I was working on the definition of the self-financing portfolio.
Say $V=\phi_tS_t+\psi_t A_t$ where $S_t$ and $A_t$ are the stock price and the money market price at time $t$, resp, and $\phi_t$ and $\psi_t$ are the shares that are invested in stock and money market at time $t$, resp.
Then I came across the formula like this: $dV_t$=$\phi_t dS_t$+$\psi_t dA_t$+ $S_t d\phi_t$+$A_t d\psi_t$+$dS_t d\phi_t$+$dA_t d\psi_t$.
I have no idea what the last four terms mean. Precisely speaking, I don't know what does $d\phi_t$ or $d\psi_t$ mean. They are just some adapted random process and nothing more I know about them.
Moreover, most of reference book or lecture notes (like this, page 16), claim the above formula is based on the Ito lemma. Doesn't Ito Lemma/formula only apply to Ito process?
I should clarify my confusion a little bit. If I understand correctly, the differential notations don't have mathematical meaning. However, for $dS_t$ and $dA_t$, their integrals do. So I understand $dS_t$ and $dA_t$. But then $d\phi_t$ or $d\psi_t$ makes no sense to me. Because neither themselves or the integrals are defined.
Moreover, the approach of @Neeraj in the equation $3$ is more like a discrete approach, which is great and I can follow the logic. On the other hand, the note says the equation follows from Ito's lemma. In fact, I even saw some notes say $dA_t d\psi_t=0$ because of Ito's lemma. Those remarks really puzzle me a lot.
I hope I illustrate my confusion a little bit better. Thanks,