The Moments for Some Hyperbolic Stochastic Differential Equations

This paper investigates moments for Ito's integral formula involving general form of hyperbolic stochastic functions, hyperbolic stochastic functions, which combine the deterministic structure of hyperbolic functions with stochastic elements such as noise and random fluctuations. The formulation of these functions through stochastic differential equations (SDEs) is discussed, along with methods for solving and analyzing such equations., with a specific focus on stochastic processes governed by Brownian motion. By utilizing Ito's integral calculus, we derive moments for integrals of hyperbolic sine and cosine functions. These results extend the classical Ito integral formula to hyperbolic trigonometric functions, providing a comprehensive understanding of their stochastic properties. These outcomes have applications in various fields such as mathematical finance, physics, and engineering, where modeling using hyperbolic functions under random effects is prevalent. Some examples are given to illustrate the theoretical results.