Deep Neural Networks for the Fractional Fokker-PlanckEquation: Application to the Heston Model with FractionalBrownian Motion

Abstract This paper investigates the application of a deep neural network approach for solving the fractionalFokker-Planck equation for the Heston model with fractional Brownian motion. In the framework of theHeston model, we first extend the Fokker-Planck equation with standard Brownian motion to a fractionalBrownian motion. Using state-of-the-art deep learning techniques, we design and train a deep neuralnetwork model and perform extensive tests. To increase the accuracy of the model, we compare the differentoptimization techniques, and Adams is more accurate for our problem in terms of losses and smoothness. Acomparison of activation functions reveals that Swish is the most effective choice to balance accuracy andcomputational efficiency. Our findings also highlight the flexibility of deep neural network-based Fokker-Planck equation models, showing that solution accuracy improves with different time points. Additionally,for various Hurst parameters 0 < H < 1, we demonstrate the robustness, scalability, and adaptabilityof the model. Furthermore, the deep neural network approach outperforms the finite difference method interms of error reduction and convergence. The results show that the solutions of both methods yield thebest agreement. In general, our results indicate that this method is effective, computationally efficient, andprovides comprehensive and standard solutions for stochastic volatility models in financial mathematics.