Some Optimal Dividend Problems in aMarkov-modulated Risk Model

Shuanming Li1 and Yi Lu21Centre for Actuarial Studies, Department of Economics

The University of Melbourne, Australia2Department of Statistics and Actuarial Science

Simon Fraser University, Canada

Abstract

In this paper, we derive some results on the dividend payments prior toruin in a Markov-modulated risk model in which the claim inter-arrivals,claim sizes and premiums are influenced by an external Markovian envi-ronment process. A system of integro-differential equations with boundaryconditions satisfied by the n-th moment of present value of the total divi-dend payments prior to ruin, given the initial environment state, is derivedand solved. We show that both the probabilities that the surplus processattains a given level from the initial surplus without first falling below zeroand the Laplace transforms of the time that the surplus process first hits abarrier without ruin occuring can be expressed in term of the solution of theabove mentioned system of integro-differential equations. In the two-statemodel, explicit solutions to the integro-differential equations are obtainedwhen both claim size distributions are exponentially distributed. Finally, anumerical example and the comparison with the results obtained from theassociated averaged compound Poisson risk model are also given.

Keywords: Markov-modulated processes; Dividend barrier; Present valueof dividend payments; Time of ruin; Integro-differential equation; Laplacetransform.

1

1 Introduction

Consider a risk model in continuous time. Denote by {I(t); t ≥ 0} the externalenvironment process, which influences the frequencies of the claims, the distribu-tions of the claims, and the premiums. As pointed out by Asmussen (1989), inhealth insurance, sojourns of {I(t); t ≥ 0} could be certain types of epidemics or,in automobile insurance, these could be weather types (for example, icy, foggy,. . .).Suppose that {I(t); t ≥ 0} is a homogeneous, irreducible and recurrent Markovprocess with finite state space J = {1, 2, . . . ,m}. Denote the intensity matrix of{I(t); t ≥ 0} by

Λ = (αij)mi,j=1 , αii := −αi = −

∑

j 6=i

αij , i ∈ J .

The transition probability matrix of the embedded Markov chain is then given by

Q = (pij)mi,j=1 , pij =

{0, i = j,αijαi

, i 6= j,i, j ∈ J . (1)

Further assume that at time t claims occur according to a Poisson process withconstant intensity rate λi ∈ R+, when I(t) = i and the corresponding claimamounts have distributions Fi(x), with density function fi(x) and finite meanµi (i ∈ J). Moreover, we assume that premiums are received continuously at apositive constant rate ci during any time interval when the environment processremains in state i. Denote by Wn and Xn, respectively, the arrival time and theamount of the n-th claim, and by Tn = Wn − Wn−1 the inter-arrival time of the(n − 1)-th claim and the n-th claim, with W0 = X0 = T0 = 0.

Let Jn be the state of the process {I(t); t ≥ 0} at the arrival of the n-th claim,i.e.,

Jn = I(Wn) , n ∈ N ,

and π = (π1, . . . , πm) be its unique stationary probability distribution. Reinhard(1984) shows that

πi =

λiηiαi∑m

k=1λkηkαk

, i ∈ J , (2)

where η = (η1, . . . , ηm) is the unique stationary probability distribution of theembedded Markov chain of process I, such that η Q = 0, in which transitionprobability matrix Q is given by (1).

2

Suppose that the sequences of random variables {Xn; n ≥ 0} and {Tn; n ≥ 0} areconditionally independent given {I(t); t ≥ 0}.

Now define N(t) = sup{n ∈ N∣∣ ∑n

i=1 Ti ≤ t} as the number of claims that haveoccured before time t. The counting process {N(t); t ≥ 0} is called a Markov-modulated Poisson process, which is a special case of Cox processes. It also canbe seen as a Poisson process with parameters driven by an external environmentprocess. The corresponding surplus process {U(t); t ≥ 0} is given by

U(t) = u + C(t)−N(t)∑

n=1

Xn , t ≥ 0 , (3)

where C(t) denotes the aggregate premium received during interval (0, t] andu(≥ 0) is the initial reserve. Let Zn be the time at which the n-th transition ofthe environment process occurs and In be the state of the environment after itsn-th transition. Reinhard (1984) shows that

C(t) =

Ne(t)∑

k=1

cIk−1(Zk − Zk−1) + cINe(t)(t− TNe(t)) , t ≥ 0 ,

where Ne(t) = sup{n ∈ N : Zn ≤ t}. We also assume that the positive loadingcondition satisfies [see Reinhard (1984)], i.e.,

d =m∑

i=1

πi( ci

λi− µi

)> 0 , (4)

where πi is given by (2).

The definition of this Markov-modulated risk model using an environmental Markovchain {J(t); t ≥ 0} is first given in Asmussen (1989), where the process J modelsthe random environment in which an insurance business is assumed to be anirreducible continuous time Markov chain, with finite state space.

Models of this type have also been investigated by some authors. Reinhard (1984)and Lu and Li (2005) consider the probability of ruin in a class of Markov-modulated risk models. Wu (1999) develops generalized bounds and Schmidli(1997) studies the estimation of the Lundberg coefficient for the probability ofruin of this model. Bäuerle (1996) also considers the expected ruin time of thesame model. The severity of ruin, in the particular case where one has two pos-sible states for the environmental process, is studied by Snoussi (2002) and Lu(2005).

3

In this paper, we consider the above defined surplus process modified by the pay-ment of dividends. When the surplus exceeds a constant barrier b ≥ u, dividendsare paid continuously so the surplus stays at the level b until a new claim occurs.Let Ub(t) be the surplus process with initial surplus Ub(0) = u under the barrierstrategy above and define Tb = inf{t ≥ 0 : Ub(t) < 0} to be the time of ruin. Letδ > 0 be the force of interest for valuation and define

Du,b =

∫ Tb0

e−δ tdD(t), 0 ≤ u ≤ b,

to be the present value of all dividends until time of ruin Tb, where D(t) is theaggregate dividends paid by time t. Define the mean of Du,b, given that the initialenvironment state is i, by

Vi(u; b) = E[Du, b|I(0) = i], 0 ≤ u ≤ b, i ∈ J.

Then the expected present value of the total dividend payments until ruin in thestationary case is

V (u; b) =m∑

i=1

ηi Vi(u; b) , 0 ≤ u ≤ b.

The barrier strategy was initially proposed by De Finetti (1957) for a binomialmodel. More general barrier strategies have been studied in a number of papersand books. These references include Bühlmann (1970), Segerdahl (1970), Gerber(1972, 1979, 1981), Paulsen and Gjessing (1997), Albrecher and Kainhofer (2002),Højgaard (2002), Lin et al. (2003), Dickson and Waters (2004), Li and Garrido(2004), Albrecher et al. (2005), and Li and Dickson (2005). The main focus ison optimal dividend payouts and problems associated with time of ruin, undervarious barrier strategies and other economic conditions. For the risk processmodeled by a Brownian motion, detailed analysis can be found in Gerber andShiu (2004).

2 Integro-differential equations with boundary

conditions

We now derive a system of integro-differential equations for Vi(u; b), i ∈ J . Con-sidering a small time interval [0, h], with h > 0, there are four possible eventsregarding to the occurrence of the claim and the change of the environment:

4

(i) no claim and no change of environment occur in [0, h],

(ii) a claim occurs in [0, h] (it can either cause the ruin or not),

(iii) the environment changes in [0, h], and

(iv) two or more events occur in [0, h].

Conditioning on the event occurs in the interval [0, h], we have

Vi(u; b) = (1 − αi h − λi h) e−δ h Vi(u + ci h; b)

+λi h e−δ h

∫ u+cih

0

Vi(u + ci h − x; b) dFi(x)

+αi h e−δ h

m∑

k=1

pik Vk(u + ci h; b) + o(h) , 0 ≤ u < b , i ∈ J ,

where o(h)/h → 0, when h → 0. Since e−δ h = 1 − δ h + o(h), we then get

Vi(u; b) = [1 − (αi + λi + δ)h]Vi(u + ci h; b) + λi h∫ u+cih

0

Vi(u + ci h − x; b) dFi(x)

+ αi hm∑

k=1

pik Vk(u + ci h; b) + o(h) , 0 ≤ u < b , i ∈ J . (5)

Equation (5) can be rewritten for 0 ≤ u < b as

Vi(u + ci h; b)− Vi(u; b)h

= (αi + λi + δ)Vi(u + ci h; b)

− λi∫ u+cih

0

Vi(u + ci h − x; b) dFi(x)

− αim∑

k=1

pik Vk(u + ci h; b) +o(h)

h, i ∈ J . (6)

Letting h → 0 in (6), we get a system of integro-differential equations satisfied byVi(u; b):

ci V′i (u; b) = (αi + λi + δ)Vi(u; b) − λi

∫ u

0

Vi(u − x; b) dFi(x)

− αim∑

k=1

pik Vk(u; b) , 0 ≤ u < b , i ∈ J . (7)

5

For the case u = b, similarly, we condition on the event occurs in [0, h] to obtain

Vi(b; b) = (1 − αi h − λi h)[e−δ hVi(b; b) + ci

∫ h

0

e−δsds

]

+λi h e−δ h

[∫ b

0

Vi(b− x; b) dFi(x) +∫ h

0

cieδsds

]

+αi h e−δ h

[m∑

k=1

pik Vk(b; b) +

∫ h

0

cieδsds

]+ o(h) , i ∈ J .

Using the same arguments as the above gives for i ∈ J ,

(αi + λi + δ)Vi(b; b) − λi∫ b

0

Vi(b− x; b) dFi(x) − αim∑

k=1

pik Vk(b; b) = ci . (8)

Setting u = b in equation (7) and using equation (8) show that Vi(u; b) satisfiesthe following boundary conditions:

V ′i (u; b)∣∣u=b

= 1, i ∈ J. (9)

We remark that if m = 1, equations (7) and (9) can be found in Bühlmann (1970)and Gerber (1979).

Now letting vi(u), 0 ≤ u < ∞, i ∈ J , be the solutions of the integro-differentialequations, we get

ci v′i(u) = (αi + λi + δ) vi(u) − λi

∫ u

0

vi(u − x) dFi(x)

−αim∑

k=1

pik vk(u) , i ∈ J. (10)

The solutions of (10) are uniquely determined by the initial conditions vi(0), i ∈ J .Further for j ∈ J , let v1,j(u), v2,j(u), . . . , vm,j(u) be the particular solutions of (10)with the following initial conditions

vi,j(0) =

{1, i = j,

0, i 6= j.

Then the general solutions of (10) are of the form

vi(u) =m∑

j=1

aj vi,j(u) , i ∈ J ,

6

where a1, a2, . . . , am are any real numbers. It follows that the solutions to theintegro-differential equations (7) with the boundary conditions (9) can be ex-pressed as

Vi(u; b) =m∑

j=1

aj(b) vi,j(u) , 0 ≤ u ≤ b, i ∈ J ,

where a1(b), a2(b), . . . , am(b) are the solutions of the following system of linearequations

m∑

j=1

aj(b) v′i,j(b) = 1 , i ∈ J .

The particular solutions vi,j(u) will be analyzed in Section 6.

3 Moment generating function of Du,b and higher

moment

In this section, we study the moment generating function of Du,b, through whichwe can analyze the higher moment of the present value of all dividend paymentsprior to ruin. Define the moment generating function of Du,b, given that the initialenvironment state is i, by

Mi(u, y; b) = E[ey Du,b|U(0) = u, I(0) = i], 0 ≤ u ≤ b, i ∈ J,

where y is such that Mi(u, y; b) exists.

Similar arguments as in Section 2, we condition on the events which could occurin the small interval [0, h],

Mi(u, y; b) = E[ey Du,b

∣∣U(0) = u, I(0) = i]

= (1 − αih − λih)Mi(u + cih, e−δhy; b)

= λih

[∫ u+cih

0

Mi(u + cih − x, e−δhy; b)dFi(x) + F̄i(u + cih)]

= αih

m∑

k=1

pikMk(u + cih, e−δhy; b) + o(h) , 0 ≤ u < b, i ∈ J ,

where F̄i = 1 − Fi is the tail of the distribution function Fi.

7

Taylor’s expansion gives

Mi(u + cih, e−δhy; b) = Mi(u, y; b) + cih

∂Mi(u, y; b)

∂u

−δyh∂Mi(u, y; b)∂y

+ o(h). (11)

Substituting (11) into the expression of Mi(u, y; b), dividing both sides by h andletting h → 0, we have

ci∂Mi(u, y; b)

∂u− δy∂Mi(u, y; b)

∂y− (λi + αi)Mi(u, y; b)

+λi

[∫ u

0

Mi(u− x, y; b)dFi(x) + F̄i(u)]

+αi

m∑

k=1

pikMk(u, y; b) = 0 , 0 ≤ u < b, i ∈ J . (12)

For the case u = b,

Mi(b, y; b) = (1 − αih − λih)ey ci hMi(b, e−δhy; b)

= λih ey ci h

[∫ b

0

Mi(b− x, e−δhy; b)dFi(x) + F̄i(b)]

= αih ey ci h

m∑

k=1

pikMk(b, e−δhy; b) + o(h) , i ∈ J .

Using Taylor’s expansion, we have, for i ∈ J ,

δy∂Mi(b, y; b)

∂y+ (λi + αi + ciy)Mi(b, y b)

= λi

[∫ b

0

Mi(b − x, y; b)dFi(x) + F̄i(b)]

+ αi

m∑

k=1

pikMk(b, y; b) = 0 .

Comparing these equations with the corresponding equations in (12) for u = b,we have the following boundary conditions

∂Mi(u, y; b)

∂u

∣∣∣u=b

= yMi(b, y; b), i ∈ J . (13)

For 0 ≤ u ≤ b and i = 1, 2, . . . ,m, define

Vi,n(u; b) = E[Dnu,b

∣∣I(0) = i], n ∈ N,

8

to be the n-th moment of Du,b, with Vi,0(u; b) = 1 and Vi,1(u; b) = Vi(u; b).Substituting Mi(u, y; b) = 1+

∑∞n=1(y

n/n!)Vi,n(u; b) into (12) and comparing thecoefficient of yn yields the following integro-differential equations

ciV′i,n(u; b) = (αi + λi + nδ)Vi,n(u; b) − λi

∫ u

0

Vi,n(u − x; b)dFi(x)

−αim∑

k=1

pikVk,n(u; b), 0 ≤ u < b, i ∈ J . (14)

It follows from (13) that

V ′i,n(u; b)∣∣u=b

= nVi,n−1(b; b), i ∈ J, n ∈ N , (15)

with Vi,0(b; b) = 1.

We remark that the way of solving the integro-differential equations (14) withboundary conditions (15) is the same as that of solving equations (7) with boun-dary conditions (9), and the only difference is to replace δ by nδ.

4 The time to reach the dividend barrier

In this section, we consider how long it takes for the surplus process to reach thedividend barrier b from the initial surplus u without ruin occuring. We defineτb to be the first time that the surplus reaches b without ruin having previouslyoccured, and for δ > 0, define

Li(u; b) = E[e−δ τb

∣∣U(0) = u, I(0) = i], 0 ≤ u ≤ b, i ∈ J.

Here Li(u; b) can be viewed as the expected present value of one dollar payableat time τb, given that the initial environment state is i, or, alternatively, it can beviewed as the Laplace transform of τb with respect to the parameter δ.

Theorem 1 Li(u; b) satisfies the following integro-differential equations

ci L′i(u; b) = (αi + λi + δ)Li(u; b)− λi∫ u

0

Li(u − x; b) dFi(x)

− αim∑

k=1

pik Lk(u; b) , 0 ≤ u < b , i ∈ J . (16)

with the following boundary conditions:

Li(b; b) = 1, i ∈ J . (17)

9

Proof: Using the same arguments as in deriving (7), we can prove that theintegro-differential equations (16) holds. The boundary conditions (17) is fromthe fact that τb = 0 and E[e

−δτb|U(0) = u, I(0) = i] = 1 when u = b. 2

The solutions to equations (16) with boundary conditions (17) are

Li(u; b) =m∑

j=1

ej(b) vi,j(u) , 0 ≤ u ≤ b, i ∈ J ,

where e1(b), e2(b), . . . , em(b) are the solutions of the following system of linearequations

m∑

j=1

ej(b)vi,j(b) = 1 , i ∈ J .

5 The probability of hitting the dividend barrier

before ruin

For b > u ≥ 0, define

ξi(u; b) = P

{sup

0≤t≤T∞U(t) < b, T∞ < ∞

∣∣∣ U(0) = u, I(0) = i}

, i ∈ J ,

to be the probability that ruin occurs from the initial surplus u without the surplusprocess reaching level b prior to ruin, given that the initial environment state is i,where T∞ is the time of ruin of the risk model (3) without a barrier. Alternatively,ξi(u; b) is the probability of ruin in the presence of an absorbing barrier at b, giventhat the initial environment state is i. Obviously, ξi(u; b) = 0, for b ≤ u.

Further define χi(u; b) to be the probability that the surplus process attains thegiven dividend barrier b from the initial surplus u without first falling below zero,given that the initial environment state is i. We have

χi(u; b) = 1 − ξi(u; b) , i ∈ J ,

since eventually either ruin occurs without the surplus process attaining b or thesurplus attains level b.

Next, we show that χi(u; b) satisfies an integro-differential equation with certainboundary conditions.

10

Theorem 2 For 0 ≤ u < b, and i ∈ J,

ci χ′i(u; b) = (λi + αi)χi(u; b)− λi

∫ u

0

χi(u − x; b)dFi(x)

−αim∑

k=1

pik χk(u; b) , (18)

with boundary conditionχi(b; b) = 1 . (19)

Proof: For a small value h > 0, conditional on the event occurs in the interval[0, h], we have for 0 ≤ u < b,

χi(u; b) = (1 − αi h − λi h)χi(u + ci h; b) + λi h∫ u+cih

0

χi(u + ci h − x; b) dFi(x)

+ αi h

m∑

k=1

pik χk(u + ci h; b) + o(h), i ∈ J .

Subtracting χi(u; b) from both sides, dividing both sides by h, and letting h → 0,we can prove that equation (18) holds. The boundary condition (19) is from thefact that the surplus hits b at the beginning without ruin occuring when u = b. 2

Let v0i,j(u) be the solutions of the integro-differential equations (10) with δ = 0.Then

χi(u; b) =m∑

j=1

hj(b) v0i,j(u) , 0 ≤ u ≤ b, i ∈ J ,

where h1(b), h2(b), . . . , hm(b) are the solutions of the following system of equations

m∑

j=1

hj(b) v0i,j(u) = 1 , i ∈ J .

We remark that when m = 1, Dickson and Gray (1984) has shown that χ(u; b),the probability that the surplus process attains the given dividend barrier b fromthe initial surplus u without first falling below zero in the classical risk model, canbe expressed as

χ(u; b) =1 − Ψ(u)1 − Ψ(b) , 0 ≤ u ≤ b ,

where Ψ(u) is the probability of ruin of the risk model (3) for m = 1 without abarrier.

11

6 Laplace transforms

We now apply Laplace transforms to find the particular solutions vi,j(u) of the

system of equations (10). Let v̂ij and f̂i be the Laplace transforms of vi,j and fi,respectively, i.e.,

v̂i,j(s) =

∫ ∞

0

e−suvi,j(u)du , f̂i(s) =

∫ ∞

0

e−sxfi(x)dx , i, j ∈ J .

Taking Laplace transforms on both sides of equation (10) yields

[s − λi + αi

ci+

λici

f̂i(s)

]v̂i,j(s) +

αici

m∑

k=1

pikvk,j(s) = vi,j(0) , i, j ∈ J ,

with vi,j(0) = I(i = j), where I(·) denotes the indicator function, or in a matrixform

A(s)v̂(s) = I ,

where

A(s) =

s − λ1[1−f̂1(s)]+α1c1

. . .

s − λm[1−f̂m(s)]+αmcm

+

α1c1

. . .αmcm

Q ,

v̂(s) = (v̂i,j(s))mi,j=1, I is an identity matrix of m by m, and Q is given by (1), with

pii = 0, for i ∈ J .

Then v̂(s) can be solved asv̂(s) = [A(s)]−1 .

We remark that when the claim sizes are rationally distributed, each element ofA(s) is a rational function, so is each element of [A(s)]−1, therefore, vi,j(u) canbe obtained by inverting v̂i,j(s) through partial fractions. This can be shown bythe examples in the section that follows.

7 Illustrations for a two-state model

In this section, we derive explicit expressions for v1,1(u), v2,1(u) v1,2(u), and v2,2(u)under some special claim size distributions when m = 2, that is, {I(t); t ≥ 0} is

12

a two-state Markov process, which reflects the random environmental effects dueto “normal” vs. “abnormal”, or “high risk” vs. “low risk” conditions. The uniquestationary probability distribution πi can be obtained from (2) as

πi =λiαi

λ1α1

+ λ2α2

, i = 1, 2 ,

and the positive loading condition (4) becomes

d =

λ1α1

(c1λ1

− µ1)

+ λ2α2

(c2λ2

− µ2)

λ1α1

+ λ2α2

> 0 .

7.1 Explicit results for exponential claims

We now consider the case where the claim size distributions f1 and f2 are expo-nentially distributed and their Laplace transformations are of the form:

f̂1(s) =β1

s + β1, f̂2(s) =

β2s + β2

,

where β1 > 0, β2 > 0. In this case matrix A(s) has the form

A(s) =

[s − λ1+α1+δ

c1+ λ1 β1

c1(s+β1)α1c1

α2c2

s − λ2+α2+δc2

+ λ2 β2c2(s+β2)

].

For simplicity, define

Qδ(s) :=

[s − λ1 + α1 + δ

c1+

λ1 β1c1(s + β1)

] [s − λ2 + α2 + δ

c2+

λ2 β2c2(s + β2)

].

Then

v̂1,1(s) =

[(s − λ2+α2+δ

c2)(s + β2) +

λ2 β2c2

](s + β1)

[Qδ(s) − α1 α2c1 c2 ](s + β1)(s + β2),

v̂1,2(s) = −α1(s + β1)(s + β2)

c1 [Qδ(s) − α1 α2c1 c2 ](s + β1)(s + β2),

v̂2,1(s) = −α2(s + β1)(s + β2)

c2 [Qδ(s) − α1 α2c1 c2 ](s + β1)(s + β2),

v̂2,2(s) =

[(s − λ1+α1+δ

c1)(s + β1) +

λ1 β1c1

](s + β2)

[Qδ(s) − α1 α2c1 c2 ](s + β1)(s + β2).

13

Since the common denominator of the above formulae is a polynomial of degree4, it has four zeros, say R1, R2, R3 and R4. It is easy to show that the four zerosare distinct, then inverting the above Laplace transforms gives

vi,j(u) =4∑

k=1

ri,j,k eRk u , i, j = 1, 2 ,

where the coefficients, ri,j,k, are given, for k = 1, 2, 3, 4, by

r1,1,k =

[(Rk−

λ2+α2+δc2

)(Rk+β2)+

λ2 β2c2

](Rk+β1)

∏4l=1,l6=k(Rk−Rl)

, r1,2,k = − α1(Rk+β1)(Rk+β2)c1 ∏4l=1,l6=k(Rk−Rl) ,

r2,2,k =

[(Rk−

λ1+α1+δc1

)(Rk+β1)+

λ1 β1c1

](Rk+β2)

∏4l=1,l6=k(Rk−Rl)

, r2,1,k = − α2(Rk+β1)(Rk+β2)c2 ∏4l=1,l6=k(Rk−Rl) .

Now we get the expected present value of the total dividend payments until ruin,given the initial state i, as follows

Vi(u; b) =2∑

j=1

aj(b)

{4∑

k=1

ri,j,k eRk u

}, 0 ≤ u ≤ b , i = 1, 2 ,

where a1(b) and a2(b) are the solutions of the following two equations:

2∑

j=1

aj(b)

{4∑

k=1

ri,j,k Rk eRk b

}= 1 , i = 1, 2 .

The higher moment of the present value of all dividend payments prior to ruin,Vi,n(u; b), can also be derived by solving similar equations.

To illustrate the results numerically, set c1 = 110, c2 = 84, λ1 = 100, λ2 = 40,α1 =

14, α2 =

34, β1 = 1, β2 = 0.5 and δ = 0.1. Then we get π1 =

1517

, π2 =217

, η1 =34, η2 =

14, the positive loading d = 0.1, and R1 = −0.121, R2 = −0.065, R3 =

0.010 and R4 = 0.074. The upper rows in Table 1 give the expected present valueof the total dividend payments until ruin V (u; b) in the stationary case, givenby η1 V1(u; b) + η2 V2(u; b), and the lower rows give the standard deviation of thepresent value of the total dividend payments until ruin in the stationary case,given by SD(u; b) =

√[η1 V1,2(u; b) + η2 V2,2(u; b)] − V (u; b)2.

7.2 Comparison with the associated averaged compound

Poisson model

We now compare some dividend related quantities in the Markov-modulated Pois-son risk process with that in the associated averaged compound Poisson model.

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Table 1: V (u; b) and SD(u; b) for b = 10, . . . , 80 and u = 10, . . . , 50

u \ b 10 20 30 40 50 60 70 8010 16.590 26.625 34.310 37.577 37.364 35.327 32.588 29.712

15.757 30.816 39.178 41.297 40.143 37.744 35.006 32.275

20 39.151 50.469 55.287 54.977 51.980 47.951 43.71931.832 40.259 41.664 40.018 37.512 34.893 32.365

30 61.683 67.593 67.220 63.558 58.631 53.45640.050 40.665 38.545 36.004 33.603 31.379

40 77.969 77.552 73.329 67.645 61.67440.288 37.674 35.061 32.837 30.876

50 87.476 82.718 76.306 69.57037.446 34.711 32.627 30.891

Asmussen et al. (1995) compared ruin functions for these two risk processes withrespect to the stochastic ordering, stop-loss ordering and ordering of adjustmentcoefficients. In their paper the arrival rate is obtained by averaging over time thearrival rate in the Markov-modulated model and the distribution of the claim sizeis obtained by averaging the ones over consecutive claim sizes.

As pointed by Asmussen et al. (1995), if the arrival rates λi, the premium ratesci, and the claim size distributions Fi in a Markov-modulated Poisson model donot fluctuate too much from the corresponding average values λ∗, c∗ and F ∗, onecan see the model as a perturbation of a classical compound Poisson risk process{U∗(t); t ≥ 0} with arrival rate λ∗, premium rate c∗ and claim size distributionF ∗. The rigorous definition of λ∗, c∗ and F ∗ in this paper for the two-state caseis as follows: λ∗ = η1 λ1 + η2 λ2 , c

∗ = η1 c1 + η2 c2 , and

F ∗(x) =1

λ∗[η1 λ1 F1(x) + η2 λ2 F2(x)] , x ≥ 0 .

Then the associated averaged compound Poisson surplus process U∗ is defined by

U∗(t) = u + c∗ t −N∗(t)∑

n=1

X∗n , t ≥ 0 ,

where {N∗(t); t ≥ 0} is Poisson with rate λ∗ and X∗n, for n ≥ 1, are i.i.d. withdistribution F ∗. Moreover, one can prove that the risk processes U , given by (3),and U∗ have the same positive safety loading d, given by (4).

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Table 2: V (u; b) (upper rows) and V ∗(u; b) (lower rows) for b = 10, . . . , 80 andu = 10, . . . , 50

u \ b 10 20 30 40 50 60 70 8010 16.590 26.625 34.310 37.577 37.364 35.327 32.588 29.712

16.322 26.613 35.174 38.817 38.366 35.938 32.893 29.823

20 39.151 50.469 55.287 54.977 51.980 47.951 43.71939.298 51.940 57.320 56.653 53.068 48.572 44.038

30 61.683 67.593 67.220 63.558 58.631 53.45663.275 69.829 69.017 64.650 59.172 53.649

40 77.969 77.552 73.329 67.645 61.67480.211 79.278 74.262 67.970 61.625

50 87.476 82.718 76.306 69.57089.157 83.515 76.439 69.304

For the numerical example showed in Section 7.1, the upper rows of Table 2give the values of V (u; b), while the lower rows give the values of the correspon-ding values of V ∗(u; b) in the associated average classical risk model. Figure 1gives the curves of V (u; b) (solid line) and V ∗(u; b) (dashed line) as functions ofb for u = 10, 20, 30, 40, 50 (from bottom to top). It can be observed that theexpected present values of the total dividend payments until ruin in the Markov-modulated risk model is overall smaller than those in the associated averaged com-pound Poisson model, which is consistent with the result, obtained in Asmussen etal. (1995), that the ruin functions related to two surplus processes {U∗(t); t ≥ 0}and {U(t); t ≥ 0} have the stochastic ordering relationship Ψ∗ ≺so Ψ under somenaturally fulfilled conditions.

Next, we compare L∗(u; b), the Laplace transform of the first time that the surplusU∗(t) reaches barrier b without ruin ever occuring, with L(u; b) = η1 L1(u; b) +η2 L2(u; b), where

Li(u; b) =2∑

j=1

ej(b)

{4∑

k=1

ri,j,k eRk u

}, 0 ≤ u ≤ b , i = 1, 2 ,

with e1(b) and e2(b) being the solutions of the following two equations:

2∑

j=1

ej(b)

{4∑

k=1

ri,j,k eRk b

}= 1 , i = 1, 2 .

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Figure 1: V (u; b) (solid lines) and V ∗(u; b) (dashed lines) for u = 10, . . . , 50 (frombottom to top)

From Figure 2, we can see that both L∗(u; b) and L(u; b) are increasing in u anddecreasing in b. Furthermore, L∗(u; b) is slightly bigger than the correspondingvalue of L(u; b).

Finally, we compare χ∗(u; b), the probability that U∗(t) attains the dividend bar-rier b from the initial surplus u without first falling below zero, with χ(u; b) =η1χ1(u; b) + η2χ2(u; b), where

χi(u; b) =

2∑

j=1

hj(b)

{4∑

k=1

r0i,j,k eR0k u

}, 0 ≤ u ≤ b , i = 1, 2 ,

with r0i,j,k and R0k being the corresponding values of ri,j,k and Rk for δ = 0, and

h1(b) and h2(b) being the solutions of the following system of equations:

2∑

j=1

hj(b)

{4∑

k=1

r0i,j,k eR0k b

}= 1 , i = 1, 2 .

Figure 3 shows that χ(u; b) is overall smaller than the corresponding value ofχ∗(u; b), this is consistent with the fact that the Markov-modulated risk modelis risky and therefore the probability of the surplus attains level b without ruin

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Figure 2: L(u; b) (solid lines) and L∗(u; b) (dashed lines) for u = 10, . . . , 50 (frombottom to top)

Figure 3: χ(u; b) (solid lines) and χ∗(u; b) (dashed lines) for u = 10, . . . , 50 (frombottom to top)

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occuring is smaller than that for the associated averaged compound Poisson riskmodel. Furthermore, we note that the two probabilities depart farther as thedividend barrier b becomes bigger.

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Shuanming LiCentre for Actuarial StudiesDepartment of EconomicsThe University of MelbourneVictoria 3010AustraliaEmail: shli@unimelb.edu.au

Yi Lu Department of Statistics and Actuarial ScienceSimon Fraser University8888 University driveBurnaby, BCV5A 1S6, CanadaEmail: yilu@sfu.ca

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