Black and Scholes Model

Question

Can someone please explain, where the term $S^2$ (while applying Ito's Lemma in the solution of a PDE of a Geometric Brownian Motion) comes from? $$ \frac{\sigma^2 \color{red}S^2}{2}\frac{\partial^2V}{\partial S^2} $$

Thank you

Answer 1

$S_t=\mu S_t\,dt+\sigma S_t\,dW_t$

Ito's Lemma yields $$ dV(S_t)=V'(S_t)\,dS_t+\frac12V''(S_t)\,d\langle S\rangle_t $$

If $dX_t=K_t\,dt+H_t\,dW_t$ then by definition $d\langle X\rangle_t=H_t^2\,dt$. So here we have $d\langle S\rangle_t=\sigma^2S_t^2\,dt$.