0, i.e.Y0n=0 with probability 1. In general,P(Ykn≥-k)=1. We should point out the assumption thatYinis restricted to be integer-valued is not completely identical with reality, but as the unit of money decreases it is closer and closer to reality. Let

gs(k)=P(Ysn=k),k=±1,±2,…,;s=0,1,2,…,

whereg0(0)=P(Y0n)=1.

LetR1(u) denote the sum of the first discounted dividend payments, andR(u) denote the sum of the discounted dividend payments up to the ruin time with a discounted factorv(0

D(i)(u)=E[R(u)|U(0)=u,J0=i],u≥0,i∈E.

Define

T=inf{t≥1:V(t)<0}(inf ∅=∞),

(3)

as the time of ruin and

τ=inf{t≥1:U(t)>b}(inf ∅=∞),

(4)

Assume

P(i)(Xn>c)=P(Xn>c|Jn-1=i)>0,i∈E.

(5)

The assumption is reasonable because an insurer should face a real risk process. Because of the assumption (5) andb<∞,P(i)(T<∞)=P(T<∞|J0=i)=1. We assume throughout this paper that 0≤u≤b.

The classical compound binomial risk model has been studied extensively in actuarial literature. For examples [1～6], [6] considered a compound binomial model with randomized decisions on paying dividends. Dividend strategy for insurance risk models were first proposed by [7] to reflect more realistically the surplus cash flows in an insurance portfolio, and the author found that the optimal strategy must be a barrier strategy. For more risk models in the presence of dividend payments, see [5～10].

Traditionally, the claim amounts are assumed to be mutually independent, identically distributed, and they are independent of the binomial processNt, however, such independence assumptions are sometimes too restrictive in practical applications, [10] proposed a more general model by introducing a Markovian environment. [11] studied the Gerber-Shiu expected discounted penalty function to the so-called compound Markov binomial model. It is worth mentioning that [13] considered a continuous time risk model with stochastic return on investments and dividend payments; [5] considered a discrete time risk process with stochastic return on investments based on the compound binomial model.

In this paper, we propose a compound binomial model defined in a markovian environment which is an extension to the compound binomial model presented by [5]. This paper is structured as follows. We introduce the model in Section 1. In Section 2, a set of linear equations for the expected present value of all dividends paid out until ruin occurs is derived. Finally, we obtain some results about all dividends up to the ruin time.

2 The Expected Present Value of Dividends

(6)

By conditioning on the time 1, we considerU(n) in the first period (0,1] and separate the five possible cases as following:

(1) no change of state in (0; 1] and no claim occurs in (0; 1];

(2) a change of state in (0; 1] and no claim occurs in (0; 1];

(3) no change of state in (0; 1] and a claim occurs in (0; 1];

(4) a change of state in (0; 1] and a claim occurs in (0; 1];

(5) except the four cases of above;

we can see that

(7)

and

(8)

Note thatgu(y)=0 fory<-u. we change equivalently Eq. (7) into

(9)

Eq. (9) can be rewritten as

(10)

N1is

andV1is the colum vector

Theorem1Under the assumption thatP(Ykn≤0)>0(k=0,1,…,b) orv<1, the set of linear equations (10) has a solution and the solution is unique, i.e.,

(11)

ProofFirst, we consider the caseP(Ykn≤0)>0 fork=0,1,…,b. Let

B=(bkj)=I-vM1-vN1

Owing to the assumption (5). We have

∃x0>c, s.t.f(i)(x0)>0,i∈E,

which leads to

f(i)(c)+f(i)(c-1)+…+f(i)(c-b)<1,i∈E,

therefore,

wheni=1, which leads to

(12)

It is easily seen that

because of (12).

We continue with the above program, and getλ3,λ4,…,λ(b+1)m-1in turn andWk=diag(1,…,1,λk,1,…,1)(b+1)m(k=3,4,…) with thekth element beingλk, Finally, we obtain

(13)

whereW=W1W2…W(b+1)m-1=diag(λ1,λ2,…,λ(b+1)m-1,1).

Thus,B(b+1)m-1is a (row) strictly diagonally dominant matrix, which is nonsingular. hence, the coeffcient matrixB=(bkj)=I-vM1-vN1(called as H-matrix) in the linear equations (10) is also a nonsingular matrix, which leads to the result.

For the casev<1,Bis a strictly diagonally dominant matrix. Owing to such a fact, the result also holds.

RemarkThe conditionP(Ykn≤0)>0 (k=0,1,…,b) implies that the insurer selects risky investments.

Similarly, note thatgu(y)=0 fory<-u. We change equivalently Eq. (8) into

(14)

Eq. (14) can be rewritten as

(I-vM-vN)D(1)(0),D(1)(1),…,D(1)(b),…,D(m)(0),…,D(m)(b))T=V1,

(15)

whereMis

Nis

Thus we have the following theorem.

Theorem2Under the assumption thatP(Ykn≤0)>0(k=0,1,…,b) orv<1, the set of linear equations (15) has a solution and the solution is unique, i.e.,

(D(1)(0),D(1)(1),…,D(1)(b),D(2)(1),…,D(2)(b),…,D(m)(0),…,D(m)(b))T=

(I-vM-vN)-1V1.

(16)

ProofSimilar to Theorem 1.

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(编辑 胡文杰)

2012-12-17

湖南省研究生创新基金资助项目(CX2011B197)

O211.62

A

1000-2537(2014)05-0070-06

带随机回报的一类离散马氏风险模型的分红问题

邓迎春*，乐胜杰，肖和录，赵昌宝

(湖南师范大学数学与计算机科学学院,高性能计算与随机信息处理省部共建教育部重点实验室, 中国 长沙 410081)

考虑了带随机回报的一类离散马氏风险模型.在此模型中，赔付的发生概率，赔付额的分布函数都是由一个离散时间的马氏链调控.当保险公司采用门槛分红策略时，通过计算得到了破产前的期望折现分红总量满足的一组线性方程.最后，给出了期望折现分红总量的显式解析式.

马氏风险模型；随机回报；门槛分红；期望折现分红量

*

,E-mail：dengyc88@yahoo.com.cn