Lévy process

A Lévy process starts at zero, has independent stationary increments, is stochastically continuous, and is conventionally taken to have càdlàg paths.

If a Lévy process has mean zero and $\mathrm{Var}(X_1)={\sigma }^2<\infty$, then $\mathrm{Var}(X_t)=t{\sigma }^2$ and $X_t^2-t{\sigma }^2$ is a martingale.