Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Numerical Method for a Markov-Modulated Risk Model with Two-Sided Jumps
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.
The risk model with two-sided jumps was first proposed by Boucherie et al.  and has been further investigated by many authors during the last few years. For example, Kou and Wang  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.  extended the results of Kou and Wang  to the case that the surplus process with phase-type downward and arbitrary upward jumps. Zhang et al.  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  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  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  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. , Ng and Yang , Li and Lu , Lu and Tsai , 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 be a homogenous, irreducible, and recurrent Markov process with finite state space . Denote the intensity matrix of by with and for . Let be a sequence of independent random variables representing the jumps, and be a standard Brownian motion with . 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 . When , the premium rate is , jumps arrive according to a Poisson process with intensity , the diffusion parameter is , and the size of the jumps which arrives at time follows the distribution with density and finite mean . Then the Markov-modulated diffusion risk model is defined by where is the initial surplus. If we denote the stationary distribution of by , 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 where , , is the indicator function, and are two arbitrary functions on .
Let ( otherwise) be the time to ruin. For , let be the Gerber-Shiu function at ruin given that the initial state is , where is a nonnegative penalty function, is the surplus immediately prior to ruin, and is the deficit at ruin. Without loss of generality, we assume that . Thus for . When , (1.4) reduces to the Laplace transform of the time to ruin when and , (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 . Numerical example is given in the last section.
2. Integrodifferential Equations and Approximate Solution
Theorem 2.1. For , satisfies the following integrodifferential equation where with boundary conditions
Proof. Similar to Ng and Yang .
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 , we will study an alternative system defined on by Chebyshev collocation method. First, we transform the interval to . Following Diko and Usábel , we set , that is, . Furthermore, we assume that is an arbitrary strictly monotone, twice continuously differentiable function throughout the paper.
Theorem 2.3. Let be a monotone increase function and for . Then satisfies the following integrodifferential equation where with boundary conditions
Proof. By the definitions of function and , we have
Substituting (2.7) and 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.
According to Akyüz-Dascioglu , and its derivatives have truncated Chebyshev series expression where , are shifted Chebyshev polynomials of the first kind and are the unknown coefficients to be determined.
Let , , . Then (2.8) can be written in the matrix form where for odd , and for even .
Similarly, the kernel functions and can be expanded to univariate Chebyshev series where with and are Chebyshev coefficients determined by Clenshaw and Curtis .
Theorem 2.5. For , an approximate expression for is given by where the column vector can be determined by the following systems where matrix with elements matrix with elements and are collocations.
Proof. Using (2.8) and (2.12), one obtains Substituting (2.18) into (2.4), we have which is identical to (2.15) in form. Substituting the collocations into (2.19) leads to (2.15). and can be obtained by (2.6).
Example 2.6. To illustration our method, we use the example of Zhang et al. . Let , , , , , , , the downward jumps are exponentially distributed with parameter , and the upward jump density is given by . We set and the collocation points are .
From Table 1 we can see that the errors between the approximate solutions and the exact solutions decrease when increases. The initial surplus can also influence the approximate solution: the bigger need a bigger to decrease the error.
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).
Y. Lu and C. C. L. Tsai, “The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion,” North American Actuarial Journal, vol. 11, no. 2, pp. 136–149, 2007.View at: Google Scholar