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Assuming that $\{W(t) | t \geq 0\}$ is a Brownian motion, I am trying to find following conditional expectation

$$\mathbb{E}\left[W^{2}(4) | W(1), W(2)\right]$$

My try:

What I think is that I should introduce the conditional terms $W(1)$ and $W(2)$ into $W^{2}(4)$ to solve the problem. I tried

$$\mathbb{E}\left[([W(4)-W(2)]+[W(2)-W(1)+W(1)])^2\right]$$

but this introduces too many terms that don't get any simpler, like $(W(4)W(2)^2)$.

Is the approach incorrect?

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    $\begingroup$ Write $$W^2(4) = (W(4)-W(2)+W(2))^2 = W(4)^2 +2 (W(4)-W(2))W(2) + W(2)^2$$ and then pull out the $W(2)$-terms from the conditional expectation; use that $W(4)-W(2)$ is independent from $W(2)$ to compute the remaining conditional expectation. $\endgroup$ – saz Mar 6 at 7:22

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