Use of Stochastic Methods as a Tool for Research: A Brief Review

Mathematical models are essential for understanding and predicting physical phenomena in both scientific and industrial contexts. While conventional differential and integral calculus is often sufficient for describing most phenomena, there are instances where systems exhibit irregular and unpredictable behaviors over time or space, making ordinary calculus tools inadequate. In such cases, stochastic formalism, which relies on probability functions, becomes a valuable alternative for modeling these complex systems. This paper aims to provide a more comprehensive exploration of the fundamental properties and concepts of stochastic processes. It highlights the importance of the master equation, a key tool for understanding how microscopic processes impact the macroscopic behavior observed. Additionally, the article outlines the methodology for deriving stochastic differential equations, which take into account the stochastic nature of external variables that interact with the system and its environment. To illustrate the practical applications of these methods, the article presents several examples of how stochastic modeling is used to describe natural and industrial processes. These examples showcase the versatility and effectiveness of stochastic formalism in addressing real-world challenges and predicting the behavior of complex systems. This paper emphasizes the significance of stochastic modeling in bridging the gap between the microscopic and macroscopic aspects of physical phenomena, providing a deeper understanding of complex systems and their behavior. Stochastic formalism has become an indispensable tool for researchers and engineers working in various fields, enabling them to tackle the challenges posed by irregular and unpredictable systems.