Front. Math. China 2012, 7(3): 437–448 DOI 10.1007/s11464-012-0176-7 A matrix operator approach to a risk model with two classes of claims Hua DONG 1 , Zaiming LIU 2 1 School of Mathematics, Qufu Normal University, Qufu 273165, China 2 School of Mathematics, Central South University, Changsha 410075, China c Higher Education Press and Springer-Verlag Berlin Heidelberg 2012 Abstract In this paper, we study a risk model with two independent classes of risks, in which both claim number processes are renewal processes with phase- type inter-arrival times. Using a generalized matrix Dickson-Hipp operator, a matrix Volterra integral equation for the Gerber-Shiu function is derived. And the analytical solution to the Gerber-Shiu function is also provided. Keywords Gerber-Shiu function, phase-type distribution, Dickson-Hipp operator MSC 91B30 1 Introduction Ruin theory has been one of the main research topics in actuarial mathematics since Lundberg [11] and Cram´ er [2]. During the last decades, the classical Lundberg problem has been extended to renewal risk model, in which the interclaim times are phase-type distributed. See, Dickson and Hipp [3], Li and Garrido [7], Ren [12], Song et al. [13], and references therein. In recent years, the risk model with multiple classes of claims has been studied extensively. Yuen et al. [14] investigated the survival probabilities for a risk model with two classes of business in which the claim number processes are Poisson and Erlang(2) process, respectively. Li and Garrido [8] studied the ruin probabilities for a risk model with two independent classes of risks in which one claim number process is Poisson process and the other claim number process is renewal process with generalized Erlang(2) inter-arrival times. Given that ruin is caused by a claim from a certain class, Li and Lu [9] further studied the Gerber-Shiu functions at ruin for the model of Li and Garrido [8]. Zhang et al. [15] extended some results in Li and Lu [9] by Received December 20, 2010; accepted November 9, 2011 Corresponding author: Hua DONG, E-mail: [email protected]
Transcript
Front. Math. China 2012, 7(3): 437–448DOI 10.1007/s11464-012-0176-7
A matrix operator approach to a risk modelwith two classes of claims
Hua DONG1, Zaiming LIU2
1 School of Mathematics, Qufu Normal University, Qufu 273165, China2 School of Mathematics, Central South University, Changsha 410075, China
Abstract In this paper, we study a risk model with two independent classes ofrisks, in which both claim number processes are renewal processes with phase-type inter-arrival times. Using a generalized matrix Dickson-Hipp operator, amatrix Volterra integral equation for the Gerber-Shiu function is derived. Andthe analytical solution to the Gerber-Shiu function is also provided.
Ruin theory has been one of the main research topics in actuarial mathematicssince Lundberg [11] and Cramer [2]. During the last decades, the classicalLundberg problem has been extended to renewal risk model, in which theinterclaim times are phase-type distributed. See, Dickson and Hipp [3], Liand Garrido [7], Ren [12], Song et al. [13], and references therein.
In recent years, the risk model with multiple classes of claims has beenstudied extensively. Yuen et al. [14] investigated the survival probabilities fora risk model with two classes of business in which the claim number processesare Poisson and Erlang(2) process, respectively. Li and Garrido [8] studiedthe ruin probabilities for a risk model with two independent classes of risksin which one claim number process is Poisson process and the other claimnumber process is renewal process with generalized Erlang(2) inter-arrival times.Given that ruin is caused by a claim from a certain class, Li and Lu [9]further studied the Gerber-Shiu functions at ruin for the model of Li andGarrido [8]. Zhang et al. [15] extended some results in Li and Lu [9] by
Received December 20, 2010; accepted November 9, 2011Corresponding author: Hua DONG, E-mail: [email protected]
438 Hua DONG, Zaiming LIU
assuming that the claim number processes are Poisson and renewal processwith generalized Erlang(n) inter-arrival times, respectively. Using interpolatingtheorem, they derived explicit expressions for the Gerber-Shiu functions whenthe claims from both classes belong to the rational family. Recently, Ji andZhang [5] studied the Gerber-Shiu functions for a risk model with twoindependent classes of risks in which both claim number processes are renewalprocesses with phase-type inter-arrival times. By using divided difference, theyderived the Laplace transforms for the Gerber-Shiu functions. In this paper,we consider the same model as Ji and Zhang [5], however, we do not distinguishthe cause of ruin. In contrast with Ji and Zhang [5], we derive a matrix Volterraintegral equation for the Gerber-Shiu function by using a generalized matrixDickson-Hipp operator. We remark that the method used in this paper can beeasily extended to the Gerber-Shiu functions analyzed in Ji and Zhang [5].
The rest of the paper is organized as follows. In Section 2, we describethe model. In Section 3, we first introduce a generalized matrix Dickson-Hippoperator, then based on the operator, we derive a matrix defective renewalequation for the Gerber-Shiu function. The analytical expression for the Gerber-Shiu function is also derived. Some explicit results are given for a two-statemodel in the last section.
2 Model
Consider the following surplus process:
U(t) = u+ ct− S(t), t � 0, (1)
where u � 0 is the amount of initial surplus, c > 0 is the constant rate ofpremium income per unit time, and {S(t); t � 0} is the aggregate claim amountprocess. In this paper, S(t) is assumed to be generated from two classes ofinsurance risks, i.e.,
S(t) = S1(t) + S2(t) =N1(t)∑
i=1
Xi +N2(t)∑
j=1
Yj , t � 0, (2)
where {Xi}i�1 are the claim sizes from the first class, assumed to be i.i.d.positive random variables with common distribution function P and density p,while {Yj}j�1 are the claim sizes from the second class, assumed to be i.i.d.positive random variables with common distribution function Q and density q.Denote by μX and μY the means of X and Y, and by
p(s) =∫ ∞
0e−sxp(x)dx, q(s) =
∫ ∞
0e−sxq(x)dx
the Laplace transforms of p and q, respectively. The claim number processes{N1(t); t � 0} and {N2(t); t � 0} are assumed to be two renewal processes
A matrix operator approach to a risk model with two classes of claims 439
with i.i.d. claim inter-arrival times {Wi}i�1 and {Vj}j�1, respectively. Weassume that Wi’s have common distribution function K1 and Vi’s have commondistribution function K2.
Furthermore, we assume that
{Xi}i�1, {Yj}j�1, {N1(t); t � 0}, {N2(t); t � 0}are mutually independent. The net profit condition is given by
c >μX
E[W1]+
μY
E[V1].
In this paper, we assume that the distribution of K1 is phase-typedistribution with representation PH(α,A,a�), where α = (α1, α2, . . . , αn) anda = (a1, a2, . . . , an) are row vectors and A = (aij)ni,j=1 is an n × n matrix. Leten be a row vector of length n with all elements being 1. Then
a� = −Ae�n , αe�n = 1.
Similarly, we assume that the distribution of K2 is phase-type distribution withrepresentation PH(β,B,b�), where β = (β1, β2, . . . , βm) and b = (b1, b2, . . . ,bm) are row vectors and B = (bij)mi,j=1 is an m ×m matrix. Let em be a rowvector of length m with all elements being 1. Then
b� = −Be�m, βe�m = 1.
From the definition of phase-type distribution, each Vk corresponds to thetime to absorption in a terminating continuous time Markov chain J1(t) with ntransient states {E1, E2, . . . , En} and one absorbing state E0. Similarly, each Wj
corresponds to the time to absorption in a terminating continuous time Markovchain J2(t) with m transient states {F1, F2, . . . , Fm} and one absorbing stateF0. Obviously, J (t) = {J1(t),J2(t)} is a 2-dimensional Markov chain withstates {(Ei, Fj), 1 � i � n, 1 � j � m} and initial distribution ν = β ⊗ α,where ⊗ is the Kronecker product of matrices (see Ji and Zhang [5]). Note thatthe sojourn time in each state of J (t) is exponentially distributed. Hence,(U(t),J (t)) is a piecewise deterministic Markov process.
Let T = inf{t � 0: U(t) < 0} (∞ otherwise) be the ruin time, and let
ψ(u) = P (T <∞ | U(0) = u), u � 0,
be the ultimate ruin probability. The Gerber-Shiu function is defined as
and (5) simplifies to Ψδ(u). When ω(x, y) = 1 and δ = 0,
ψ(u) = νΨ0(u).
Throughout this paper, D denotes the differentiation operator; I denotesthe identity operator; Ij is the identity matrix of order j; L −1 is the inverseLaplace transform; ⊗ denotes the Kronecker product of matrices; ⊕ denotes theKronecker sum of matrices; the boldface letter i denotes row vector; ei denotesrow vector of length i with all elements being 1; and “�” denotes the transpose.
3 Main results
3.1 Dickson-Hipp operator
The Dickson-Hipp operator for a real-valued integrable function was firstintroduced by Dickson and Hipp [3] and more properties were given by Liand Garrido [7]. Later, Li and Lu [10] extended the definition of Dickson-Hippoperator to a matrix function where each element being a real-valued integrablefunction. Feng [4] developed a new matrix version of the Dickson-Hippoperator:
TSf(x) = eSx
∫ ∞
xe−Suf(u)du,
where S is a matrix, f is a function, and the integral exists. Following Li andLu [10] and Feng [4], we will introduce a new operator which will facilitate thecalculation in this paper.
Definition 3.1 For any n-dimensional vector function f�, define
TSf�(x) = eSx
∫ ∞
xe−Suf�(u)du, (6)
where S is an n× n matrix and all the integrals exist. When x = 0, we write
TSf�(0) = f�(S). (7)
A matrix operator approach to a risk model with two classes of claims 441
Remark 3.1 1) When n = 1, (6) is the Dickson-Hipp operator and (7) is theLaplace transform.
2) Let s be a nonnegative real number, and let f� be an n-dimensionalvector function. Then
Tsf�(x) = TsInf�(x).
Definition 3.2 For matrix functions Fl×n and Kn×m, define
F ∗K(x) =∫ x
0F(x− y)K(y)dy, x � 0, (8)
where all the entries of F and K are defined on [0,∞). For n = l, define
F∗k = F∗k−1 ∗ F, F0 = I.
Similar to Feng [4], we have the following properties.
Property 3.1 For all matrices S1 and S2, we have
TS1(S2 − S1)TS2f� = TS1f
� − TS2f�.
Property 3.2 Suppose that f� is an n-dimensional vector function and g isa function. Then
TS{f� ∗ g}(x) = g(S)TSf�(x) + TS g ∗ f�(x).
Property 3.3 Let S be an n × n matrix with n positive (or having positivereal part) eigenvalues, and let f� be an n-dimensional vector function. Then
TS(SI − D)f� = f�.
Proof We have
TS(SI − D)f�(x) =∫ ∞
xe−S(y−x)Sf�(y)dy − e−S(y−x)f�(y) |∞x
+∫ ∞
xe−S(y−x)Sf�(y)dy
= f�(x). �3.2 Matrix Volterra integral equation
In this subsection, our goal is to derive a matrix Volterra integral equation forthe Gerber-Shiu function Φ(u).
Theorem 3.1 For u � 0, the Gerber-Shiu function Φ(u) satisfies the followingmatrix Volterra integral equation:
Φ(u) =∫ u
0ϑ(x)Φ(u− x)dx+ TR�(u), (9)
442 Hua DONG, Zaiming LIU
whereϑ(u) = c−1(TRp(u)Gp + TRq(u)Gq),
Gp = Im ⊗ (a�α), Gq = (b�β) ⊗ In,
H = c−1(δImn −A ⊕ B),
ωp(u) =∫ ∞
uω(u, x− u)p(x)dx,
ωq(u) =∫ ∞
uω(u, x− u)q(x)dx,
�(u) = c−1(e�m ⊗ a�ωp(u) + b� ⊗ e�nωq(u)),
and R is an mn×mn matrix satisfying
c(H − R) = p(R)Gp + q(R)Gq. (10)
Proof We consider the infinitesimal interval from t to t+dt. Conditioning, oneobtains
where the second equality follows from (11), the fourth equality follows fromProperty 3.2, and the last equality follows from (10). This completes the proof.
�Remark 3.2 When m = 1, Q(0) = 1, (9) is in accordance with [4, (3.1)].
where Δ is the Dirac delta function. In fact, fij(x) can be seen as the discounteddensity function of deficit x given that the initial surplus is 0 and initial stateis (Ei, Fj). On the other hand, it is a special case of the Gerber-Shiu functionφij(0) with ω(x, y) = Δ(y−x). It follows from the expression for ω and (9) that
TR�(0) =∫ ∞
0e−Ry�(y)dy
= c−1
∫ ∞
0e−Ry(p(y + x)e�m ⊗ a� + q(y + x)b� ⊗ e�n )dy
= c−1(TRp(x)e�m ⊗ a� + TRq(x)b� ⊗ e�n ).
Clearly,(TR�(0))ij = fij(x), ϑ(x)e�mn = TR�(0).
Therefore, ∫ ∞
0ϑ(x)dxe�mn � e�mn.
Since ϑ is a semi-Markov kernel, we have
Φ(u) =∞∑
k=0
ϑ∗k ∗ TR�(u).
3.3 Representation of RLet
Lδ(s) = sImn − H + c−1Gpp(s) + c−1Gq q(s), s � 0,
where Gp, Gq, and H are given in Theorem 3.1.It follows from Ji and Zhang [5] that det(Lδ(s)) = 0 has mn roots with
positive real part, denoted by ρ1, ρ2, . . . , ρmn. In the sequel, we assume thatthese mn roots are distinct.
Following the idea of Li [6], we assume that d�i is a eigenvector of the
transpose of L(ρi) corresponding to the eigenvalue 0 for i = 1, 2, . . . ,mn. Then
0 = diL(ρi)
= diρi − diH +1cdi
∫ ∞
0e−ρixp(x)dxGp +
1cdi
∫ ∞
0e−ρixq(x)dxGq,
where 0 is a row vector of length mn with all elements being 0. Letting
Λ = diag(ρ1, . . . , ρmn),
and rearranging the above equation in matrix, we have
cΛD = cDH−∫ ∞
0exp(−Λx)Dp(x)dxGp −
∫ ∞
0exp(−Λx)Dq(x)dxGq, (13)
A matrix operator approach to a risk model with two classes of claims 445
whereD = (d�
1 ,d�2 , . . . ,d
�mn)�
and Λ is given in Corollary 3.1. We assume that di’s are linearly independentand D is invertible. Left-multiplying both sides of (13) by D−1 yields
cD−1ΛD = cH −∫ ∞
0D−1 exp(−Λx)Dp(x)dxGp
−∫ ∞
0D−1 exp(−Λy)Dq(y)dyGq. (14)
Comparing (14) with (10), we have R = D−1ΛD.
3.4 Analytical expression for Φ(u)Theorem 3.2 The analytical expression for Φ(u) is given by
where the last equality follows from (10). According to Burton [1], the solutionto (15) can be written as
Φ(u) = Z(u)Φ(0) −∫ u
0Z(u− t)�(t)dt,
where Z(u) is an mn ×mn matrix whose columns are particular solutions tothe following equation:
Φ′(u) = HΦ(u) −∫ u
0c−1(p(x)Gp + q(x)Gq)Φ(u− x)dx.
ThenZ′(u) = HZ(u) −
∫ u
0c−1(p(x)Gp + q(x)Gq)Z(u− x)dx.
Taking Laplace transforms on both sides of the above equation and noting thatZ(0) = Imn, we have Z(s) = L−1
δ (s). Inverting it leads to the desired result. �
446 Hua DONG, Zaiming LIU
Remark 3.3 Corollary 3.2 and Theorem 3.2 provide two solutions to Φ(u).In fact, they are the same solution in different forms, because they have thesame Laplace transform:
Corollary 3.3 The Laplace transform of the ruin time Ψδ(u) admits thefollowing expression:
Ψδ(u) =∫ u
0ϑ(s)Ψδ(u− s)ds+
∫ ∞
uϑ(y)dye�mn, u � 0. (17)
And Ψδ(u) also has the following matrix geometric series expression:
Ψδ(u) =∞∑
n=0
V∗n(u) (Imn − Σ) e�mn, (18)
whereV∗n(u) =
∫ ∞
uϑ∗n(y)dy, Σ =
∫ ∞
0ϑ(y)dy.
Proof When ω(x, y) = 1, we have
�(u) = c−1(e�m ⊗ a�P (u) + b� ⊗ e�nQ(u))
= c−1(Gpe�mnP (u) + Gqe�mnQ(u)).
Therefore,
TR�(u) =∫ ∞
uϑ(y)dye�mn.
Then (9) reduces to (17). Taking Laplace transforms on both sides of (17)yields
Ψδ(s) = ϑ(s)Ψδ(s) +Σ − ϑ(s)
se�mn.
Rearranging the above equation leads to
Ψδ(s) = (Imn − ϑ(s))−1 Σ − ϑ(s)s
e�mn
= (Imn − ϑ(s))−1 Imn − ϑ(s) − (Imn − Σ)s
e�mn
=Σ − ((Imn − ϑ(s))−1 − Imn)(Imn − Σ)
se�mn. (19)
Inverting (19) yields (18).
Remark 3.4 When ω(x, y) = 1, we have
TsTR�(0) =1s
(Σ − �(s))e�mn.
A matrix operator approach to a risk model with two classes of claims 447
Then (16) reduces to
Φ(s) =(Imn − ϑ(s))−1(Σ − ϑ(s))
se�mn,
which is just the Laplace transform of (18).
4 Example
In this section, we assume that N1(t) is a generalized Erlang(2) process withparameters λ1 and λ2, and N2(t) is a Poisson process with parameter λ. Then
m = 1, n = 2, α = (1, 0), β = 1, ν = (1, 0), B = −λ,
with ρi are roots of det(Lδ(s)) = 0, Re(ρi) > 0, i = 1, 2.For illustration, we set
p(s) =ζ
s+ ζ, q(s) =
γ
s+ γ, δ = 0, ω(x, y) = 1,
c =32, λ = 1, λ1 =
710, λ2 =
52, ζ =
32, γ = 1.
Then we have det(D) = 0.8778 and the ruin probability is ψ(0) = 0.8976 whichis in accordance with the result of Li and Lu [9].
Acknowledgements The authors thank the anonymous referees for detailed comments
and valuable suggestions that improved the contents of this paper. The authors would like to
thank Dr. Runhuan Feng for his valuable discussion. This work was supported by the National
Natural Science Foundation of China (Grant Nos. 10971230, 11171179), the Tianyuan Fund
for Mathematics (No. 11126232), and the Natural Science Foundation of Shandong Province
(Nos. ZR2009AL015, ZR2010AQ015).
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