Question on variance and expectation of Brownian Motion related things

In a mathematical finance text by Ubbo F Wiersema, I came across the following

Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$. $Var[\Delta t\Delta B(t)]=(\Delta t)^2$.

It is also written that $Var[(\Delta B(t))^2]=2(\Delta t)^2$.

How was the calculations of the above two variances done?

• Are you sure you are reading correctly? Assuming $\Delta t$ is a constant, $Var[\Delta t \Delta B(t)] = (\Delta t)^2\,Var[\Delta B(t)] = (\Delta t)^3$? – Juho Kokkala Nov 3 '17 at 6:22