I'm having a hard time understanding a passage in a proof of Ito's lemma provided in this link.
In theorem 6.3, to derive formula (6.18), the third passage is what troubles me. I see that the author does a Taylor polynomial expansion, but I fail to glue all the pieces together. Things seem to pop out of nowhere. Could anybody be so kind as to indicate from where each piece comes from? Why are some of the terms with a tilde and others not?
It uses the first and second degree remainder terms of the Taylor expansion in \begin{align} f(t+Δt,x+Δx)-f(t,x+Δx)&=f_t(\tilde t,x+Δx),\\& \tilde t\in(t,t+Δt)\\~\\ f(t,x+Δx)-f(t,x) &=f_x(t,x)Δx+\frac12f_{xx}(t,\tilde x)(Δx)^2,\\&\tilde x\in(x,x+Δx) \end{align} This still holds when $x$ is a vector (multi-dimensional Brownian motion), but can not stated this way when $f$ is vector-valued. In the vector case this computation still can be applied for each component separately, the final result can then again assembled in vector terms.