Mean, variance and SDE of root-t times Brownian motion

Let $B$ be standard Brownian motion and $Z_t=\sqrt t\,B_t$. Find $\mathbb{E}[Z_t]$ and $\operatorname{Var}(Z_t)$, and derive the SDE satisfied by $Z$.

Mean $0$, variance $t^2$, and $dZ_t=\dfrac{B_t}{2\sqrt t}dt+\sqrt t\,dB_t=\dfrac{Z_t}{2t}dt+\sqrt t\,dB_t$. The moments follow from $B_t\sim N(0,t)$. For the SDE, apply the product rule to $\sqrt t\cdot B_t$; there is no Ito correction because $\sqrt t$ is deterministic and differentiable.