I am confused and don't get the big picture concerning the connection between


* [Ito integral](http://en.wikipedia.org/wiki/Ito_integral)
* [Stratonovich integral](http://en.wikipedia.org/wiki/Stratonovich_integral)
* Standard results in probability theory concerning skewed distributions.


**Example:** Take e.g. the mean of the [log-normal distribution](http://en.wikipedia.org/wiki/Log-normal_distribution) $e^{\mu+{\sigma^2\over 2}}$. The extra term $\sigma^2\over 2$ is a result of [Jensen's inequality](http://en.wikipedia.org/wiki/Jensen%27s_inequality#Proofs). The same result can be produced by Ito's lemma, it is the result of the famous extra term. Therefore this term is sometimes also called the Ito correction term.


**Question:** What I don't understand is why we need a strange integral which can't be used in the classical way (e.g. the standard chain rule doesn't hold any more) to come to a result which could as well be derived by standard techniques. On the other hand if we use an integral where the classical rules still apply (Stratonovich integral) we miss this additional term.


Why can't we just transform the stochastic process into an equivalent probability distribution and use the classical integrals to arrive at the right results (and, too, stick to the classical calculus rules). In a way we then would integrate not with respect to a stochastic process but with respect to the resulting probability distribution. It all seems like outsmarting ourselves and overcomplicating matters?!?


**Addendum:** Don't get me wrong: I think I understand (most) of these results piece by piece, what I miss is the bigger picture how everything fits together - perhaps somebody can enlighten me - thank you!



Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
It's not quite clear where exactly you have a difficulty. Of course, using stochastic calculus and Ito's integral (which is central to the modern theory of stochastic processes) to derive properties of the log-normal distribution is an overkill, but it might be a nice exercise.


Some random quick points on why Ito's integral is important:


1. Ito introduced his integrals to describe continuous Markov processes. This was successfully carried out so that now we have a nice view of Markov processes as solutions of stochastic differential equations.
2. Ito's integral seen as a process is a martingale which is very convenient since martingales "have a lot of structure", and there are lots of martingale tools. As for Stratonovich's integral, the change of variables formula for it is the same as in classical calculus, so it might appear more natural, but it is harder to work with due to the lack of martingale structure.


