Variance of integrated Brownian motion

MediumStochastic Processes~6m

Let $W_t$ be a standard Brownian motion. The variance $\operatorname{Var}\!\left(\int_0^t W_s\,ds\right)$ has the form $k\,t^3$ . Find the constant $k$ .

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The integral is Gaussian, mean $0$ : its variance is $\int_0^t\int_0^t \operatorname{Cov}(W_s,W_u)\,ds\,du$ .
Use $\operatorname{Cov}(W_s,W_u)=\min(s,u)$ and integrate.

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0.3333 (± 0.001)

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Variance of integrated Brownian motion

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