In the book about Stochastic Differential Equations, the term $dX(t)$ appears everywhere and I have a question about the meaning of it.
Because for every $t\in [0,T]$, $X(t)$ is a random variable instead of a number, compared to a function $f:R\to R$, I don't understand the $dX(t)$, is it a differetial of $X(t)$?
Also I see the Itô's Integral$\int_{0}^{t}fdW(t)$, where $W(t)$ is the Brownian motion and we can get $\int_{0}^{t}fdW(t) = I(f)(t,\omega)$ by definition.
I also wonder what is the definition of $fdW(t)$, because $I(f)$ is also a random variable.