Say I have two Brownian motion processes $B=\left\{B_t:t\ge 0\right\}$ and $W=\left\{W_t:t\ge 0\right\}$, with means $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$, respectively. Also, say that they both have random initial distributions $B(0)$ and $W(0)$.
I want to compute the correlation, $\rho$, between $B$ and $W$. I know this is probably very basic, but I have not been able to find a nice formula or expression to be able to compute $\rho$. (Nor do I know it's definition in the case for random processes.) Answers to these questions would be appreciated!