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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.

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  • $\begingroup$ Consider an iid sequence as a candidate for martingale under the proposed definition. $\endgroup$
    – Calculon
    Commented Dec 12, 2017 at 12:56

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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.

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