Stock Price Forecasting Using Gaussian-ADAM Optimized Long Short-Term Memory Neural Networks

Abstract

This study introduces a novel optimization algorithm, \textbf{ItoAdam}, which integrates stochastic differential calculus—specifically Itô's lemma—into the standard Adam optimizer. Although Adam is widely used for training deep learning models, it may fail to converge reliably in complex, non-convex settings. To address this, ItoAdam injects Brownian noise into the gradient updates, enabling probabilistic exploration of the loss surface and improving the ability to escape poor local minima. ItoAdam is applied to train Long Short-Term Memory (LSTM) neural networks for stock price forecasting using historical data from 13 major companies, including Google, Nvidia, Apple, Microsoft, and JPMorgan. Theoretical analysis confirms the \textbf{convergence} of the proposed method under mild conditions, ensuring its robustness for deep learning applications. In addition, a \textbf{Differential Evolution (DE)} algorithm is employed to automatically optimize critical LSTM hyperparameters such as hidden size, number of layers, bidirectionality, and noise standard deviation. Experimental results show that the ItoAdam-LSTM model consistently outperforms the standard Adam-LSTM approach across evaluation metrics including RMSE, MAE, and $R^2$. A detailed sensitivity analysis reveals that optimal forecasting accuracy is typically achieved when the noise standard deviation lies between \textbf{$2.1 \times 10^{-4}$} and \textbf{$2.9 \times 10^{-4}$}. These findings highlight the effectiveness of combining Itô-driven stochastic optimization with evolutionary search and recurrent architectures for robust financial time series prediction in noisy and nonstationary environments.