Unpacking Ito's Lemma to its Construction

[Nusc]

Does anyone know a document that explains the construction of Ito's lemma? In most financial mathematics textbooks, it's poorly motivated.

Thanks!

 

[HallsofIvy]

Try this:
http://www.contingencyanalysis.com/archive/archive99-4/00000264.htm

 

[Nusc]

Wow nice, is there more?

 

[Nusc]

Actually nevermind about that, what about the stochastic perturbation of the chain rule?

 

[Jademonkey]

I'm taking an undergrad course on stochastic processes right now, and all of our course materials are online. They aren't necessarily the best, in my opinion, but you may find them useful. If his notes themselves aren't useful to you, he usually includes very specific references so you can find the info elsewhere.

I'm pretty sure he doesn't give the most general version of Ito's Lemma/Formula here, but since you mentioned mathematical finance in your post, I think it'll probably be good enough. Anyway, here it is:

http://www.math.unl.edu/~sdunbar1/MathematicalFinance/Lessons/StochasticCalculus/ItosFormula/itosformula.xml

In case you're interested, here's a page with all of the materials from the course: http://www.math.unl.edu/~sdunbar1/MathematicalFinance/mathfinance.shtml

Hope that helps!

 

[Nusc]

I just want to know the motivation behind the stochastic perturbation of the chain rule.

Ask your prof and see if he knows, lol

 

[Jademonkey]

If I understand what you are asking, the link I posted explains it. Read "Example 1" where he shows how using the ordinary chain rule fails. He goes on to show that Ito figured out that he could use an algebraic identity and the quadratic variation of Brownian Motion to derive a new chain rule that accounts for the stochasticity.