Recognizing a Martingale

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!

• 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 $\mathbb{E}_0[X_t] = X_0$. Oct 8, 2020 at 0:31
• For two try $W_t^3$. For the other please see here: math.stackexchange.com/questions/1186859/… Oct 16, 2020 at 22:22