Sub-Martingale and Martingale

Question

An integrable sub-martingale $S_t$ with $\mathbb E(S_t)$ being a constant is a martingale.

Is this statement true, please? I think so.

Answer 1

Yes it is.

You know that $$ E[S_{t+h}|S_t] \ge E[S_t] $$

Th expectation of both terms if $$ E[S_{t+h}] \ge E[S_t] $$and as this is an euqality, you have a.s. equality in the first equation.

Answer 2

If $t>s$, then ${\rm E}[S_t\mid \mathcal{F}_s]-S_s\geq 0$ a.s. by the sub-martingale property. Since this random variable also has mean zero, we may conclude that it is zero a.s.