Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. [Wikipedia:Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum).


The [Itô integral](https://en.wikipedia.org/wiki/Ito_integral) has due to the unbounded total variation but bounded quadratic variation an extra term (sometimes called Itô correction term). The standard intuition for this is a [Taylor expansion](https://en.wikipedia.org/wiki/Taylor_series), sometimes [Jensen's inequality](https://en.wikipedia.org/wiki/Jensen_inequality#Proofs).


But normally there is more than one intuition for a mathematical phenomenon, e.g. in Thurston's paper, ["On Proof and Progress in Mathematics"](https://arxiv.org/abs/math/9404236), he gives seven different elementary ways of thinking about the derivative.


**My question**  

Could you give me some other intuitions for the Itô integral (and/or Itô's lemma as the so called "chain rule of stochastic calculus"). The more the better and from different fields of mathematics to see the big picture and connections. I am esp. interested in new intuitions and intuitions that are not so well known.



Intuition and/or visualisation of Itô integral/Itô's lemma
I find the intuitive explanation in [Paul Wilmott on Quantitative Finance](https://www.amazon.co.uk/Paul-Wilmott-Quantitative-Finance-2nd/dp/0470018704) particularly appealing.


Fix a small $h>0$. The stochastic integral
$$\int\_0^{h} f(W(t))\ dW(t)=\lim\limits\_{N\to\infty}\sum\limits\_{j=1}^{N}
f\left(W(t\_{j-1})\right)\left(W(t\_{j})-W({t\_{j-1}})\right),\quad t\_j= h\frac{j}{N},$$
involves adding up an infinite number of random variables. Let's substitute every term $f\left(W(t\_{j-1})\right)$ with its formal Taylor expansion. Then there are several contributions to the sum: those that are *a sum of random variables* and those that are *a sum of the squares of random variables*, and then there are *higher-order terms*.


Add up a large number of independent random variables and the Central Limit Theorem kicks in, the end result being a normally distributed random variable. Let's calculate its mean and standard deviation.


When we add up $N$ terms that are normal, each with a mean of $0$ and a standard deviation of $\sqrt{h/N}$, we end up with another normal, with a mean of $0$ and a standard deviation of $\sqrt{h}$. This is our $dW$. Notice how the $N$ disappears in the limit.


Now, if we add up the $N$ *squares* of the same normal terms then we get something which is normally distributed with a mean of
$$N\left(\sqrt{\frac{h}{N}}\right)^2=h$$
and a standard deviation which is
$h\sqrt{2/N}.$
This tends to zero as $N$ gets larger. In this limit we end up with, in a sense, our $dW^2(t)=dt$, because *the randomness as measured by the standard deviation disappears leaving us just with the mean $dt$*.


The higher-order terms have means and standard deviations that are too small, disappearing
rapidly in the limit as $N\to\infty$.


