 Martingale

A discrete-time martingale relative to a filtration $({\mathcal{F}}_n)$ is an adapted process $(M_n)$ with $\mathbb{E}|M_n|<\infty$ and $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$ almost surely. Given all information available at time $n$, the current value is the conditional prediction of the next value. A simple symmetric random walk is an example: its next independent increment has mean zero. The tower property implies $\mathbb{E}[M_m\mid {\mathcal{F}}_n]=M_n$ for $m\ge n$, hence a constant expected value. Constant unconditional means alone do not establish the conditional property. The choice of information is part of the definition.^[ref]