BREAKING BOUNDARIES: EULER-MARUYAMA METHODOLOGY FOR UNCERTAIN STOCHASTIC SYSTEMS OF THE FIRST ORDER

Abstract

Dynamical systems, serving as indispensable tools for modeling complex phenomena, find widespread application in diverse fields such as biology, engineering, physics, and finance. These systems, characterized by their time-dependent nature, are foundational in understanding the evolving dynamics of various processes. This study delves into the intricate realm of dynamical systems, with a particular focus on addressing uncertainties that inherently influence their behavior. Two primary forms of uncertainty emerge as pivotal factors in the study of dynamical systems: randomness and belief degree. These uncertainties add layers of complexity to the modeling process, requiring specialized mathematical frameworks. To tackle the stochastic nature of certain systems, the stochastic differential equation (SDE) emerges as a crucial class of differential equations. SDEs provide a robust methodology for capturing and analyzing the inherent randomness prevalent in dynamic processes. On the other front, another significant class of differential equations, the uncertain differential equation (UDE), comes into play when dealing with dynamical systems influenced primarily by belief degrees. Belief degrees introduce a subjective aspect to uncertainties, reflecting varying levels of confidence or belief in the evolving dynamics. UDEs, tailored to model these scenarios, provide a mathematical foundation for addressing uncertainties stemming from subjective beliefs. This abstract aims to highlight the pivotal roles played by stochastic and uncertain differential equations in modeling dynamical systems. Through a comprehensive exploration of SDEs and UDEs, this study contributes to a deeper understanding of how randomness and belief degrees individually shape the intricate dynamics of systems across different disciplines. The practical implications of this research extend across various domains. In biology, where processes often exhibit unpredictable behavior, the application of SDEs proves instrumental. In contrast, UDEs find application in scenarios such as decision-making processes, where uncertainties arise from subjective beliefs. The engineering field benefits from a combined approach, utilizing both SDEs and UDEs to address diverse uncertainties in system dynamics. Physics, with its intricate phenomena, gains a more accurate representation through the nuanced description provided by these differential equations. Furthermore, in the domain of finance, where the dynamics of markets are influenced by both stochastic elements and subjective beliefs, the application of SDEs and UDEs becomes indispensable. In conclusion, this abstract provides a comprehensive overview of the role of differential equations in modeling dynamical systems, emphasizing the distinct contributions of stochastic and uncertain differential equations in addressing randomness and belief degrees, respectively.