Solution
= Solution
{c}
The <Itô formula> for $f(x)=x^4$ gives
$$
d(B_t^4)=4B_t^3dB_t+6B_t^2dt.
$$
The stochastic integral has expectation zero: its square integrability on a finite horizon follows from the finite sixth moments of a <normal distribution>. Since $\mathbb EB_s^2=s$, integration and expectation yield
$$
\boxed{\mathbb EB_t^4=6\int_0^t s\,ds=3t^2.}
$$