I have the following two stochastic volatility models:
$ d S_1 = r_1 S_1 dt + \sqrt{v_1} S_1 d W_1 \\ d v_1 = a_1 dt + b_1 d Z_1 \\ $
together with $d W_1 d Z_1 = \rho_1 dt$ and
$ d S_2 = r_2 S_2 dt + \sqrt{v_2} S_2 d W_2 \\ d v_2 = a_2 dt + b_2 d Z_2 \\ $,
with $d W_2 d Z_2 = \rho_2 dt$
If I define $X_1 := \left(\begin{matrix} S_1 \\ v_1 \end{matrix} \right)$ and $X_2 := \left(\begin{matrix} S_2 \\ v_2 \end{matrix} \right)$, can I consider the quantity $d (X_1 X_2)$ ?
I would write
$ d (X_1 X_2) = X_1 d X_2 + X_2 d X_1 + d <X_1,X_2>$,
but I am not really sure how to do all the computations explicitly.
Any help or hint is very much appreciated! :)