Predicting path-dependent processes by deep learning

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.