Moment Analysis for Some Nonlinear (Inverse Trigonometric) SDE Using Itô Calculus: Case Studies in Gene Regulation and Robotic Navigation
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Abstract
This paper explain the application of Ito-integral formula to finding the moments for nonlinear stochastic differential equations (SDEs) with coefficients (inverse trigonometric function) such as , and , this method naturally constrains state variables within predefined physical or operational limits. The study derives moment equations to explain and analyze statistical properties, including mean and variance, as well as higher-order moments( the moment generation function ), while addressing the challenges of nonlinear drift, diffusion interactions, and multiplicative noise and the main results show that finite terms such as enforce stability in robotic angular control, yielding error bounds , while gene regulatory models with the function ensure protein concentrations remain biologically viable, limiting transcriptional noise to , With inherent limitations, multiplicative noise can push moments toward saturation. Systematic trade-offs in moment closure approximations and numerical verification (e.g., Euler-Maruyama, stochastic Runge-Kutta) are critically evaluated, highlighting their effectiveness in biological and engineering applications and the framework is extended to fractional SDEs and multidimensional systems, proposing machine learning techniques to solve moment hierarchies and enhance predictive modeling by uniting theoretical rigor with practical insights and this work advances robust stochastic modeling in constrained systems, providing scalable solutions for gene networks, financial markets, and autonomous navigation under uncertainty.
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