An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise

Xiao Qi; Tianyao Duan; Huan Guo; Xiao Qi; Tianyao Duan; Huan Guo

doi:10.3934/math.2024585

AIMS Mathematics

2024, Volume 9, Issue 5: 11975-11991. doi: 10.3934/math.2024585

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An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise

School of Artificial Intelligence, Jianghan University, 430056 Wuhan, China

Received: 16 January 2024 Revised: 12 March 2024 Accepted: 20 March 2024 Published: 27 March 2024
MSC : 62M45, 60H35, 82C32

This paper studied the numerical approximation of the stochastic differential equations driven by non-global Lipschitz drift coefficient and multiplicative noise. An efficient data-driven method, called extended continuous latent process flow, was proposed for the underlying problem. Compared with the piecewise construction of a variational posterior process used in the classical continuous latent process flow developed by Deng et al. ^[13], the principle idea of our method was to derive a variational lower bound by constructing a posterior latent process conditional on all information over the whole time interval to maximize the log-likelihood generated by the observations, which reduces the computational cost and, thus, provides a convenient way to approximate the considered equation. Particularly, our new method showed a better approximation to the underlying equation than the classical drift-$ \theta $ discretization scheme through numerical error comparison. Numerical experiments were finally reported to demonstrate the effectiveness and generalization performance of the proposed method.
Citation: Xiao Qi, Tianyao Duan, Huan Guo. An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise[J]. AIMS Mathematics, 2024, 9(5): 11975-11991. doi: 10.3934/math.2024585

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Abstract

This paper studied the numerical approximation of the stochastic differential equations driven by non-global Lipschitz drift coefficient and multiplicative noise. An efficient data-driven method, called extended continuous latent process flow, was proposed for the underlying problem. Compared with the piecewise construction of a variational posterior process used in the classical continuous latent process flow developed by Deng et al. ^[13], the principle idea of our method was to derive a variational lower bound by constructing a posterior latent process conditional on all information over the whole time interval to maximize the log-likelihood generated by the observations, which reduces the computational cost and, thus, provides a convenient way to approximate the considered equation. Particularly, our new method showed a better approximation to the underlying equation than the classical drift-$ \theta $ discretization scheme through numerical error comparison. Numerical experiments were finally reported to demonstrate the effectiveness and generalization performance of the proposed method.

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