I've been studying Ito's Lemma for 1D case with 1D Brownian Motion. Here's the lemma just to remember:
$$dX_t=\big(\frac{\partial g}{\partial t}+\frac{1}{2}\frac{\partial^2g}{\partial x^2}|_{x=B_t}\big)dt+\frac{\partial g}{\partial x}|_{x=B_t}dB_t$$
where $X_t=g(t,x)|_{x=B_t}$.
I know the extra term $\frac{1}{2}\frac{\partial^2g}{\partial x^2}|_{x=B_t}dt$ comes from quadratic variation of Brownian Motion but does it have a geometric interpretation on its own?
Thank you.