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Re: Derivation of Ito Lemma
Posted:
Sep 18, 2012 3:12 PM


On 20120917, 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 chisquared 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)4946054 FAX: (765)4940558



