# The product rule for Brownian Motion

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.

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