The product rule for Brownian Motion

Question

I am reading Martin Baxter's book on Financial Calculus and in it, the product rule for the common Brownian motion case is described. I cannot understand how the Ito's formula applies here.

First, we start with $$ dX_t = \sigma_t dW_t + \mu_t dt $$

$$ dY_t = \rho_t dW_t + \nu_t dt $$

The book now says that by applying Ito's lemma to $f(X_t, Y_t) = \frac{1}{2}\left( (X_t + Y_t)^2 - X_t^2 - Y_t^2 \right) = X_t Y_t$ we can see that:

$$ d(X_t Y_t) = X_t dY_t + Y_t dX_t + \sigma_t \rho_t dt $$

How is Ito's lemma applied to $f(X_t, Y_t)$? Ito's lemma states that if $dX_t = \sigma_t dW_t + \mu_t dt$ and $Y_t = f(X_t)$ then:

$$ dY_t = \sigma_t f'(X_t) dW_t + \left( \mu_t f'(X_t) + \frac{1}{2} \sigma_t^2 f''(X_t) \right) dt $$

but I don't yet know how to apply this to a function of 2 stochastic variables and the book hasn't mentioned anything about the application in this case.

I see the result for $d(X_tY_t)$ but I don't understand how we got it by applying Ito's lemma.

Answer 1

Set $Z_t = f(X_t, Y_t)$ for processes $X$ and $Y$ and $f\in C^2(\mathbb{R}^2)$. Itô's lemma is

$dZ_t = f_x(X_t, Y_t)dX_t + f_y(X_t, Y_t) dY_t + \frac{1}{2} f_{xx}(X_t,Y_t)d\langle X\rangle_t + \frac{1}{2} f_{yy}(X_t,Y_t)d\langle Y\rangle_t + f_{xy} (X_t,Y_t)d\langle X, Y \rangle_t$

By applying this to $f(x,y) = xy$, you obtain Itô's product rule:

$$d(XY)_t = X_t dY_t + Y_t dX_t + d\langle X, Y \rangle_t$$

Answer 2

Note that the polarization formula is used to avoid using the extension of the Ito formula to several variables. The only formula you need to apply is for $h(x)=x^2$ where you get $$ dh(X)=d(X^2)=2XdX+d⟨X⟩. $$ Then $f(X,Y)=\frac12(h(X+Y)-h(X)-h(Y))$ requires the threefold application of the Ito formula for the square.