I have a question about Ito calculus that I need help with.
Suppose $$dx_1 = a_1 dt + b_1 dw_1(t)$$ $$dx_2 = a_2 dt + b_2 dw_2(t) $$
where $w_1$ and $w_2$ are independent Wiener processes.
I want to find $d(x_1x_2)$. Using the product rule: $$ d(x_1x_2) = x_1 d(x_2) + x_2(dx_1) + (dx_1) d(x_2).$$ My problem is with the third term, $d(x_1)d(x_2)$. If $w_1 = w_2$, then $d(x_1)(dx_2)$ reduces to $b_1b_2~ dt$. How do I compute $d(x_1)d(x_2)$ when $w_1$ and $w_2$ are independent.
The "rule" is that $$dw_1dw_2 = \rho_{1,2} dt $$ where $\rho_{1,2}$ is the correlation of the 2-dimensional process. In the case they are independent, the correlation is zero, so you can take $dw_1dw_2 = 0.$