Conditioning Brownian motion on its future

MediumStochastic Processes~4m

Standard Brownian motion is observed at time $5$ to be $W_5 = 3$ . Given only that information, what is the expected value of $W_2$ (the earlier position, to two decimals)?

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$(W_2,W_5)$ are jointly Gaussian; regress $W_2$ on $W_5$ .
$E[W_2\mid W_5]=\tfrac{\operatorname{Cov}(W_2,W_5)}{\operatorname{Var}(W_5)}W_5=\tfrac{2}{5}W_5$ .

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1.2 (± 0.001)

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Conditioning Brownian motion on its future

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