Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. [Wikipedia:Riemann sum][1].

The [Itô integral][2] 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][3], sometimes [Jensen's inequality][4].

But normally there is more than one intuition for a mathematical phenomenon, e.g. in Thurston's paper, ["On Proof and Progress in Mathematics"][5], he gives seven different elementary ways of thinking about the derivative.

**My question**<br/>
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.


  [1]: https://en.wikipedia.org/wiki/Riemann_sum
  [2]: https://en.wikipedia.org/wiki/Ito_integral
  [3]: https://en.wikipedia.org/wiki/Taylor_series
  [4]: https://en.wikipedia.org/wiki/Jensen_inequality#Proofs
  [5]: https://arxiv.org/abs/math/9404236