Variance of a time-weighted Itô integral

MediumCalculus Stochastic Processes~5m

Let $B_t$ be a standard Brownian motion. Compute the variance of $X_t = \displaystyle\int_0^t s\,dB_s$ . The answer has the form $k\,t^3$ ; find $k$ .

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Itô integrals have mean $0$ ; variance $=$ second moment $=$ Itô isometry $\int_0^t f^2\,ds$ .
With $f(s)=s$ : $\int_0^t s^2\,ds=t^3/3$ , so $k=\tfrac13$ .

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0.3333 (± 0.005)

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Variance of a time-weighted Itô integral

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