Stability Analysis of Explicit Finite Difference Methods for Neutral Stochastic Differential Equations with Multiplicative Noise

In this study, stability analysis of the first order of explicit finite difference methods (EFDM) for multiplicative noise NSDEs is examined. Numerical methods ND FM is an established technique that has already found applications for solving PDEs in heat conduction, fluid dynamics, and wave propagation, among others. Stability concerns are vital for precise numerical solutions; therefore, their performance depends on them. This study analyzes the numerical stability of the EFDM to determine its stability under specified conditions utilizing stability criteria, including the CFL condition and von Neumann stability analysis. The paper examines how spatial domain discretization parameters like time step size and spatial resolution affect the numerical stability of solutions. NSDEs need unique stability criteria because wrong parameter choice causes numerical instabilities, as shown by sample comparative investigations. The presented work also examines the stability effects of stochastic integrators like Itô and Stratonovich ones and their strengths and weaknesses. Discretization options affect EFDM system stability and stochastic integrators, as shown by numerical simulations. Results are tabulated and graphed to highlight solution unpredictability and parameter dependency on stability. The results demonstrate the necessity to balance arithmetic speed and numeric precision and offer methodological insights into stochastic analysis using EFDM. Through a detailed analysis at high resolution, the present work improves the existing information on the stability of EFDM and the identification of novel computationally efficient numerical approaches for large-scale problems.