Suppose that the value of a variable \(x\) follows the \(Ito\) process
\[dx = a(x,t) dt + b(x,t) dz\]
where \(dz\) is a Wiener process and \(a\) and \(b\) are functions of \(x\) and \(t\). The variable \(x\) has a drift rate of \(a\) and a variance rate of \(b^2\). \(Ito\) lemma shows that a function \(G(x,t)\) follows the process
\[dG = (\frac{\partial G}{\partial x} a + \frac{\partial G}{\partial t} + \frac{1}{2} \frac{\partial^2 G}{\partial x^2} b^2) dt + \frac{\partial G}{\partial x} b dz.\]
Thus \(G\) also follows an \(Ito\) process, with a drift rate of \(\frac{\partial G}{\partial x} a + \frac{\partial G}{\partial t} + \frac{1}{2} \frac{\partial^2 G}{\partial x^2} b^2\) and a variance rate of \((\frac{\partial G}{\partial x} b)^2\).
In stock price process \(a(x,t)=\mu S\) and \(b(x,t)=\sigma S\).