Local martingale convergence theorem

Question

I really need help, suppose we have $$E[x(t)^2]\leq C\exp(Dt),$$ where $C$ and $D$ are positive constants, Is x^2 a martingale? Can we apply the martingale convergence theorem?

Answer 1

The answer would typically be no. Take $x(t) := \exp(t/2)$ to be deterministic. Then clearly $x(t)^2=\exp(t)$ is a submartingale with respect to any filtration, but not a martingale, and it does not converge in $\mathbb{R}$.