What are the processes whose second moment equal their quadratic variation (like the Brownian motion)?

Question

If $B_t$ is a standard Brownian motion, we know that

$$[B]_t=t=E[B_t^2]$$

where $[.]_t$ is the quadratic variation at time $t$.

Can we characterize all processes such that

$$ [X]_t=E[X_t^2]$$

If not, what are large classes of processes that have this property ?