 Solution

Relative to a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a filtration $({\mathcal{F}}_n)$, the real stochastic process $(M_n)$ is a martingale if it is adapted, each $M_n$ is an integrable random variable, andHere adapted means $M_n$ is ${\mathcal{F}}_n$-measurable, and integrability means $\mathbb{E}|M_n|<\infty$. By the tower property of conditional expectation, the equivalent multi-step condition is $\mathbb{E}[M_j\mid {\mathcal{F}}_i]=M_i$ for all $i\le j$. The filtration is part of the definition; specifying only the marginal means is insufficient.