So with the definition of quadratic covariation between two processes given here, in the case when $Y$ is a Weiner process, is it true that
$$\lim_{||P|| \to 0} \sum_{i=1}^n (X_{(t_k + t_{k-1})/2} - X_{t_{k-1}}) (Y_{t_k} - Y_{t_{k-1}}) = \frac{1}{2} [X,Y]_t$$ when $[X,Y]_t$ exists?
I've tried breaking $[X,Y]_t$ down into $$\sum_{i=1}^n (X_{(t_k + t_{k-1})/2} - X_{t_{k-1}}) (Y_{t_k} - Y_{t_{k-1}}) + \sum_{i=1}^n (X_{t_k} - X_{(t_k + t_{k-1})/2} ) (Y_{t_k} - Y_{t_{k-1}}) $$ so it reduces down to proving the limits of these two sums are equal. At least from my intuition, since $(-Y_t)$ is a Weiner process, the seccond sum is essentially going backwards from $t$ to 0 with $t_k$ being the left point but I don't know how to justify this properly or if this is even true at all.