Unpacking Ito's Lemma: A Guide to its Construction

In summary, the conversation is about finding resources that explain the construction of Ito's lemma and the stochastic perturbation of the chain rule. One person suggests a link to an online course on stochastic processes, which includes specific references for further information. The link provided explains how Ito's lemma accounts for the stochasticity in the chain rule. The conversation ends with a suggestion to ask the professor for more clarification.
  • #1
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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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  • #2
Try this:
http://www.contingencyanalysis.com/archive/archive99-4/00000264.htm [Broken]
 
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  • #3
Wow nice, is there more?
 
  • #4
Actually nevermind about that, what about the stochastic perturbation of the chain rule?
 
  • #5
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 [Broken]

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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  • #6
I just want to know the motivation behind the stochastic perturbation of the chain rule.

Ask your prof and see if he knows, lol
 
  • #7
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.
 

1. What is Ito's Lemma and why is it important in finance?

Ito's Lemma is a mathematical formula used in finance to calculate the change in a stochastic process. It is important because it allows for the calculation of expected values of complex financial instruments, such as derivatives.

2. How is Ito's Lemma constructed?

Ito's Lemma is constructed using the Taylor series expansion and the multidimensional Itô integral. It involves taking the partial derivatives of the stochastic process and using the chain rule to calculate the change in the process over a small time interval.

3. What are the assumptions made in Ito's Lemma?

Ito's Lemma assumes that the stochastic process is continuous and differentiable, and that the increments of the process are normally distributed with mean 0 and constant variance. It also assumes that the process is not affected by external factors such as interest rates or market conditions.

4. How is Ito's Lemma used in practice?

Ito's Lemma is used in finance to calculate the expected value of a financial instrument, such as a stock price or option, over a small time interval. It is also used in risk management to calculate the potential impact of small changes in the underlying asset.

5. What are some common applications of Ito's Lemma?

Ito's Lemma is commonly used in the pricing of options and other derivatives, as well as in the study of stochastic processes in finance. It is also used in the Black-Scholes model for option pricing and in the analysis of financial time series data.

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