Here is some intuition for Ito's formula.
The Taylor expansion for a function $f$ about a point $y$ is
$$ f(y) = f(x) + f'(x)(x-y) + \frac12 f''(x)(x-y)^2 + \dots \,.$$
If you replace $x-y$ with $dx$ and $f(y) – f(x)$ with $df(x)$, then
$$ df(x) = f'(x)dx + \frac12 f''(x) dx^2 + \dots \,.$$
If you keep only the first term, you have the formula for the differential, $df = f' dx$.
If you keep the first two terms, you have Ito's formula, $ df = f'dx + \frac12 f'' dx^2$.
Is there some explanation for why functions of stochastic processes need the second derivative term when taking the differential of $f$? I know that we use the fact that "$dz^2 = dt$", where $z$ is a Brownian motion, but I don't fully understand that. I know $\mathbb{E}[B_t^2] = t$, is that related? Why is the second-order term in the Taylor expansion for the regular differentual zero?
Edit: on slide 8 of these lecture notes, we have
If $dX_t = a dt + b dB_t$ is an Ito process, then
\begin{align*}
(dX_t)^2 &= (adt + b dB_t)^2
\\
&= a^2 dt^2 + 2(adt)(bdB_t) + (bdB_t)^2
\\
&= bdB_t^2 \,.
\end{align*}
Why are the first two terms zero? (I also don't understand why the term $dB_t dt =0$, on page 10.)
Best Answer
As explained here Intuition between Ito-Formula, after we Taylor expand (using the Mean-value remainder) we have
$$f(B_t)=f(B_0)+\sum_{i=0}^{n-1}f'(B_{t_{i}})(B_{t_{i+1}}-B_{t_{i}})+\frac{1}{2}\sum_{i=0}^{n-1}f''(\theta_i)(B_{t_{i+1}}-B_{t_{i}})^2.$$
So now if we had that Brownian motion was of bounded variation then the last term is bounded by
$$\frac{1}{n}|\sum_{i=0}^{n-1}|f''(\theta_i)|B_{t_{i+1}}-B_{t_{i}}|\approx \frac{1}{n}\int |f''(s)||dB_s|\to 0 $$
and so it disappears. However, the reality is that Brownian motion is not of bounded variation but it is of finite quadratic-variation i.e. $\sum (B_{t_{i+1}}-B_{t_{i}})^2<\infty$ in $L^2$, and so Itô instead worked to define his integral in terms of the quadratic variation.
The same idea works if we have p-variation, see rough paths eg. Multidimensional Stochastic Processes as Rough Paths.
See also geometric intuition here https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma.