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

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Herman Rubin

Posts: 325
Registered: 2/4/10
Re: Derivation of Ito Lemma
Posted: Sep 18, 2012 3:12 PM
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On 2012-09-17, Paul <paul.domaskis@gmail.com> wrote:
> 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.


One way to see this is to compute the mean and variance for
a computation of the sum of deltaB^2 for a fine partition
of an interval of length T. If the partition is of equal
intervals on the t axis, the sum will be a chi-squared
distribution with the number N of intervals divided by N,
which has variance 2T/N.

Another way to look at this is to note that when X and Y
are independent random variables with mean 0,

E(X^2 + Y^2) = E((X+Y)^2).

and as powers of X and Y are also independent,

V(X^2 + Y^2) = V((X+Y)^2) - 4 E(X^2) E(Y^2)

Apply this to the normal case, and this shows that
the variance decreases. Again, calculation will
give the result that the variance converges to 0
in probability, and if one uses refinements, the
convergence is even with probability one.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558



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