Does local martingale have the same mean value as well?

Question

We know that if $\{M_n\}$ is a martingale, we know from definition of martingale that $E(M_n) = M_0$ for all $n \geq 0$. However, if we only know that a sequence of random variables $\{X_n\}$ is a local martingale, do we still have the result that $$E(X_n) = X_0, \forall n\geq 0?$$

And why?

Answer 1

Based on the first paragraph of Wiki, no:

Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.