Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network

Abstract

The ruin probability is used to determine the overall operating risk of an insurance company. Modeling risks through the characteristics of the historical data of an insurance business, such as premium income, dividends and reinvestments, can usually produce an integral differential equation that is satisfied by the ruin probability. However, the distribution function of the claim inter-arrival times is more complicated, which makes it difficult to find an analytical solution of the ruin probability. Therefore, based on the principles of artificial intelligence and machine learning, we propose a novel numerical method for solving the ruin probability equation. The initial asset u is used as the input vector and the ruin probability as the only output. A trigonometric exponential function is proposed as the projection mapping in the hidden layer, then a block trigonometric exponential neural network (BTENN) model with a symmetrical structure is established. Trial solution is set to meet the initial value condition, simultaneously, connection weights are optimized by solving a linear system using the extreme learning machine (ELM) algorithm. Three numerical experiments were carried out by Python. The results show that the BTENN model can obtain the approximate solution of the ruin probability under the classical risk model and the Erlang(2) risk model at any time point. Comparing with existing methods such as Legendre neural networks (LNN) and trigonometric neural networks (TNN), the proposed BTENN model has a higher stability and lower deviation, which proves that it is feasible and superior to use a BTENN model to estimate the ruin probability.

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine

“…Many neural network methods based on the improved extreme learning machine algorithm for solving ordinary differential equations (Yang et al, 2018 ; Lu et al, 2022 ), partial differential equations (Sun et al, 2019 ; Yang et al, 2020 ), the ruin probabilities of the classical risk model and the Erlang (2) risk model in Zhou et al ( 2019 ); Lu et al ( 2020 ), and one-dimensional asset-pricing (Ma et al, 2021 ) have been developed. Chen et al ( 2020 , 2021 , 2022 ) proposed the trigonometric exponential neural network, Laguerre neural network, and neural finite element method for ruin probability, generalized Black–Scholes differential equation, and generalized Black–Scholes–Merton differential equation. Inspired by these studies, the motivation of this research is to present the sine-cosine ELM (SC-ELM) algorithm to solve linear Volterra integral equations of the first kind, linear Volterra integral equations of the second kind, linear Fredholm integral equations of the first kind, linear Fredholm integral equations of the second kind, and linear Volterra–Fredholm integral equations.…”

MPC Based Trajectory Tracking for An Automonous Deep-Sea Tracked Mining Vehicle

“…The modern control theory, together with intelligent algorithms, is increasingly being used to solve engineering problems [12,13].…”