Vtyjkyuk

$text Let W text and widetilde W text be two independent Brownian motion and rho text is a constant in (0,1).$
$text For all t geq 0 , text let X _ t = rho W _ t + sqrt 1 - rho ^ 2 widetilde W _ t text and forall t geq 0 , X _ t sim N ( 0 , t ).$ Show: $$operatorname Cov left[ X _ t _ 2 - X _ t _ 1 ; X _ t _ 4 - X _ t _ 3 right] = 0$$

My attempt:

$operatorname Cov left[ X _ t _ 2 - X _ t _ 1 ; X _ t _ 4 - X _ t _ 3 right] = \ operatorname Cov left[ rho left( W _ t _ 2 - W _ t _ 1 right) + sqrt 1 - rho ^ 2 left( widetilde W _ t _ 2 - widetilde W _ t _ 1 right) ; rho left( W _ t _ 4 - W _ t _ 3 right) + sqrt 1 - rho ^ 2 left( widetilde W _ t _ 4 - widetilde W _ t _ 3 right) right] = \
rho ^ 2 operatorname Cov left[ W _ t _ 2 - W _ t _ 1 ; W _ t _ 4 - W _ t _ 3 right] + rho sqrt 1 - rho ^ 2 operatorname Cov left[ W _ t _ 2 - W _ t _ 1 ; widetilde W _ t _ 4 - widetilde W _ t _ 3 right] + \
rho sqrt 1 - rho ^ 2 operatorname Cov left[ widetilde W _ t _ 2 - widetilde W _ t _ 1 ; W _ t _ 4 - W _ t _ 3 right] + left( 1 - rho ^ 2 right) operatorname Cov left[ widetilde W _ t _ 2 - widetilde W _ t _ 4 ; widetilde W _ t _ 4 - widetilde W _ t _ 3 right]$

$hspace2mm$

Now, I know $rho sqrt 1 - rho ^ 2 operatorname Cov left[ W _ t _ 2 - W _ t _ 1 ; widetilde W _ t _ 4 - widetilde W _ t _ 3 right] = 0$ and $rho sqrt 1 - rho ^ 2 operatorname Cov left[ widetilde W _ t _ 2 - widetilde W _ t _ 1 ; W _ t _ 4 - W _ t _ 3 right] = 0$ because of independence between $W$ and $widetilde W$. But I get troubles showing the last two terms is zero. Assume $t_2 - t_1 < t_4 - t_3:$

$rho ^ 2 operatorname Cov left[ W _ t _ 2 - W _ t _ 1 ; W _ t _ 4 - W _ t _ 3 right] + left( 1 - rho ^ 2 right) operatorname Cov left[ widetilde W _ t _ 2 - widetilde W _ t _ 4 ; widetilde W _ t _ 4 - widetilde W _ t _ 3 right] =\ t_2 - t_1 (rho^2 + (1 - rho^2)) = t_2 - t_1$.

I dont't know what I did wrong. I made my calculations several times, and I dont think I did anything wrong. Any help is appreciated.

You did not state what the $t_i$ 's are. I think this result is for $t_1<t_2<t_3<t_4$. If that is the case then $Cov(W_t_2-W_t_1;W_t_4-W_t_3)=0$ by independence. Similarly for $widetilde W$.