@article{Qi2024, 
author = {Xiao Qi and Tianyao Duan and Huan Guo},
title = {An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {11975-11991},
keywords = {non-global Lipschitz, drift-θ Milstein scheme, neural stochastic differential equations, extended continuous latent process flow},
url = {https://www.sciopen.com/article/10.3934/math.2024585},
doi = {10.3934/math.2024585},
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-   θ discretization scheme through numerical error comparison. Numerical experiments were finally reported to demonstrate the effectiveness and generalization performance of the proposed method.}
}