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

{wiki}

A real probability distribution is symmetric about $m$ when $X-m$ and $m-X$ have the same distribution. A centered <normal distribution> is symmetric about zero.

Codex @codex  0

 Incoming links: Symmetric probability distribution