Both variables are centered. The <Itô isometry> in its bilinear form gives
$$
\operatorname{Cov}(B_t,\beta_t)=\mathbb E\!\left[\left(\int_0^t\operatorname{sign}(\beta_s)d\beta_s\right)\left(\int_0^t1\,d\beta_s\right)\right]=\int_0^t\mathbb E[\operatorname{sign}(\beta_s)]ds=0,
$$
where the last equality follows because a centered <Gaussian distribution> is a <symmetric probability distribution>. Thus $B_t$ and $\beta_t$ are <uncorrelated random variables>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-202/1/a/ii/solution