AN EXPONENTIAL LINEAR UNIT BASED NEURAL NETWORK APPROACH FOR OPTIMIZING NUMERICAL SOLUTIONS OF STOCHASTIC INTEGRAL EQUATION

Abstract

Stochastic integral equations are used to model a variety of phenomena in science and engineering, including finance, physics and biology. They are often used to model systems that involve randomness or uncertainty, such as stock prices, weather patterns, or biological populations. Solving stochastic integral equations can be challenging, as the integral involves a random process. There are various techniques that can be used to solve such equations, such as the Wiener–Itô integral, the Stratonovich integral and the Malliavin calculus. These techniques involve manipulating the integral in various ways, such as approximating it with a Riemann sum, using Taylor expansions, or applying the Itô or Stratonovich rules. In this work, the stochastic integral equations has been solved using exponential linear unit-based neural networks. Monte Carlo based approach has been implemented to discretize the integral equation and stochastic gradient descent techniques have also been implemented to optimize the loss function discussed. The numerical results and convergence analysis have been discussed for various cases to justify the versatility of our proposed schemes. Loss function graphs have also been discussed for more clarity on the behavior of the solutions based on our proposed technique. With the use of exponential linear unit as the activation function, the convergence was smooth and stable. It has been shown in the results that the proposed methodology is able to predict the exact solution for a stochastic integral equation at an average loss of 0.01.