Neural network stochastic differential equation models with applications to financial data forecasting

In this article, we employ a collection of stochastic differential equations with drift and diffusion coefficients approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions are, first, we propose a model called L\évy induced stochastic differential equation network, which explores compounded stochastic differential equations with $\alpha$-stable L\évy motion to model complex time series data and solve the problem through neural network approximation. Second, we theoretically prove that the numerical solution through our algorithm converges in probability to the solution of corresponding stochastic differential equation, without curse of dimensionality. Finally, we illustrate our method by applying it to real financial time series data and find the accuracy increases through the use of non-Gaussian L\évy processes. We also present detailed comparisons in terms of data patterns, various models, different shapes of L\évy motion and the prediction lengths.