Recent Advances in Numerical Methods for Stochastic Differential Equations: From Classical Schemes to Wavelet-Based Techniques

Stochastic processes prevail throughout fields as diverse as finance, physics, and biology, demanding increasingly sophisticated mathematical tools. These are the principal components of modeling systems influenced by random forces. The present work offers a structured examination of the solution landscape, progressing from foundational classical techniques to the advanced wavelet-based alternatives that have gained considerable traction in recent years. The paper initiates by establishing the foundations of stochastic processes, succeeded by a discussion of classical numerical techniques, encompassing their strengths and inherent limitations. Thereafter, different wavelet approaches are developed for diverse types of stochastic differential equations. In particular, the stochastic ordinary, partial, delay, and fractional differential equations are systematically examined with specific attention to their accuracy, computational efficiency, and multiresolution capabilities. This comparative synthesis of existing studies highlights the advantages and challenges associated with different wavelet frameworks. Drawing upon observations from the literature, potential research directions, key insights, and emerging opportunities are outlined. This survey is aimed at providing researchers with a unified perspective on the current status, development, and future prospects of wavelet methods for stochastic differential equations.