The moment for nonlinear (exponential) stochastic differential equations

in this article we study the moments for the exponential stochastic differential equations by using Ito's-formula. First we find the general form for exponential stochastic differential equation. After we obtain the exact solution for some stochastic differential equation, we derive their moments (mean, variance, and k.th moments). We also provided some examples to explain the method. then the exact solution will be needed in order to find the moments to the solution of the exponential stochastic differential equation Stochastic differential equations (SDEs) are frequently employed in various modeling applications due to their ability to incorporate randomness or uncertainty into ordinary differential equations. By introducing a random or stochastic component, these equations can capture unexpected phenomena. Consequently, SDEs are also known as stochastic or random differential equations, with the noise term representing the random element. In this way, an SDE comprises multiple random processes, leading to the solution itself being a stochastic process. Consider the ordinary differential equation