Predicting path-dependent processes by deep learning

TL;DR

This paper presents a deep learning approach for predicting path-dependent stochastic processes, demonstrating convergence and accuracy through theoretical proofs and numerical simulations on fractional Brownian motion and related models.

Contribution

The paper introduces a nonparametric deep learning method for path-dependent process prediction with proven convergence and practical effectiveness demonstrated through simulations.

Findings

The method achieves convergence of $L_2$ errors to zero.

Predictions based on discrete observations approximate continuous ones as observation frequency increases.

Numerical results show high accuracy in predicting fractional Brownian motion and related processes.

Abstract

In this paper, we investigate a deep learning method for predicting path-dependent processes based on discretely observed historical information. This method is implemented by considering the prediction as a nonparametric regression and obtaining the regression function through simulated samples and deep neural networks. When applying this method to fractional Brownian motion and the solutions of some stochastic differential equations driven by it, we theoretically proved that the $L_2$ errors converge to 0, and we further discussed the scope of the method. With the frequency of discrete observations tending to infinity, the predictions based on discrete observations converge to the predictions based on continuous observations, which implies that we can make approximations by the method. We apply the method to the fractional Brownian motion and the fractional Ornstein-Uhlenbeck process as examples. Comparing the results with the theoretical optimal predictions and taking the mean square error as a measure, the numerical simulations demonstrate that the method can generate accurate results. We also analyze the impact of factors such as prediction period, Hurst index, etc. on the accuracy.