Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Derivation of Ito Lemma
Replies: 12   Last Post: Feb 7, 2013 1:37 PM

 Messages: [ Previous | Next ]
 Paul Posts: 513 Registered: 2/23/10
Derivation of Ito Lemma
Posted: Sep 17, 2012 6:19 PM

I'm looking at http://en.wikipedia.org/wiki/It%C5%8D%27s_formula#Informal_derivation
where it says dB^2 tends to E(dB^2). I followed the link to the basic
properties for Wiener processes, but I can't find why dB^2 tends to
E(dB^2). I am guessing that it has to do with the limit as dt
approaches zero. The closest thing seems to be that the variance of a
Wiener process is t, but that's not quite the same thing. dB is a
sampling of a normal random variable, it is not a summary statistic.

For context, I am looking at the Ito Lemma for Geometric Brownian
motion (immediately above the Ito derivation link above). In the
second line, there is a -(1/2)(sigma^2)dt. This is a direct result of
the fact that the random variable dB^2 gets replaced by dt. It seems
to be a pivotal change, so I'd like to understand it.

Thanks.

Date Subject Author
9/17/12 Paul
9/18/12 Herman Rubin
9/18/12 Paul
9/20/12 Paul
9/20/12 Paul
9/21/12 Herman Rubin
9/23/12 Paul
9/19/12 divergent.tseries@gmail.com
9/28/12 Paul
9/28/12 Paul
9/29/12 divergent.tseries@gmail.com
9/30/12 Paul
2/7/13 voyteg