Let J be the random process defined by Jt := Xt Yt. At any time, the product of X and Y is J’s value. (It’s often instructive to put aside the “process” and regard J as a derived random VARIABLE.) 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 also 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.