Hostname: page-component-7f64f4797f-rcs6z Total loading time: 0 Render date: 2025-11-04T00:51:08.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Gerber-Shiu analysis in the compound Poisson model with constant inter-observation times

Published online by Cambridge University Press:  06 June 2022

Jiayi Xie ,
Wenguang Yu ,
Zhimin Zhang Open the ORCID record for Zhimin Zhang [Opens in a new window]  and
Zhenyu Cui Open the ORCID record for Zhenyu Cui [Opens in a new window]
Jiayi Xie
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China. E-mail: zmzhang@cqu.edu.cn
Wenguang Yu
Affiliation:
School of Insurance, Shandong University of Finance and Economics, Jinan 250014, P.R. China.
Zhimin Zhang
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China. E-mail: zmzhang@cqu.edu.cn
Zhenyu Cui
Affiliation:
School of Business, Stevens Institute of Technology, Hoboken, NJ, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

Keywords

Compound Poisson modelGerber-Shiu functionLaguerre series expansionPeriodic observation

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Special Issue 2: Probability and stochastic modeling in actuarial science and related fields , April 2023 , pp. 324 - 356
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. & Stegun, I.A. (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Applied mathematics series 55. New York: Dover Publications.Google Scholar
Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70(1): 223–262.CrossRefGoogle Scholar
Albrecher, H., Cheung, E.C.K., & Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model: Dividend. ASTIN Bulletin 41(2): 645–672.Google Scholar
Albrecher, H., Cheung, E.C.K., & Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: The discounted penalty function. Scandinavian Actuarial Journal 2013(6): 424–452.CrossRefGoogle Scholar
Asmussen, S. & Albrecher, H. (2010). Ruin probabilities, 2nd ed. New Jersey: World Scientific.CrossRefGoogle Scholar
Asmussen, S., Avram, F., & Usabel, M. (2002). Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bulletin 32(2): 267–291.CrossRefGoogle Scholar
Avanzi, B., Cheung, E.C.K., Wong, B., & Woo, J.-K. (2013). On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insurance: Mathematics and Economics 52(1): 98–113.Google Scholar
Bongioanni, B. & Torrea, J.L. (2009). What is a Sobolev space for the Laguerre function systems?. Studia Mathematica 192(2): 147–172.CrossRefGoogle Scholar
Cheung, E.C.K. & Zhang, Z. (2021). Simple approximation for the ruin probability in renewal risk model under interest force via Laguerre series expansion. Scandinavian Actuarial Journal. 2021(9): 804–831.CrossRefGoogle Scholar
Comte, F. & Genon-Catalot, V. (2015). Adaptive Laguerre density estimation for mixed Poisson models. Electronic Journal of Statistics 9(1): 1113–1149.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): 259–279.CrossRefGoogle Scholar
Dickson, D.C.M. & Li, S. (2010). Finite time ruin problems for Erlang(2) risk models. Insurance: Mathematics and Economics 46(1): 12–18.Google Scholar
Dickson, D.C.M. & Willmot, G.E. (2005). The density of the time to ruin in the classical Poisson risk model. ASTIN Bulletin 35(1): 45–60.CrossRefGoogle Scholar
Dupuis, P. & Wang, H. (2002). Optimal stopping with random intervention times. Advances in Applied Probability 34(1): 141–157.CrossRefGoogle Scholar
Garcia, J.M.A. (2005). Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin 35(1): 113–130.CrossRefGoogle Scholar
Gerber, H.U. & Shiu, E.S.W. (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72.CrossRefGoogle Scholar
Kuznetsov, A. & Morales, M. (2014). Computing the finite-time expected discounted penalty function for a family of Lévy risk processes. European Actuarial Journal 2014(1): 1–31.CrossRefGoogle Scholar
Lee, W.Y., Li, X., Liu, F., Shi, Y., & Yam, S.C.P. (2021). A Fourier-cosine method for finite-time ruin probabilities. Insurance: Mathematics and Economics 99: 256–267.Google Scholar
Li, S. & Lu, Y. (2017). Distributional study of finite-time ruin related problems for the classical risk model. Applied Mathematics and Computation 315: 319–330.CrossRefGoogle Scholar
Li, S. & Sendova, K.P. (2013). Finite-time ruin probability for the compound binomial risk model. European Actuarial Journal 3: 249–271.CrossRefGoogle Scholar
Li, S., Lu, Y., & Sendova, K.P. (2019). The expected discounted penalty function: From infinite time to finite time. Scandinavian Actuarial Journal 2019(4): 336–354.CrossRefGoogle Scholar
Li, X., Shi, Y., Yam, S.C.P., & Yang, H. (2021). Fourier-Cosine method for finite-time Gerber–Shiu functions. SIAM Journal on Scientific Computing 43(3): B650–B677.CrossRefGoogle Scholar
Lian, G., Zhu, S.P., Elliott, R.J., & Cui, Z. (2017). Semi-analytical valuation for discrete barrier options under time-dependent Lévy processes. Journal of Banking and Finance 75: 167–183.CrossRefGoogle Scholar
Prahbu, H. (1961). On the ruin problem of collective risk theory. The Annals of Mathematical Statistics 32(3): 757–764.Google Scholar
Ramsay, C.M. (2003). A solution to the ruin problem for Pareto distributions. Insurance: Mathematics and Economics 33(1): 106–119.Google Scholar
Seal, H.L. (1974). The numerical calculation of $U(w, t)$, the probability of non-ruin in an interval $(0, t)$. Scandinavian Actuarial Journal 1974(3): 121–139.CrossRefGoogle Scholar
Shimizu, Y. & Zhang, Z. (2019). Asymptotically normal estimators of the ruin probability for Lévy insurance surplus from discrete samples. Risks 7(2): 37.CrossRefGoogle Scholar
Stenger, F. (1993). Numerical methods based on Sinc and analytic functions. Springer Series in Computational Mathematics, Vol. 20. New York: Springer-Verlag.CrossRefGoogle Scholar
Xie, J. & Zhang, Z. (2021). Finite-time dividend problems in a Lévy risk model under periodic observation. Applied Mathematics and Computation 398(1): 125981.CrossRefGoogle Scholar
Xie, J. & Zhang, Z. (2021). Recursive approximating to the finite-time Gerber-Shiu function in Lévy risk models under periodic observation. Journal of Computational and Applied Mathematics 399: 113703.CrossRefGoogle Scholar
Zhang, Z. (2014). On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization 10(4): 1041–1058.CrossRefGoogle Scholar
Zhang, Z. & Cheung, E.C.K. (2016). The Markov additive risk process under an Erlangized dividend barrier strategy. Methodology and Computing in Applied Probability 18: 275–306.CrossRefGoogle Scholar
Zhang, Z. & Cheung, E.C.K. (2018). A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial and Management Optimization 14(1): 35–63.CrossRefGoogle Scholar
Zhang, Z. & Su, W. (2018). A new efficient method for estimating the Gerber-Shiu function in the classical risk model. Scandinavian Actuarial Journal 2018(5): 426–449.CrossRefGoogle Scholar
Zhang, Z. & Su, W. (2019). Estimating the Gerber-Shiu function in a Lévy risk model by Laguerre series expansion. Journal of Computational and Applied Mathematics 346: 133–149.CrossRefGoogle Scholar
Zhang, Z., Cheung, E.C.K., & Yang, H. (2017). Lévy insurance risk process with Poissonian taxation. Scandinavian Actuarial Journal 2017(1): 51–87.CrossRefGoogle Scholar
Zhang, Z., Cheung, E.C.K., & Yang, H. (2018). On the compound poisson risk model with periodic capital injections. ASTIN Bulletin 48(1): 435–477.CrossRefGoogle Scholar