# Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

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

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!

• At least in one of the main applications of stochastic calculus, mathematical finance, it is important that one takes the evolution of available information into account. That is, one is working with an adapted stochastic process, and there is no clear way how one could do the same with "classical" methods. – Michael Greinecker Apr 28 '10 at 9:22
• Well, yes formally you are right about the filtration. But please keep in mind that we still work with very heavy assumptions which in any case result in a unique probability distribution (e.g. GBM -> log-normality) and which enable the ito correction term to become deterministic here. – vonjd Apr 28 '10 at 9:40