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

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