Gerber-Shiu function is the joint distribution of the time to ruin, the surplus before ruin and the deficit at ruin. In this paper, we propose a non-parametric estimator of the expected discounted penalty function; the Gerber-Shiu function for a compound Poisson risk model perturbed by diffusion is also called the Wiener-Poisson risk model. The estimator is based on the Fourier cosine series expansion method. It shows that our estimator has a fast convergence rate. We also derived some simulation examples to show the effectiveness of the estimator under a finite sample.
Lundberg, F. (1903) 1. Approximerad framställning af sannolikhetsfunktionen: 2. Återförsäkring af kollektivrisker. Almqvist &Wiksells, Stockholm.
Lundberg, F. and Riskutjämning, F. (1926) I: Teori. II: Statistik (Insurance Technical Smoothing of Risks) F. Englunds Boktryckeri AB, Stockholm.
Constantinescu, C., Samorodnitsky, G. and Zhu, W. (2018) Ruin Probabilities in Classical Risk Models with γ Claims. Scandinavian Actuarial Journal Taylor & Francis, 2018, 555-575. https://doi.org/10.1080/03461238.2017.1402817
Gerber, H.U. and Shiu, E.S.W. (1998) On the Time Value of Ruin. North American Actuarial Journal, 2, 48-72. https://doi.org/10.1080/10920277.1998.10595671
Mnatsakanov, R., Ruymgaart, L.L. and Ruymgaart, F.H. (2008) Nonparametric Estimation of Ruin Probabilities Given a Random Sample of Claims. Mathematical Methods of Statistics, 17, 35-43. https://doi.org/10.3103/S1066530708010031
He, Y., Kawai, R., Shimizu, Y. and Yamazaki, K. (2022) The Gerber-Shiu Discounted Penalty Function: A Review from Practical Perspectives. Insurance: Mathematics and Economics, 109, 1-28. https://doi.org/10.1016/j.insmatheco.2022.12.003
Wang, W.Y. and Zhang, Z.M. (2019) Computing the Gerber-Shiu Function by Frame Duality Projection. Scandinavian Actuarial Journal, 2019, 291-307. https://doi.org/10.1080/03461238.2018.1557739
Xie, J.Y. and Zhang, Z.M. (2022) Recursive Approximating to the Finite-Time Gerber-Shiu Function in Lévy Risk Models under Periodic Observation. Journal of Computational and Applied Mathematics, 399, Article ID: 113703. https://doi.org/10.1016/j.cam.2021.113703
Li, X.L., Shi, Y.F., Phillip, Y., Sheung, C. and Yang, H.L. (2021) Fourier-Cosine Method for Finite-Time Gerber-Shiu Functions. Scandinavian Actuarial Journal, 43, B650-B677. https://doi.org/10.1137/20M1328580
Gerber, H.U. (1970) An Extension of the Renewal Equation and Its Application in the Collective Theory of Risk. Scandinavian Actuarial Journal, 1970, 205-210. https://doi.org/10.1080/03461238.1970.10405664
Dufresne, F. and Gerber, H.U. (1991) Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion. Insurance: Mathematics and Economics, 10, 51-59. https://doi.org/10.1016/0167-6687(91)90023-Q
Cai, J. and Yang, H.L. (2005) Ruin in the Perturbed Compound Poisson Risk Process under Interest Force. Advances in Applied Probability, 37, 819-835. https://doi.org/10.1239/aap/1127483749
Li, S.M. (2006) The Distribution of the Dividend Payments in the Compound Poisson Risk Model Perturbed by Diffusion. Scandinavian Actuarial Journal, 2006, 73-85. https://doi.org/10.1080/03461230600589237
Yin, C.C. and Wang, C.W. (2010) The Perturbed Compound Poisson Risk Process with Investment and Debit Interest. Methodology and Computing in Applied Probability, 12, 391-413. https://doi.org/10.1007/s11009-008-9109-z
Zhang, Z.M., Yang, H. and Li, S.M. (2010) The Perturbed Compound Poisson Risk Model with Two-Sided Jumps. Journal of Computational and Applied Mathematics, 233, 1773-1784. https://doi.org/10.1016/j.cam.2009.09.014
Tsai, C.C.L. (2001) On the Discounted Distribution Functions of the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 28, 401-419. https://doi.org/10.1016/S0167-6687(01)00067-1
Tsai, C.C.L. and Willmot, G.E. (2002) A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 30, 51-66. https://doi.org/10.1016/S0167-6687(01)00096-8
Zhang, Z.M. and Yang, H. (2011) Gerber-Shiu Analysis in a Perturbed Risk Model with Dependence between Claim Sizes and Interclaim Times. Journal of Computational and Applied Mathematics, 235, 1189-1204. https://doi.org/10.1016/j.cam.2010.08.003
Masiello, E. (2014) On Semiparametric Estimation of Ruin Probabilities in the Classical Risk Model. Scandinavian Actuarial Journal, 2014, 283-308. https://doi.org/10.1080/03461238.2012.690247
Zhang, Z.M., Yang, H.L. and Yang, H. (2014) On a Nonparametric Estimator for Ruin Probability in the Classical Risk Model. Scandinavian Actuarial Journal, 2014, 309-338. https://doi.org/10.1080/03461238.2012.691427
Zhang, Z.M. (2017) Nonparametric Estimation of the Finite Time Ruin Probability in the Classical Risk Model. Scandinavian Actuarial Journal, 2017, 452-469. https://doi.org/10.1080/03461238.2016.1174876
Shimizu, Y. (2012) Non-Parametric Estimation of the Gerber-Shiu Function for the Wiener—Poisson Risk Model. Scandinavian Actuarial Journal, 2012, 56-69. https://doi.org/10.1080/03461238.2010.523515
Zhang, Z.M. (2017) Estimating the Gerber-Shiu function by Fourier-Sinc Series Expansion. Scandinavian Actuarial Journal, 2017, 898-919. https://doi.org/10.1080/03461238.2016.1268541
Su, W., Yong, Y.D. and Zhang, Z.M. (2019) Estimating the Gerber-Shiu Function in the Perturbed Compound Poisson Model by Laguerre Series Expansion. Journal of Mathematical Analysis and Applications, 469, 705-729. https://doi.org/10.1016/j.jmaa.2018.09.033
Su, W. and Wang, Y.Y. (2021) Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion. Mathematics, 9, Article 1402. https://doi.org/10.3390/math9121402
Zhang, Z.M. (2017) Approximating the Density of the Time to Ruin via Fourier-Cosine Series Expansion. ASTIN Bulletin: The Journal of the IAA, 47, 169-198. https://doi.org/10.1017/asb.2016.27
Huang, Y.J., Yu, W.G., Pan, Y. and Cui, C.R. (2019) Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model. Discrete Dynamics in Nature and Society, 2019, Article ID: 3607201. https://doi.org/10.1155/2019/3607201
Xie, J.Y. and Zhang, Z.M. (2020) Statistical Estimation for Some Dividend Problems under the Compound Poisson Risk Model. Insurance: Mathematics and Economics, 95, 101-115. https://doi.org/10.1016/j.insmatheco.2020.09.002
Morales, M. (2007) On the Expected Discounted Penalty Function for a Perturbed Risk Process Driven by a Subordinator. Insurance: Mathematics and Economics, 40, 293-301. https://doi.org/10.1016/j.insmatheco.2006.04.008
Shimizu, Y. (2011) Estimation of the Expected Discounted Penalty Function for Lévy Insurance Risks. Mathematical Methods of Statistics, 20, 125-149. https://doi.org/10.3103/S1066530711020037
Yang, Y., Xie, J.Y. and Zhang, Z.M. (2023) Nonparametric Estimation of Some Dividend Problems in the Perturbed Compound Poisson Model. Probability in the Engineering and Informational Sciences, 37, 418-441. https://doi.org/10.1017/S0269964822000298
Fang, F. and Oosterlee, C.W. (2009) A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM Journal on Scientific Computing, 31, 826-848. https://doi.org/10.1137/080718061