Friday, February 20, 2015

My take on Ito's, using d(X*Y) as example

Let J be the random process defined by Jt := Xt Yt. At any time, the product of X and Y is J's value. Ito's formula says

    dJ := d(Xt Yt) = Xt dY + Yt dX + dX dY

Note this is actually a stoch integral equation. If there's no dW term hidden in dX, then this reduces to an ordinary integral equation. Is this a differential equation? No. There's no differential here.

Note that X and Y are random processes with some diffusion i.e. dW elements.

I used to see it as an equation relating multiple unknowns – dJ, dX, dY, X, Y. Wrong! Instead, it describes how the Next increment in the process J is Determined and precisely predicted

Ito's formula is a predictive formula, but it's 100% reliable and accurate. Based on info revealed so far, this formula specifies exactly the mean and variance of the next increment dJ. Since dJ is Guassian, the distribution of this rand var is fully described. We can work out the precise probability of dJ falling into any range.

Therefore, Ito's formula is the most precise prediction of the next increment. No prediction can be more precise. By construction, all of Xt, Yt, Jt ... are already revealed, and are potential inputs to the predictive formula. If X (and Y) is a well-defined stoch process, then dX (and dY) is predicted in terms of Xt , Yt , dB and dt, such as dX = Xt2 dt + 3Yt dB

The formula above actually means "Over the next interval dt, the increment in X has a deterministic component (= current revealed value of X squared times dt), and a BM component ~ N(0, variance = 9 Yt2 dt)"

Given 1) the dynamics of stoch process(es), 2) how a new process is composed therefrom, Ito's formula lets us work out the deterministic + random components of __next_increment__.

We have a similarly precise prediction of dY, the next increment in Y. As such, we already know
Xt, Yt -- the Realized values
dt – the interval's length
dX, dY – predicted increments

Therefore dJ can be predicted.

For me, the #1 take-away is in the dX formula, which predicts the next increment using Realized values.

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New York (Time Square), NY, United States
http://www.linkedin.com/in/tanbin