## Abstract and Applied Analysis

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

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 401562, 9 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365168877

Digital Object Identifier
doi:10.1155/2012/401562

Mathematical Reviews number (MathSciNet)
MR3004923

Zentralblatt MATH identifier
1264.91137

#### Citation

Dong, Hua; Zhao, Xianghua. Numerical Method for a Markov-Modulated Risk Model with Two-Sided Jumps. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 401562, 9 pages. doi:10.1155/2012/401562. https://projecteuclid.org/euclid.aaa/1365168877

#### References

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• S. Asmussen, “Risk theory in a Markovian environment,” Scandinavian Actuarial Journal, no. 2, pp. 69–100, 1989.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numerische Mathematik, vol. 2, pp. 197–205, 1960.