A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management

This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.