Can a negative local martingale start from 0?

Question

In a recent paper that I've found it is written that if $M_t$ is a local martingale such that $M_0=0$ and $M_t<0$ for all $t>0$ then this is a contradiction.

I don't see why. How this can be true?

Answer 1

$X:=-M$ is then a non-negative local martingale, let's say localized by the sequence $(T_n)$ of stopping times. You have $0=\Bbb E[X_0]=\Bbb E[X_{t\wedge T_n}]$ for each $n\ge 1$ and $t>0$, by optional stopping. Now let $n\to\infty$ and use Fatou's lemma; conclusion: $\Bbb E[X_t]=0$ for each $t>0$.