L ÉVY I NDUCED S TOCHASTIC D IFFERENTIAL E QUATION E QUIPPED WITH N EURAL N ETWORK FOR T IME S ERIES F ORECASTING APREPRINT Luxuan Yang 1, † , Ting Gao 1, * , Yubin Lu 1, ‡ , Jinqiao Duan 2, § and Tao Liu 3, ¶ 1 School of Mathematics and Statistics & Center for Mathematical Sciences,Huazhong University of Science and Technology, Wuhan 430074, China. 2 Department of Applied Mathematics, College of Computing, Illinois Institute of Technology, Chicago, IL 60616, USA 3 Securities Finance Department, China Securities Co., Ltd ABSTRACT With the fast development of modern deep learning techniques, the study of dynamic systems and neural networks is increasingly benefiting each other in a lot of different ways. Since uncertainties often arise in real world observations, SDEs (stochastic differential equations) come to play an important role. To be more specific, in this paper, we use a collection of SDEs equipped with neural networks to predict long-term trend of noisy time series which has big jump properties and high probability distribution shift. Our contributions are, first, we explored SDEs driven by α-stable Lévy motion to model the time series data and solved the problem through neural network approximation. Second, we theoretically proved the convergence of the model and obtained the convergence rate. Finally, we illustrated our method by applying it to stock marketing time series prediction and found the convergence order of error. Keywords Stochastic Differential Equations, ResNet and α-stable Lévy motion Lead Paragraph Time series analysis with various advanced deep learning mechanisms has been significantly investigated recently. Financial data, especially stock price prediction is considered to be an important task for many quan- titative researchers. However, as there always exists various kinds of uncertainties in the forecasting problems, how to accurately capture the intrinsic volatile nature of financial time series data is still challenging. In this paper, we try to improve prediction performance from the viewpoint of stochastic dynamical systems by em- bedding neural network approximation into the classical numerical SDE (stochastic differential equations) al- gorithms driven by α-stable Lévy motion. In this way, our method will appropriately capture the uncertainties from both the model and the data, which leads high-accurate prediction results even for data that has distribu- tion discrepancy on training and testing sets. Besides, we also prove the convergence of our network structure, which will facilitate the interdisciplinary study between conventional numerical analysis and deep learning theory. † Email: luxyang@hust.edu.cn * Email: tgao0716@hust.edu.cn ‡ Email: yubin_lu@hust.edu.cn § Email: duan@iit.edu ¶ Email:liutao@csc.com.cn * is the corresponding author arXiv:2111.13164v3 [cs.LG] 7 Feb 2022