Martingale

Let $\{X_t\}_{t \geq 0}$ be an adapted process (Definition~Filtration and Adapted Process) with $X_t \in L^1$ (Definition~Expectation) for all $t$. The process is a martingale with respect to $(\mathcal{F}_t)$ if \[ \mathbb{E}[X_t \mid \mathcal{F}_s] = X_s \quad \mathbb{P}-a.s.\ for all 0 \leq s \leq t. \] It is a supermartingale (resp.\ submartingale) if the equality is replaced by $\leq$ (resp.\ $\geq$).