NUMERICAL METHODS APPLIED TO RUIN PROBABILITY IN AN ERLANG(2) RISK PROCESS WITH WEIBULL LOSS DISTRIBUTION
Keywords:
finite difference method, Laplace transform, ruin probability, trapezoid rule, Weibull distributionDOI:
https://doi.org/10.17654/0975045224008Abstract
In the Erlang(2) risk process, ruin occurs when the surplus falls below zero, indicating potential bankruptcy for the insurance company. Predicting bankruptcy is crucial, and ruin probability serves as a key metric for this purpose. It involves solving an Integral-Differential equation derived from the Erlang(2) risk process. This study focuses on computing ruin probability under the assumption of a Weibull claim distribution. The analysis is divided into two scenarios based on the Weibull parameter: when $\alpha$ equals 1 and when it differs. While the Laplace transform offers an analytical solution for the Integral-Differential equation, its applicability diminishes when faced with an improper integral that defies analytical resolution. Consequently, for greater than 1, finite difference method is employed to obtain a numerical solution. The accuracy of this numerical approach is verified by comparing it against the analytical solution when equals 1. Subsequently, leveraging the accuracy established in the first scenario, the finite difference method is applied to compute the numerical solution for the differing scenario. The numerical method is satisfactory when calculated from $u = 0$ to $u = 100$, matching the analytical solution.
Received: April 12, 2024
Accepted: May 27, 2024
References
D. C. M. Dickson and C. Hipp, On the time to ruin for Erlang(2) risk processes, Insurance Mathematics and Economics 29(3) (2001), 333-344.
D. C. M. Dickson and C Hipp, Ruin probabilities for Erlang(2) risk processes, Insurance: Mathematics and Economics 22 (1998), 251-262.
D. A. Hamzah and R. Febrianti, Solvability conditions of integro-differential equation on classical risk models with exponential claims via Laplace transform, Jurnal Rekayasa Teknologi Dan Sains Terapan 4(2) (2023), 14-19.
D. A. Hamzah, T. S. A. Siahaan and V. C. Pranata, Ruin probability in the classical risk process with Weibull claims distribution, BAREKENG: J. Math. and App. 17(4) (2023), 2351-2358.
M. Goovaerts and F. De Vylder, A stable recursive algorithm for evaluation of ultimate ruin probabilities, ASTIN Bulletin 14(1) (1984).
N. K. Boots and P. Shahabuddin, Simulating ruin probabilities in insurance risk processes with subexponential claims, Winter Simulation Conference Proceedings, Vol. 1, 2001.
C. Constantinescu, G. Samorodnitsky and W. Zhu, Ruin probabilities in classical risk models with gamma claims, Scand. Actuar. J. 2018(7) (2018), 555-575.
doi: 10.1080/03461238.2017.1402817.
P. O. Goffard, S. Loisel and D. Pommeret, A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model, J. Comput. Appl. Math. 296 (2016).
J. M. Sánchez and F. Baltazar-Larios, Approximations of the ultimate ruin probability in the classical risk model using the banach’s fixed-point theorem and the continuity of the ruin probability, Kybernetika 58(2) (2022), 254-276.
doi: 10.14736/kyb-2022-2-0254.
D. J. Santana and L. Rincón, Approximations of the ruin probability in a discrete time risk model, Modern Stochastics: Theory and Applications 7(3) (2020), 221-243. doi: 10.15559/20-VMSTA158.
F. Dufresne and H. U. Gerber, Three methods to calculate the probability of ruin, ASTIN Bulletin 19(1) (1989), 71-90. doi: 10.2143/AST.19.1.2014916.
K. W. Chau, S. C. P. Yam and H. Yang, Fourier-cosine method for ruin probabilities, J. Comput. Appl. Math. 281 (2015), 94-106.
doi: 10.1016/j.cam.2014.12.014.
Z. G. Ignatov and V. K. Kaishev, A finite-time ruin probability formula for continuous claim severities, J. Appl. Probab. 41(02) (2004).
doi: 10.1017/s0021900200014510.
H. You, J. Guo and J. Jiang, Interval estimation of the ruin probability in the classical compound Poisson risk model, Comput. Stat. Data Anal. 144 (2020).
doi: 10.1016/j.csda.2019.106890.
D. C. M. Dickson and H. R. Waters, The probability and severity of ruin in finite and infinite time, ASTIN Bulletin 22(2) (1992), 191-207.
M. A. Diasparra and R. Romera, Bounds for the ruin probability of a discrete-time risk process, J. Appl. Probab. 46(1) (2009). doi: 10.1239/jap/1238592119.
J. Das and D. C. Nath, Weighted quantile regression theory and its application Weibull distribution as an actuarial risk model: Computation of its probability of ultimate ruin and the moments of the time to ruin, deficit at ruin and surplus prior to ruin, Journal of Data Science 17(1) (2021).
doi: 10.6339/jds.201901_17(1).0008.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pushpa Publishing House for more info or permissions.
Publication count: 
Citation count: 
Google h-index: 
Downloads: 