Prove that the process is a martingale

Question

I know that the process $X_t$ is a martingale when:

• $\mathbb{E}[|X_t|]<\infty$ for each $t$,
• $\mathbb{E}[X_t|\mathcal{F_s}] = X_s$ for $s<t$.

Using properties of the conditional expectation we can show that:

$\mathbb{E}[X_s] = \mathbb{E}[X_t]$.

My question is:

Can we use this property ($\mathbb{E}[X_s] = \mathbb{E}[X_t]$) to prove that the process is a martingale?

I noted that in many sources people show that $\mathbb{E}[X_t|\mathcal{F_s}] = X_s$ to prove that the process is a martingale.

I think that these conditions are equivalent and showing the first condition should be easier. Am I right?

Answer 1

Fractional Brownian motion with $H\neq 1/2$ is not a martingale yet $E(B_s)=E(B_t)=0$ for all $s,t$.