Numerical solving for generalized Black-Scholes-Merton model with neural finite element method

Chen, Yinghao; Lei, Wei; Cao, Shan; Liu, Fan; Cheng, Yangjin

doi:10.1016/j.dsp.2022.103757

Cited by 7 publications

(1 citation statement)

References 58 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang²,

Weng³

et al. 2023

Front. Comput. Neurosci.

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In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra–Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.

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