I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of
- Geometric Brownian Motion,
- Diffusion Process.
For example, for a diffusion process that satisfies $dX_t = \mu(t, X_t) dt + \sigma (t, X_t) dW_t$, I've heard that $d\langle X\rangle_t = \sigma^2dt$. But I've tried looking everywhere for a proof or verification of this and not found anything. I attempted it myself but no luck so far; I vaguely think Ito's lemma may be needed.
I already know that the QV of standard Brownian motion is simply the time $t$.
I'd appreciate if anyone could show me how the equation for QV of Diffusion and GBM comes about, or point out a reference to me that deals with this.
Thanks in advance!