Numerical Method for a Markov-Modulated Risk Model with Two-Sided Jumps

and Applied Analysis 3 when ω 1 and δ 0, 1.4 reduces to the probability of ruin ψi u P T <∞ | J 0 i,U 0 u , u ≥ 0. 1.6 The purpose of this paper is to present some numerical results on the GerberShiu 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 Solution Theorem 2.1. For u ≥ 0, φi u i 1, 2, . . . , n satisfies the following integrodifferential equation σ2 i 2 φ′′ i u ciφ ′ i u − λi δ φi u n ∑


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  with n j 1 α ij 0 and α ii : . .} 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 c i , jumps arrive according to a Poisson process with intensity λ i , the diffusion parameter is σ i > 0, and the size of the jumps which arrives at time t follows the distribution F i with density f i and finite mean μ i . Then the Markov-modulated diffusion risk model {U t , t ≥ 0} is defined by where u ≥ 0 is the initial surplus. If we denote the stationary distribution of {J t , t ≥ 0} by π π 1 , π 2 , . . . , π n , then the positive security loading condition is given by 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 be the Gerber-Shiu function at ruin given that the initial state is i, where ω x 1 , x 2 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 φ i 0 1 for i 1, 2, . . . , n. When ω 1, 1.4 reduces to the Laplace transform of the time to ruin Abstract and Applied Analysis 3 when ω 1 and δ 0, 1.4 reduces to the probability of ruin 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 φ i u . Numerical example is given in the last section.

Integrodifferential Equations and Approximate Solution
. . , n satisfies the following integrodifferential equation with boundary conditions

2.3
Proof. Similar to Ng and Yang 10 . 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 χ i x φ i h x for x ∈ 0, 1 . Then χ i x satisfies the following integrodifferential equation

2.5
with boundary conditions

2.6
Proof. By the definitions of function h and χ i , we have  Let for even N. Similarly, the kernel functions K i x, t and L i x, t can be expanded to univariate Chebyshev series Abstract and Applied Analysis where with κ i,r and i,r are Chebyshev coefficients determined by Clenshaw and Curtis 16 .
Theorem 2.5. For 0 ≤ x ≤ 1, an approximate expression for χ i x is given by where the column vector A i can be determined by the following systems

2.15
where matrix Z z ij with elements for even i j, for odd i j
Proof. Using 2.8 and 2.12 , one obtains

2.18
Substituting 2.18 into 2.4 , we have   with parameter 0.3, and the upward jump density is given by f 1,u x 0.08e −0.4x 0.64e −0.8x . We set u h x − ln 1 − x and the collocation points are x j 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 .
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.