Variance of the Wiener integral of s dB

Question

Let $B$ be standard Brownian motion and $X_t=\int_0^t s\,dB_s$ . Its variance can be written $k\,t^3$ for a constant $k$ . Find $k$ .

Short answer

$1/3$ . A Wiener integral of a deterministic integrand is Gaussian with mean $0$ and variance equal to the integral of the squared integrand: $\int_0^t s^2\,ds=t^3/3$ .

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Lessons behind this question

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

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