Variance of a Brownian combination

MediumStochastic Processes~4m

$W(t)$ is a standard Brownian motion, and $W(5)$ and $W(2)$ are read off the same path. What is $\operatorname{Var}\!\big(W(5) - 2\,W(2)\big)$ ?

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$W(5)$ and $W(2)$ are dependent — split $W(5)=W(2)+(W(5)-W(2))$ into independent pieces.
Then variances of the independent terms add, coefficients squared: $1^2\cdot 3 + (-1)^2\cdot 2$ .

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Asked at: Citadel, Two Sigma

Variance of a Brownian combination

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