Assume $M$ is a continuous, local martingale s.t. for a single given $T$ we have $M(T)\geq 0$ and $P(M(T)>0)>0$. Can we then deduce $M(t)\geq 0$ for $t\leq T$?
I'm trying to use the good old Fatou trick showing a nonnegative local martingale is a supermartingale, but I only have a single point of non-negativity as above.
Thanks.
