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If a stochastic process has the same expectation value for all pisitive t, then is it a martingale? I don’t know how to show it whether that is right.

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The Brownian motion $W_t$ is a martingale because for all $u>t$: $$ W_t = E(W_u | \mathcal{F}_t), $$

However, the process $X_t$
$$X_t = 1_{\{a<t<b\}} (2 . 1_{\{W_a < 0\}} -1) $$ has 0 expectation but is not a martingale.

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  • $\begingroup$ Hmm... Why 21? If you give more information , I'll very thanks for you. $\endgroup$ – Hobong Dec 10 '18 at 15:33
  • $\begingroup$ I just added the dot between 2 and 1, this process takes value +1 or -1 between a and b, but it is not a martingale as it goes back to 0 for $t>b$ $\endgroup$ – Sebapi Dec 10 '18 at 15:50

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