Unpacking Ito's Lemma to its Construction

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The discussion revolves around finding a comprehensive explanation of Ito's Lemma, particularly its construction and motivation, which is often lacking in financial mathematics textbooks. Participants share useful resources, including links to course materials and specific notes that address Ito's Lemma and its application in stochastic calculus. There is a focus on the stochastic perturbation of the chain rule, with one user highlighting an example that illustrates the limitations of the ordinary chain rule in this context. The conversation emphasizes the importance of understanding the algebraic identity and quadratic variation of Brownian Motion in deriving a new chain rule suitable for stochastic processes. Overall, the thread serves as a resource for those seeking clarity on Ito's Lemma and its foundational concepts.
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Does anyone know a document that explains the construction of Ito's lemma? In most financial mathematics textbooks, it's poorly motivated.

Thanks!
 
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Try this:
http://www.contingencyanalysis.com/archive/archive99-4/00000264.htm
 
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Wow nice, is there more?
 
Actually nevermind about that, what about the stochastic perturbation of the chain rule?
 
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!
 
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I just want to know the motivation behind the stochastic perturbation of the chain rule.

Ask your prof and see if he knows, lol
 
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.
 
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