Given a continuous-time stochastic process X(t), can we say that X(t) is a martingale with respect to its natural filtration only if its infinitesimal generator satisfies
$$ L(X) = \lim_{s \downarrow 0} \frac{\mathbb{E} [ X(t+ s)|X(t)] - X(t)}{s} = 0 \qquad $$ for all $X(t)$?