**Commenced**in January 2007

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**Edition:**International

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##### A Hyperexponential Approximation to Finite-Time and Infinite-Time Ruin Probabilities of Compound Poisson Processes

**Authors:**
Amir T. Payandeh Najafabadi

**Abstract:**

**Keywords:**
Ruin probability,
compound Poisson processes,
mixture exponential (hyperexponential) distribution,
heavy-tailed
distributions.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1339712

**References:**

[1] Albrecher, H., Teugels, J. L., & Tichy, R. F. (2001). On a gamma series expansion for the time-dependent probability of collective ruin. Insurance: Mathematics and Economics, 29(3), 345–355.

[2] Avram, F., Chedom, D. F., & Horváth, A. (2011). On moments based Padé approximations of ruin probabilities. Journal of computational and applied mathematics, 235(10), 3215–3228.

[3] Asmussen, S., & Albrecher, H. (2010). Ruin probabilities (Vol. 14). World Scientific, New Jersey.

[4] Belding, D. F., & Mitchell, K. J. (2008). Foundations of analysis (Vol. 79). Courier Corporation, New York.

[5] Bilik, A. (2008). Heavy-tail central limit theorem. Lecture notes from University of California, San Diego.

[6] Champeney, D. C. (1987). A handbook of Fourier theorems. Cambridge University press, New York.

[7] Feldmann, A., & Whitt, W. (1998). Fitting mixtures of exponentials to long–tail distributions to analyze network performance models. Performance evaluation, 31(3-4), 245–279.

[8] Feller, W. (1971). An introduction to probability and its applications, Vol. II. Wiley, New York.

[9] Jewell, N. P. (1982). Mixtures of Exponential Distributions. The Annals of Statistics, 10(2), 479–484.

[10] Kaas, R., Goovaerts, M., Dhaene, J., & Denuit, M. (2008). Modern actuarial risk theory: using R (Vol. 128). Springer-Verlag, Berlin Heidelberg.

[11] Kucerovsky, D., & Payandeh Najafabadi, A. T. (2009). An approximation for a subclass of the Riemann-Hilbert problems. IMA journal of applied mathematics, 74(4), 533–547.

[12] Kucerovsky, D. Z. & Payandeh Najafabadi, A. T. (2015). Solving an integral equation arising from the Ruin probability of long-term Bonus–Malus systems. Preprinted.

[13] Lukacs, E. (1987). Characteristic Functions. Oxford University Press, New York.

[14] Makroglou, A. (2003). Integral equations and actuarial risk management: some models and numerics. Mathematical Modelling and Analysis, 8(2), 143–154.

[15] Melnikov, A. (2011). Risk analysis in finance and insurance. CRC Press.

[16] Merzon, A., & Sadov, S. (2011). Hausdorff-Young type theorems for the Laplace transform restricted to a ray or to a curve in the complex plane. arXiv preprint arXiv:1109.6085.

[17] Pandey, J. (1996). The Hilbert transform of Schwartz distributions and application. John Wiley & Sons, New York.

[18] Pervozvansky, A. A. (1998). Equation for survival probability in a finite time interval in case of non-zero real interest force. Insurance: Mathematics and Economics, 23(3), 287–295.

[19] Qin, L., & Pitts, S. M. (2012). Nonparametric estimation of the finite-time survival probability with zero initial capital in the classical risk model. Methodology and Computing in Applied Probability, 14(4), 919-936.

[20] Rachev, S. T. (Ed.). (2003). Handbook of Heavy Tailed Distributions in Finance: Handbooks in Finance (Vol. 1). Elsevier, Netherlands.

[21] Sadov, S. Y. (2013). Characterization of Carleson measures by the Hausdorff-Young property. Mathematical Notes, 94(3-4), 551-558.

[22] Schmidli, H (2006). Lecture notes on Risk Theory. Institute of Mathematics University of Cologne, German.

[23] Sylvia, F. S. (2008). Finite mixture and Markov switching models: Implementation in MATLAB using the package bayesf Version 2.0. Springer-Verlag, New York.

[24] Tran, D. X. (2015). Padé approximants for finite time ruin probabilities. Journal of Computational and Applied Mathematics, 278, 130-137.

[25] Walnut, D. F. (2002). An introduction to wavelet analysis. 2nd ed. Birkhäuser publisher, New York.