Jump activity of a Lévy process

The jump activity of a Lévy process describes whether it has finitely or infinitely many jumps in a bounded time interval. Its Lévy measure $\nu$ decides this: finite activity corresponds to $\nu ({\mathbb{R}}^d\setminus \{0\})<\infty$, in which case the jump component is a Compound Poisson process. Infinite mass gives infinitely many small jumps in every positive-length time interval, almost surely. For every $\epsilon >0$, however, jumps larger than $\epsilon$ still have a finite rate.

Activity and total variation are different. With zero Brownian component, sample paths have finite variation on compact time intervals exactly whenFor example, $\nu (dx)=c{x}^{-1-\alpha }dx$ on $(0,\infty )$, with $c>0$ and $0<\alpha <1$, has infinite total mass but a finite integral of $x$ near zero. It therefore permits infinite jump activity with finite variation. Conversely, Brownian motion has no jumps but infinite total variation on every positive-length time interval, almost surely.^[ref]