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

- Ito integral
- Stratonovich integral
- Standard results in probability theory concerning skewed distributions.

**Example:** Take e.g. the mean of the log-normal distribution $e^{\mu+{\sigma^2\over 2}}$. The extra term $\sigma^2\over 2$ is a result of Jensen's inequality. 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!