An improved optimal trigonometric ELM algorithm for numerical solution to ruin probability of Erlang(2) risk model
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An improved optimal trigonometric ELM algorithm for numerical solution to ruin probability of Erlang(2) risk model Yang-Jin Cheng 1 & Muzhou Hou 2
& Juan Wang
3
Received: 21 September 2019 / Revised: 1 July 2020 / Accepted: 16 July 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
In this paper, we focus on accurately calculating the numerical solution of the integral-differential equation for ruin probability in Erlang(2) renewal risk model with arbitrary claim distribution. Because the analytical solutions of the equation do not usually exist, firstly, using machine learning method in modern artificial intelligence, the activation functions in the ELM model are changed to trigonometric function, the initial conditions in the integral-differential equation are added to the ELM linear solver to get the ITELM model, and the steps and feasibility of the algorithm are strictly deduced in theory. As the analytic solution for the integral-differential equation only exists when the claim is subject to exponential distribution, and the numerical solution can be gotten with the pareto distribution. And, since the number of hidden neurons in the ITELM model is uncertain, a good numerical value of hidden neurons can only be determined through a large number of iterative tests and comparisons in the actual calculation. Then, we construct a multi-objective optimization model and algorithm, which can get the optimal number of hidden neurons to obtain the IOTELM model and algorithm. Then, in the above two cases for exponential distribution and pareto distribution, the optimal number of hidden neurons is calculated by IOTELM model and algorithm, and then corresponding ITELM models and algorithms are constructed to calculate the corresponding ruin probability. Compared with the previous numerical experiments, it can be seen that the numerical accuracy is greatly improved, which verified the versatility, feasibility and superiority of the proposed IOTELM model and algorithm. Keywords Erlang(2) risk model . Ruin probability . Renewal integral-differential equation . IOTELM algorithm
* Muzhou Hou [email protected] Extended author information available on the last page of the article
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1 Introduction Risk theory is one of the important components of insurance mathematics, and is a hot research topic in the financial and mathematics circles [63]. The core issue of risk theory research is how much insurance companies may face bankruptcy. We use the ruin probability to estimate the magnitude of this possibility, so the ruin probability has not only become an important indicator to ensure the normal and smooth operation of insurance companies, but also a quantitative criterion for insurance companies to effectively control risks. In the classical risk model, the arrival of a claim is a Poisson process [6, 40]. Poisson process has many excellent features [8, 18, 52, 67, 81, 82], which bring a lot of conveniences for the study of classical risk model [14], but it als
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