My guess is this holds, for example, if $X(t) = c$ is constant, $Y(t) = W(t)$ is Brownian motion, then their quadratic variation is $\langle X, Y \rangle (t) = W(t)$ which has mean of 0; $X(t) = W_1(t)$ and $Y(t) = W_2(t)$ also work; $X(t) = W(t)$ and $Y(t) = N(t)$ a poisson process, also works...
But is it really true? can it be proved?
