# Tag Info

A key property of Brownian motion is independent increments. So if $x-1 > y$, then $$\mathbb{E}[\Delta W_x \Delta W_y] = 0$$ because the time intervals [x-1,x] and [y-1,y] do not overlap. If they do overlap, i.e. $x-1 \leq y < x$, then \begin{align} \mathbb{E}[\Delta W_x \Delta W_y] =&\ \mathbb{E}[(W_x - W_{x-1}) (W_y-W_{y-1})] \\ =&\ ...