I need to compute $\mathbb{E}(W_{2}W_{6}|W_{3})$, where $W_{i}$ is a Wiener process. How do I do this?
2026-03-26 11:05:36.1774523136
Conditional expected value of Wiener process
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$E(W_2W_6|W_2,W_3)=W_2E(W_6|W_2,W_3)=W_2E(W_6-W_3|W_2,W_3)+W_2W_3=W_2W_3$. Now condition oc $W_3$ to get $E(W_2W_6|W_3)=E(W_2W_3|W_3)=W_3E(W_2|W_3)$. Can you compute this? Hint: there exists a constant $a$ such that $W_2-aW_3$ is independent of $W_3$.