This paper considers a perturbed Markov-modulated risk model with two-sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber-Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate the method.

1. Introduction

The risk model with two-sided jumps was first proposed by Boucherie et al. [1] and has been further investigated by many authors during the last few years. For example, Kou and Wang [2] studied the Laplace transform of the first passage time and the overshoot for a perturbed compound Poisson model with double exponential jumps. Xing et al. [3] extended the results of Kou and Wang [2] to the case that the surplus process with phase-type downward and arbitrary upward jumps. Zhang et al. [4] assumed that the downward jumps follow arbitrary distribution and the upward jumps have a rational Laplace transform. They derived the Laplace transform of the Gerber-Shiu function by using the roots of the generalized Lundberg equation. Under the assumption that the upward jumps follow Laplace distribution and arbitrary downward jumps, Chi [5] obtained a closed-form expression for the Gerber-Shiu function by applying Wiener-Hopf factorization technique. The applications of the model in finance were also discussed. Jacobsen [6] studied a perturbed renewal risk model with phase-type interclaim times and two-sided jumps, where both the jumps have rational Laplace transforms. Based on the roots of the Cramér-Lundberg equation, the joint Laplace transform on the time to ruin, and the undershoot at ruin were given. However, in all the aforemetioned papers, the topic that the jumps in both directions are arbitrary distributions is still not discussed. The Markov-modulated risk model (Markovian regime switching model) was first proposed by Asmussen [7] to extend the classical risk model. Since then, it has received remarkable attention in actuarial mathematics, see, for example, Zhu and Yang [8, 9], Zhang et al. [4], Ng and Yang [10], Li and Lu [11], Lu and Tsai [12], and references therein. Motivated by the papers mentioned above, in this paper, we will study the Markov-modulated risk model with two-sided jumps.

Let {J(t),t≥0} be a homogenous, irreducible, and recurrent Markov process with finite state space E={1,2,…,n}. Denote the intensity matrix of {J(t),t≥0} by A=(αij)i,j=1n with ∑j=1nαij=0 and αii:=-αi=-∑j≠inαij for i∈E. Let {Xi,i=1,2,…} be a sequence of independent random variables representing the jumps, and B(t) be a standard Brownian motion with B(0)=0. Here we assume that the premium rates, claim interarrival times, the distributions of the jumps, and the diffusion parameter are all influenced by the environment process {J(t),t≥0}. When J(t)=i, the premium rate is ci, jumps arrive according to a Poisson process with intensity λi, the diffusion parameter is σi>0, and the size of the jumps which arrives at time t follows the distribution Fi with density fi and finite mean μi. Then the Markov-modulated diffusion risk model {U(t),t≥0} is defined by
(1.1)U(t)=u+∫0tcJ(s)ds-∑j=1N(t)Xj+∫0tσJ(s)dB(s),
where u≥0 is the initial surplus. If we denote the stationary distribution of {J(t),t≥0} by π=(π1,π2,…,πn), then the positive security loading condition is given by
(1.2)∑i=1nπi(ci-λiμi)>0.

In this paper, we further assume that the jumps in (1.1) are two-sided. The upward jumps can be explained as the random income (premium or investment), while the downward jumps are interpreted as the random loss. In this case, the density function is given by
(1.3)fi(x)=pifi,d(x)I(x≥0)+qifi,u(-x)I(x<0),forJ(t)=i,i=1,2,…,n,
where 0<pi≤1, pi+qi=1, I(·) is the indicator function, fi,d and fi,u are two arbitrary functions on [0,∞).

Let T=inf{t≥0:U(t)≤0} (∞ otherwise) be the time to ruin. For δ≥0, let
(1.4)ϕi(u)=E[e-δTω(U(T-),|U(T)|)I(T<∞)∣J(0)=i,U(0)=u],u≥0,
be the Gerber-Shiu function at ruin given that the initial state is i, where ω(x1,x2) is a nonnegative penalty function, U(T-) is the surplus immediately prior to ruin, and |U(T)| is the deficit at ruin. Without loss of generality, we assume that ω(0,0)=1. Thus ϕi(0)=1 for i=1,2,…,n. When ω=1, (1.4) reduces to the Laplace transform of the time to ruin
(1.5)ψδ,i(u)=E[e-δTI(T<∞)∣J(0)=i,U(0)=u],u≥0,
when ω=1 and δ=0, (1.4) reduces to the probability of ruin
(1.6)ψi(u)=P(T<∞∣J(0)=i,U(0)=u),u≥0.

The purpose of this paper is to present some numerical results on the Gerber-Shiu function for the Markov-modulated diffusion risk model with arbitrary upward and downward jumps. In Section 2 we derive a system of integrodifferential equations and approximate solutions for ϕi(u). Numerical example is given in the last section.

2. Integrodifferential Equations and Approximate SolutionTheorem 2.1.

For u≥0, ϕi(u)(i=1,2,…,n) satisfies the following integrodifferential equation
(2.1)σi22ϕi′′(u)+ciϕi′(u)-(λi+δ)ϕi(u)+∑k=1nαikϕk(u)=-λi∫-∞∞ϕi(u-y)f(y)dy=-λipi∫0uϕi(u-y)fi,d(y)dy-λiqi∫0∞ϕi(u+y)fi,u(y)dy-ωi(u),
where
(2.2)ωi(u)=λipi∫u∞ω(u,y-u)fi,d(y)dy,
with boundary conditions
(2.3)ϕi(0)=1,ϕi(∞)=0.

Proof.

Similar to Ng and Yang [10].

Remark 2.2.

When n=1, (2.1) is identical to (3.2) in Zhang et al. [4].

Clearly, (2.1) is a system of second order linear integrodifferential equations of Fredholm-Volterra type. As is well known, it is very difficult to find analytical solution of this system. Motivated by Akyüz-Dascioglu [13], we will study an alternative system defined on [0,1] by Chebyshev collocation method. First, we transform the interval [0,∞) to [0,1]. Following Diko and Usábel [14], we set u=h(x), that is, h:[0,1]→[0,∞). Furthermore, we assume that h is an arbitrary strictly monotone, twice continuously differentiable function throughout the paper.

Theorem 2.3.

Let h(x) be a monotone increase function and χi(x)=ϕi(h(x)) for x∈[0,1]. Then χi(x) satisfies the following integrodifferential equation
(2.4)σi22(h′(x))2χi′′(x)+(cih′(x)-σi2h′′(x)2(h′(x))3)χi′(x)-(λi+δ)χi(x)+∑k=1nαikχk(x)+∫0xKi(x,t)χi(t)dt+∫01Li(x,t)χi(t)dt+Wi(x)=0,
where
(2.5)Ki(x,t)=λi(pifi,d(h(x)-h(t))-qifi,u(h(t)-h(x)))h′(t),Li(x,t)=λiqifi,u(h(t)-h(x))h′(t),Wi(x)=ωi(h(x)),
with boundary conditions
(2.6)χi(0)=1,χi(1)=0.

Proof.

By the definitions of function h and χi, we have
(2.7)pi∫0uϕi(u-y)fi,d(y)dy+qi∫0∞ϕi(u+y)fi,u(y)dy=qi∫01ϕi(h(t))fi,u(h(t)-h(x))h′(t)dt+∫0xϕi(h(t))(pifi,d(h(x)-h(t))-qifi,u(h(t)-h(x)))h′(t)dt=qi∫01χi(t)fi,u(h(t)-h(x))h′(t)dt+∫0xχi(t)[pifi,d(h(x)-h(t))-qifi,u(h(t)-h(x))]h′(t)dt.

Substituting (2.7) and χi(x)=ϕi(h(x)) into (2.1) and simplifying lead to (2.4). The boundary conditions are direct result of the boundary conditions in Theorem 2.1. This completes the proof.

Remark 2.4.

The existence of the solution for the system of integrodifferential equations (2.4) can be found in Fariborzi and Behzadi [15].

According to Akyüz-Dascioglu [13], χi(x) and its derivatives have truncated Chebyshev series expression
(2.8)χi(j)(x)=∑r=0Nair(j)Tr*(x),i=1,2,…,n,j=0,1,2,…,x∈[0,1],
where χi(0)(x)=χi(x), air(0)=air,Tr*(x) are shifted Chebyshev polynomials of the first kind and air(j) are the unknown coefficients to be determined.

Let T*(x)=(T1*(x),T2*(x),…,TN*(x))⊤, Ai=(ai0,ai1,…,aiN)⊤, Ai(j)=(ai0(j),ai1(j),…,aiN(j))⊤. Then (2.8) can be written in the matrix form
(2.9)χi(x)=T*(x)Ai,χi(j)(x)=T*(x)Ai(j)=4jTr*(x)MnAi(j),j=1,2,…,
where
(2.10)M=(012032052⋯N2002040⋯0000305⋯N⋮⋮⋮⋮⋮⋮⋱⋮000000⋯N000000⋯0)(N+1)×(N+1),
for odd N, and
(2.11)M=(012032052⋯0002040⋯N000305⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯0000000⋯N)(N+1)×(N+1),
for even N.

Similarly, the kernel functions Ki(x,t) and Li(x,t) can be expanded to univariate Chebyshev series
(2.12)Ki(x,t)=κi(x)Tr*(t),Li(x,t)=li(x)Tr*(t),
where
(2.13)κi(x)=(12κi,1(x),κi,2(x),…,12κi,n(x)),li(x)=(12ℓi,1(x),ℓi,2(x),…,12ℓi,n(x)),
with κi,r and ℓi,r are Chebyshev coefficients determined by Clenshaw and Curtis [16].

Theorem 2.5.

For 0≤x≤1, an approximate expression for χi(x) is given by
(2.14)χi(x)=T*(x)Ai,i=1,2,…,n,
where the column vector Ai can be determined by the following systems
(2.15)2σi2(h′(xj))2T*(xj)MAi+(cih′(xj)-σi2h′′(xj)2(h′(xj))3)T*(xj)Ai-(λi+δ)T*(xj)Ai+∑k=1nαikT*(xj)Ak+12κi(xj)Z(2xj-1)Ai+12li(xj)ZAi+Wi(xj)=0,T*(0)Ai=1,T*(1)Ai=0,
where matrix Z=(zij) with elements
(2.16)zij={11-(i-j)2+11-(i+j)2,foreveni+j,0,foroddi+j,
matrix Z(x)=(zij(x)) with elements
(2.17)zij(x)=14{2x2-2,i+j=1,Ti+j+1i+j+1-Ti+j-1i+j-1-1i+j+1+1i+j-1+x2-1,|i-j|=1,Ti+j+1i+j+1+T1-i-j1-i-j+T1+i-j1+i-j+T1-i+j1-i+j+2(11-(i+j)2+11-(i-j)2),foreveni+j,Ti+j+1i+j+1+T1-i-j1-i-j+T1+i-j1+i-j+T1-i+j1-i+j-2(11-(i+j)2+11-(i-j)2),foroddi+j
and xj(j=0,1,2,…,N-1) are collocations.

Proof.

Using (2.8) and (2.12), one obtains
(2.18)∫0xKi(x,t)χi(t)dt=∫0xκi(x)(T*(t))⊤T*(t)Aidt=12κi(x)Z(2x-1)Ai,∫01Li(x,t)χi(t)dt=∫01li(x)(T*(t))⊤T*(t)Aidt=12li(x)ZAi,
Substituting (2.18) into (2.4), we have
(2.19)2σi2(h′(x))2T*(x)MAi+(cih′(x)-σi2h′′(x)2(h′(x))3)T*(x)Ai-(λi+δ)T*(x)Ai+∑k=1nαikT*(x)Ak+12κi(x)Z(2x-1)Ai+12li(x)ZAi+Wi(x)=0,
which is identical to (2.15) in form. Substituting the collocations xj(j=0,1,…,N-1) into (2.19) leads to (2.15). T*(0)Ai=1 and T*(1)Ai=0 can be obtained by (2.6).

Example 2.6.

To illustration our method, we use the example of Zhang et al. [4]. Let n=1, c=2, σ2=2, λ=1, p1=0.6, q1=0.4, δ=0.3, the downward jumps are exponentially distributed with parameter 0.3, and the upward jump density is given by f1,u(x)=0.08e-0.4x+0.64e-0.8x. We set u=h(x)=-ln(1-x) and the collocation points are xj=((1+cos(sπ/N))/2)(i=1,2,…,N).

Figure 1 shows that the approximate solution is very near to the exact solution for any initial surplus u. We remark that the horizontal axis in Figure 1 is cos(sπ/N)(i=1,2,…,N) and u=-ln((1-cos(sπ/N))/2).

Laplace transform of the time to ruin.

From Table 1 we can see that the errors between the approximate solutions and the exact solutions decrease when N increases. The initial surplus u can also influence the approximate solution: the bigger u need a bigger N to decrease the error.

Laplace transform of the time to ruin.

u

n=100

n=200

n=300

n=400

n=500

exact

1

0.4656

0.46863

0.47008

0.47107

0.47177

0.4718

2

0.36228

0.36702

0.36918

0.37053

0.37149

0.3720

3

0.30111

0.30774

0.31072

0.31251

0.31371

0.3152

4

0.24968

0.25892

0.26227

0.26427

0.26594

0.2690

5

0.2054

0.21521

0.22034

0.22272

0.22445

0.2297

6

0.16376

0.17695

0.18356

0.18607

0.18856

0.1961

7

0.12624

0.1439

0.15074

0.15459

0.15713

0.1675

8

0.082052

0.11061

0.12152

0.12482

0.129

0.1430

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong (no. ZR2010AQ015), the Tianyuan fund for Mathematics (no. 11226251), and the Natural Science Foundation of Qufu Normal University (no. 2012ZRB01473).

BoucherieR. J.BoxmaO. J.SigmanK.A note on negative customers, GI/G/1 workload, and risk processesKouS. G.WangH.First passage times of a jump diffusion processXingX.ZhangW.JiangY.On the time to ruin and the deficit at ruin in a risk model with double-sided jumpsZhangZ.YangH.LiS.The perturbed compound Poisson risk model with two-sided jumpsChiY.Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in financeJacobsenM.The time to ruin for a class of Markov additive risk process with two-sided jumpsAsmussenS.Risk theory in a Markovian environmentZhuJ.YangH.Ruin theory for a Markov regime-switching model under a threshold dividend strategyZhuJ.YangH.On differentiability of ruin functions under Markov-modulated modelsNgA. C. Y.YangH.On the joint distribution of surplus before and after ruin under a Markovian regime switching modelLiS.LuY.The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk modelLuY.TsaiC. C. L.The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusionAkyüz-DasciogluA.A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general formDikoP.UsábelM.A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion processFariborzi AraghiM. A.BehzadiS. S.Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using homotopy analysis methodClenshawC. W.CurtisA. R.A method for numerical integration on an automatic computer