Variance of the time integral of Brownian motion

Question

For a standard Brownian motion $W_t$ , the random variable $\displaystyle\int_0^t W_s\,ds$ has variance $kt^3$ . What is $k$ ?

Short answer

$k=\tfrac13$ . Integrating by parts, $\int_0^tW_s\,ds=\int_0^t(t-s)\,dW_s$ , a Gaussian with variance $\int_0^t(t-s)^2ds=t^3/3$ by the Ito isometry. Alternatively, $\int_0^t\!\int_0^t\min(u,v)\,du\,dv=t^3/3$ .

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Lessons behind this question

The Four Faces of Brownian Motion
Covariance and Correlation
Itô's Formula

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