= Path and moment criteria from a Lévy triplet
{c}
For the small-jump cutoff convention in the <Lévy–Khintchine formula>[Lévy-Khintchine formula], let $(a,b,K)$ be the triplet of a real <Lévy process>[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 interval>[compact time intervals] has the different criterion
$$
b=0,\qquad \int_{|x|\leq1}|x|K(dx)<\infty.
$$
Under these conditions the uncompensated <Lévy–Khintchine formula>[Lévy-Khintchine formula] gives a drift plus an absolutely summable small-jump process and finitely many large jumps. A nonzero <Brownian motion>[Brownian component] precludes <finite variation>, while an infinite absolute small-jump sum already forces infinite variation.
Integrability requires only the large-jump first <moment>:
$$
\mathbb E|X_1|<\infty\quad\Longleftrightarrow\quad
\int_{|x|>1}|x|K(dx)<\infty.
$$
The small <compensated Poisson integral>[compensated jumps] are square <integrable> and hence <integrable>. When this condition holds, $\mathbb EX_t=t[a+\int_{|x|>1}xK(dx)]$; the small-jump compensation convention is why no small-jump <mean> appears.
Likewise a finite <second moment> is equivalent to $\int x^2K(dx)<\infty$, and then
$$
\operatorname{Var}(X_t)=t\left(b+\int x^2K(dx)\right).
$$
The <Lévy–Itô decomposition>[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.
Back to article page
