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 $k$ .

Show hints (2)+

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 $k=\tfrac13$ .

Answer

Reveal answer →

0.3333 (± 0.005)

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Citadel, Two Sigma

Variance of a time-weighted Itô integral

Answer

Related questions