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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 401562, 9 pages
http://dx.doi.org/10.1155/2012/401562
Research Article

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

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 24 August 2012; Accepted 30 October 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Hua Dong and Xianghua Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 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 [10].

Remark 2.2. When , (2.1) is identical to 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 by Chebyshev collocation method. First, we transform the interval to . Following Diko and Usábel [14], 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.

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], 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 [16].

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. [4]. Let , , , , , , , the downward jumps are exponentially distributed with parameter , and the upward jump density is given by . We set and the collocation points are .

Figure 1 shows that the approximate solution is very near to the exact solution for any initial surplus . We remark that the horizontal axis in Figure 1 is and .

401562.fig.001
Figure 1: 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 increases. The initial surplus can also influence the approximate solution: the bigger need a bigger to decrease the error.

tab1
Table 1: Laplace transform of the time to ruin.

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).

References

  1. R. J. Boucherie, O. J. Boxma, and K. Sigman, “A note on negative customers, GI/G/1 workload, and risk processes,” Probability in the Engineering and Informational Sciences, vol. 11, no. 3, pp. 305–311, 1997. View at Publisher · View at Google Scholar
  2. S. G. Kou and H. Wang, “First passage times of a jump diffusion process,” Advances in Applied Probability, vol. 35, no. 2, pp. 504–531, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. X. Xing, W. Zhang, and Y. Jiang, “On the time to ruin and the deficit at ruin in a risk model with double-sided jumps,” Statistics & Probability Letters, vol. 78, no. 16, pp. 2692–2699, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Z. Zhang, H. Yang, and S. Li, “The perturbed compound Poisson risk model with two-sided jumps,” Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1773–1784, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Y. Chi, “Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance,” Insurance: Mathematics & Economics, vol. 46, no. 2, pp. 385–396, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. Jacobsen, “The time to ruin for a class of Markov additive risk process with two-sided jumps,” Advances in Applied Probability, vol. 37, no. 4, pp. 963–992, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. Asmussen, “Risk theory in a Markovian environment,” Scandinavian Actuarial Journal, no. 2, pp. 69–100, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. Zhu and H. Yang, “Ruin theory for a Markov regime-switching model under a threshold dividend strategy,” Insurance: Mathematics & Economics, vol. 42, no. 1, pp. 311–318, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Zhu and H. Yang, “On differentiability of ruin functions under Markov-modulated models,” Stochastic Processes and Their Applications, vol. 119, no. 5, pp. 1673–1695, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. C. Y. Ng and H. Yang, “On the joint distribution of surplus before and after ruin under a Markovian regime switching model,” Stochastic Processes and Their Applications, vol. 116, no. 2, pp. 244–266, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. Li and Y. Lu, “The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model,” ASTIN Bulletin, vol. 38, no. 1, pp. 53–71, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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.
  13. A. Akyüz-Dascioglu, “A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 103–112, 2007. View at Publisher · View at Google Scholar
  14. P. Diko and M. Usábel, “A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process,” Insurance: Mathematics & Economics, vol. 49, no. 1, pp. 126–131, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. A. Fariborzi Araghi and S. S. Behzadi, “Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using homotopy analysis method,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 1–12, 2011. View at Publisher · View at Google Scholar
  16. C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numerische Mathematik, vol. 2, pp. 197–205, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH