A stochastic process $(X_n)_{n \in \mathbb{N}}$ is a martingale if:

\[\mathbb{E}[X_{n+1} | \mathcal{F}_{n}] = X_n\]

$\mathcal{F}_n$ is a filtration designating all information we have a time $n$.

Thus, a martingal is a process whose expectation is equal to the current value. In other words, the current value is the best estimate of its future value. This process has no drift (otherwise the best estimate of the future value would be the current value + drift).

A random walk is a martingale.

A Brownian motion with no drift is a martingale.

A process $K$ if $\mathcal{F}$-adapted if at all time $t$, $K_t$ depends only on all information available until time $t$.

A semimartingale is a process that is not always a martingale (e.g. stock A can be correctly predicted only the first week). Mathematically:

\[X_t = M_t + A_t\]

where $M$ is a martingale and $A$ is a continuous at right and has a limit at left (“càdlag”).