I wonder what is covariance of two different wiener processes, say $W_t$ and $V_t$:
$$Cov(W_t,V_t) = \text{ }?$$
If it was one process we could use the property that $\Delta = W_t – W_s \text{~} N(0, t-s)$ to get that
$$Cov(W_t,W_s) = Cov( W_s + (W_t – W_s) ,W_s) = s, \text{ } s \leq t$$
However, I don't know, whether it's possible to do it in the same way. Intuitively, the $Cov(W_t,V_t)$ is zero.
Best Answer
Indeed, it depends on the task. Usually, it is stated in the problem set whether the the processes are independent.
Roughly, if $X$ and $Y$ are independent, then $ \Bbb Corr(X,Y) = 0$ or $$\Bbb Corr(X,Y) = \frac{ \Bbb Cov(X,Y)}{\sigma_X\sigma_Y} \implies \Bbb Cov(X,Y) = 0$$ If nothing about independence is said, we cannot say the $\Bbb Cov$ is zero.