[944] Correction and updated notes

[Gautam Iyer]

Hi Guys,

Quick correction: In the notes in Definition 5.5 (chapter 3) I
*incorrectly* used the term Itô integral. It should simply be
"integral", and is corrected now.

An Itô integral is what we constructed in class (using limit of I_P),
with respect to Brownian motion. This is always a martingale. We
define integrals with respect to an Itô process X, as the sum of a
specific Riemann integral and a Itô integral (as in Definition 5.5).
This will only be a martingale if the Riemann integral part vanishes.

Also -- when doing conditional expectation, I didn't state the
independence lemma explicitly. We used it in the proof of Theorem 4.23
(in chapter 2), and I explained it in class without explicitly stating
it. I thought it would be a bit too abstract at that stage; but given
that Ryan stated it in the recitation and it appears to be showing up on
interviews, I decided to include it. I updated chapter 2 and stated the
independence lemma there (it is now Lemma 4.12). I will do at least one
example using this on Monday.

Finally, about the question I avoided (from NY): The quick explanation
as to why the third order terms don't matter is as follows: All these
terms will have

    (W(t+dt) - W(t))^3

in front of them. The random variable (W(t+dt)-W(t))^3 has mean 0 and
variance of order (dt)^3, and so we expect that (W(t+dt) - W(t)) will
typically have size (dt)^{3/2}.

Now our sums have order 1/dt terms in them, and when we add 1/dt terms
of size (dt)^{3/2} we will get something that vanishes as dt goes to 0.

Best,

Gautam

PS: I updated the notes online with the above corrections. The only
    other change I will likely make before your midterm is to include a
    few examples (without proof) of what I might review on Monday.

-- 
Emacs is a nice operating system, but I prefer UNIX.