Under which conditions is the stochastic process $\{X_t\}_{t=0,1,...,T}$ a martingale? Demonstrate and explain clearly for each case below. If it is not necessarily a martingale, provide a counterexample.

$\mathbb{E}_t[X_s] = X_t$ for $0 \leq t \leq s \leq T$

$\mathbb{E}_0[X_t] = X_0$ for $0 \leq t \leq T$

$\mathbb{E}_t[X_T] = X_t$ for $0 \leq t \leq T$

I am having a bit of a hard time figuring this out. My hunch is that, as long as (1) $\mathbb{E}[|X_t|] < \infty$ and (2) $X_t$ is adapted to a filtration $\mathcal{F}_T$, then all of the stochastic processes listed above are Martingales because they are just equivalent ways of stating the first property of Martingales, that is, $\mathbb{E}[X_T|\mathcal{F}_t] = X_t$ for $0 \leq t \leq T$ . Therefore, a counterexample could be anything that does not satisfy properties (1) and (2). Is this correct? If not, why? Thanks!

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    $\begingroup$ The second is definitely not sufficient. If the marginal distribution at any time in the future has any weight at $X_t = 0$, but subsequent times have any probability of becoming non-zero, then it fails the martingale property. But such a process could easily satisfy $\mathbb{E}_0[X_t] = X_0$. $\endgroup$ – StackG Oct 8 '20 at 0:31
  • $\begingroup$ For two try $W_t^3$. For the other please see here: math.stackexchange.com/questions/1186859/… $\endgroup$ – Magic is in the chain Oct 16 '20 at 22:22

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