A Novel Method for Optimizing Numerical Solutions of Multi-Dimensional Itô-Volterra Stochastic Integral Equation Using Recurrent Neural Network

This research study proposes a deep learning approach based on a recurrent neural network and it utilizes two popular types of RNNs - Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) and Monte Carlo simulation to effectively solve multi-dimensional stochastic Itô-Volterra integral equations (MSIVIE). In numerical experiments and practical applications, the proposed approach achieves reliability and great accuracy compared to current methods. Using optimization techniques such as Adam further enhances the model's performance. The adaptability of the approach is demonstrated through various scenarios, and error analysis is conducted to validate its efficacy. The numerical computations are implemented in Python and MATLAB. Consequently, provided visualizations such as loss function and actual and predicted function value graphs for a better understanding of the solution behaviour.