Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 1 / a / iii / Solution

They are not independent. The Itô formula gives ${\beta }_t^2=t+2{\int }_0^t{\beta }_{s}d{\beta }_s$, and the bilinear Itô isometry therefore givesIf $B_t$ and ${\beta }_t$ were independent random variables, then $B_t$ would also be independent of the measurable function ${\beta }_t^2$, and centeredness would instead give $\mathbb{E}[B_t{\beta }_t^2]=\mathbb{E}[B_t]\mathbb{E}[{\beta }_t^2]=0$. This contradiction disproves independence.