Path and moment criteria from a Lévy triplet

For the small-jump cutoff convention in the Lévy-Khintchine formula, let $(a,b,K)$ be the triplet of a real Lévy process. Its paths are continuous almost surely exactly when $K=0$. They are everywhere differentiable almost surely exactly when $K=0$ and $b=0$, giving the deterministic path $X_t=at$. A compound Poisson path can have a derivative between its jumps without being an everywhere differentiable path.

finite variation on compact time intervals has the different criterionUnder these conditions the uncompensated Lévy-Khintchine formula gives a drift plus an absolutely summable small-jump process and finitely many large jumps. A nonzero Brownian component precludes finite variation, while an infinite absolute small-jump sum already forces infinite variation.

Integrability requires only the large-jump first moment:The small compensated jumps are square integrable and hence integrable. When this condition holds, $\mathbb{E}{X}_t=t[a+{\int }_{|x|>1}xK(dx)]$; the small-jump compensation convention is why no small-jump mean appears.

Likewise a finite second moment is equivalent to $\int x^{2}K(dx)<\infty$, and thenThe Lévy-Itô decomposition proves these moment statements by independent Brownian, compensated-small-jump and compound-Poisson-large-jump parts. The large-jump one-event case also shows why the tail moment conditions are necessary.