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