Stochastic Process and Martingale

Question

I am confused about the following question. Can anyone explain me me? Thank you

Is the stochastic process a martingale if it has the same expectation value for all $t>= 0$?

the explanation of $t$ is given like this : $W(t), t>=0$, be a Brownian motion.

I would really appreciate your help.

Answer 1

No it's not enough for $(X_t)_{t\ge 0}$ being a martingale that $$E[X_t] = E[X_0]$$ holds for all $t \ge 0$.

But it is sufficient if you have an adapted process that for all bounded stopping times $\tau$ it holds $$E[X_\tau] = E[X_0]$$ then the process is a martingale.