Itô's Lemma, Part 2: Product Rule and Isometry

In ordinary calculus $d(xy) = x\,dy + y\,dx$. The piece you throw away is $dx\,dy$, a product of two small changes, which is second order. For random processes that piece can be the same size as $dt$, so it stays:

You get it by expanding $(X + dX)(Y + dY) - XY$, keeping every term, and simplifying $dX\,dY$ with the box rules from the Quadratic Variation lesson. The only new rule covers two Brownian motions $W$ and $B$ with correlation $\rho$: $dW\,dB = \rho\,dt$. The term $dX\,dY$ is the cross-variation of $X$ and $Y$. Setting $Y = X$ gives $d(X^2) = 2X\,dX + (dX)^2$, which is Itô's lemma for $f(x) = x^2$, so the two rules agree.

If one factor is smooth, the extra term drops out: $d(tW_t) = W_t\,dt + t\,dW_t$, because $dt\,dW = 0$. If both factors are Brownian, it survives:

The two $dW$-type terms have mean zero, so $E[W_tB_t] = \rho t$. With $\rho = 0.5$ and $t = 2$ that is $1$, which matches $\mathrm{Cov}(W_t, B_t) = \rho t$ computed directly. With $\rho = 0$ the rule collapses to the ordinary one and $W_tB_t$ is a martingale.