# 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!

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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

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.

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Concerning the overkill: Isn't it strange that the original and still current standard derivation of Black-Scholes is done via Ito's lemma? Definitely much easier is doing that via properties of the log-normal distribution (see e.g. arxiv.org/abs/physics/0612022 or bus.lsu.edu/hillebrand/Hillebrand_BlackScholesNote_2005.pdf) My point is: Many people in fact done need Ito's lemma but it is treated like the holy grail! –  vonjd Apr 28 '10 at 18:29
BTW: I in fact consider this derivation of BS even easier than binomial trees where you have some intuition but the tricky part comes when transforming the discrete into a continuous case. It is not well known (e.g. it is not even mentioned by Wilmott in his FAQ QuantFinance - where he mentiones 12 (!) ways of deriving BS (included such esoteric concepts like local time). –  vonjd Apr 28 '10 at 18:29
@vonjd: The standard setting for BS is exactly an SDE, so that Ito's calculus and the martingale approach are truly natural. –  Yuri Bakhtin Apr 28 '10 at 19:09
Although you can define the Black-Scholes model by just specifying the distribution, and even prove some basic facts about it, that just doesn't seem very satisfactory. I mean, BS isn't even a very good model. And if you try to use something better, you're going to need at least some stochastic calculus. Besides, ideas such as martingale representation (Market completeness in math finance lingo) need some stochastic integration, even in BS. –  George Lowther Apr 28 '10 at 20:51