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APPROXIMATING THE DENSITY OF THE TIME TO RUIN VIA FOURIER-COSINE SERIES EXPANSION

Published online by Cambridge University Press:  19 September 2016

Zhimin Zhang
Zhimin Zhang*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China
*
E-Mail: zmzhang@cqu.edu.cn
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Abstract

In this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.

Keywords

Density of the time to ruinCOSFourier transformRuin
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 1 , January 2017 , pp. 169 - 198
Copyright
Copyright © Astin Bulletin 2016 

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References

Borovkov, K.A. and Dickson, D.C.M. (2008) On the ruin time distribution for a Sparre Andersen process with exponential claim sizes. Insurance: Mathematics and Economics, 42 (3), 1104–1108.Google Scholar
Chau, K.W., Yam, S.C.P. and Yang, H. (2015a) Modern Fourier-cosine method for Gerber-Shiu function. Insurance: Mathematics and Economics, 61, 170–180.Google Scholar
Chau, K.W., Yam, S.C.P. and Yang, H. (2015b) Modern Fourier-cosine method for ruin probabilities. Journal of Computational and Applied Mathematics, 281, 94–106.Google Scholar
Delbaen, F. (1990) A remark on the moments of ruin time in classical risk theory. Insurance: Mathematics and Economics, 9, 121–126.Google Scholar
Dickson, D.C.M. (2007) Some finite time ruin problems. Annals of Actuarial Science, 2, 217–232.CrossRefGoogle Scholar
Dickson, D.C.M. (2008) Some explicit solutions for the joint density of the time of ruin and the deficit at ruin. ASTIN Bulletin, 38 (1), 259–276.Google Scholar
Dickson, D.C.M., Hughes, B.D. and Zhang, L. (2005) The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scandinavian Actuarial Journal, 2005 (5), 358–376.CrossRefGoogle Scholar
Dickson, D.C.M. and Li, S. (2010) Finite time ruin problems for the Erlang(2) risk model. Insurance: Mathematics and Economics, 46 (1), 12–18.Google Scholar
Dickson, D.C.M. and Li, S. (2012) Erlang risk models and finite time ruin problems. Scandinavian Actuarial Journal, 3, 183–202.CrossRefGoogle Scholar
Dickson, D.C.M. and Willmot, G.E. (2005) The density of the time to ruin in the classical Poisson risk model. ASTIN Bulletin, 35 (1), 45–60.Google Scholar
Drekic, S. and Willmot, G.E. (2003) On the density and moments of the time of ruin with exponential claims. ASTIN Bulletin, 33, 11–21.CrossRefGoogle Scholar
Fang, F. and Oosterlee, C.W. (2008) A novel option pricing method based on Fourier cosine series expansions. SIAM Journal on Scientific Computing, 31 (2), 826–848.CrossRefGoogle Scholar
Fang, F. and Oosterlee, C.W. (2009) Pricing early-exercise and discrete barrier options by Fourier cosine series expansions. Numerische Mathematik, 114, 27–62.Google Scholar
Garcia, J.M.A. (2005) Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin, 35 (1), 113–130.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin; with discussion and a reply by the authors. North American Actuarial Journal, 2 (1), 48–78.Google Scholar
Landriault, D., Shi, T. and Willmot, G.E. (2011) Joint densities involving the time to ruin in the Sparre-Andersen risk model with exponential claim sizes. Insurance: Mathematics and Economics, 49, 371–379.Google Scholar
Lee, W.Y. and Willmot, G.E. (2014) On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times. Insurance: Mathematics and Economics, 59, 1–10.Google Scholar
Lee, W.Y. and Willmot, G.E. (2016) The moments of the time to ruin in dependent Sparre Andersen models with Coxian claim sizes. Scandinavian Actuarial Journal, 2016 (6), 550–564.Google Scholar
Lin, X.S. and Willmot, G.E. (2000) The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 27 (1), 19–44.Google Scholar
Meng, Q.-J. and Ding, D. (2013) An efficient pricing method for rainbow options based on two-dimensional modified sine-sine series expansions. International Journal of Computer Mathematics, 90 (5), 1096–1113.Google Scholar
Pitts, S.M. and Politis, K. (2008) Approximations for the moments of ruin time in the compound Poisson model. Insurance: Mathematics and Economics, 42, 668–679.Google Scholar
Pivato, M. (2010) Linear Partial Differential Equations and Fourier Theory. New York: Cambridge University Press.CrossRefGoogle Scholar
Ruijter, M.J. and Oosterlee, C.W. (2012) Two-dimensional Fourier cosine series expansion method for pricing financial options. SIAM Journal on Scientific Computing, 34, 642–671.CrossRefGoogle Scholar
Shi, T. and Landriault, D. (2013) Distribution of the time to ruin is some Sparre Andersen risk models. ASTIN Bulletin, 43 (1), 39–59.Google Scholar
Willmot, G.E. and Woo, J.-K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 2007.Google Scholar
Zhang, B. and Oosterlee, C.W. (2013) Efficient pricing of European-style Asian options under exponential Lévy processes based on Fourier cosine expansions. SIAM Journal on Financial Mathematics, 4 (1), 399–426.Google Scholar
Zhang, B. and Oosterlee, C.W. (2014) Pricing of early-exercise Asian options under Lévy processes based on Fourier cosine expansions. Applied Numerical Mathematics, 78, 14–30.CrossRefGoogle Scholar
Zhang, Z. and Yang, H. (2013) Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model. Insurance: Mathematics and Economics, 53, 24–35.Google Scholar
Zhang, Z. and Yang, H. (2014) Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation. Insurance: Mathematics and Economics, 59, 168–177.Google Scholar