Applications of Fuzzy Sets in Stochastic Differential Equations: A Novel Functional Analysis Perspective

Abstract

This research focuses on applying fuzzy sets with stochastic differential equations (FSDEs) to model and analyze the uncertainty and randomness that characterize systems. The inability to capture the two forms of variability is an inherent weakness in conventional approaches and hinders their applicability in addressing practical problems in cases where ideal measurements cannot be obtained. This weakness is solved by incorporating fuzzy logic into FSDEs to provide a high degree of randomness and a more suitable framework for the emulation of various systems. This paper uses and implements FSDEs in a variety of areas such as financial derivatives pricing, ecological population dynamics, climatic modeling, industry automation, and epidemic prediction. The fuzzy Black-Scholes model gives a wider range of option prices under conditions of higher volatility as compared to other models in financial modeling to better manage risks. Introducing environmental stochasticity into the differential equation models of ecology and climate selectively adds uncertainty to FSDEs and yields forecasts of population growth and future temperature shifts that are more realistic. FSDE-based control systems can handle sensor measurement errors and process noise well in industrial automation which reduces operational mistakes. It is observable that the fuzzy-stochastic SIR model for epidemics yields flexibility in projecting uncertainties in disease transmission and intervention measures.These findings show that incorporated FSDEs greatly improve the performance of models in dealing with variability and uncertainty characteristics ideal for complex environments. As to providing a range of outcomes that is broader and more reliable, FSDEs become helpful in fields where accurate prediction is a relevant factor. This research provides directions for advancements of FSDE in artificial intelligence, robotics, and sustainability and reveals the versatility of the fuzzy-stochastic modeling paradigm in dealing with complex issues.