Ito-Levy decomposition for $\alpha$-stable processes?

Question

The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?

Answer 1

$\newcommand\al\alpha\newcommand\R{\mathbb R}$Let $(X_t)_{t\ge0}$ be a nondegenerate $\al$-stable Lévy process (so that $P(X_t=a)\ne1$ for all $t\in(0,\infty)$ and all $a\in\R$). According to (say) Theorem 2.2.1, if $\al\in(0,2)$, then
$$Ee^{isX_t}=\exp\Big\{t\Big(ics+\int_{-\infty}^0g(s,u)dM(u) +\int_0^\infty g(s,u)dN(u)\Big)\Big\}$$ for all real $t\ge0$ and all real $s$, where $g(s,u):=e^{isu}-1-isu\,1(|u|<1)$, $M(u):=c_1(-u)^{-\al}$, $N(u):=-c_2 u^{-\al}$, $c\in\R$, $c_1\in[0,\infty)$, $c_2\in[0,\infty)$, and $c_1+c_2>0$; and if $\al=2$, then $$Ee^{isX_t}=\exp\big\{t(ics-\sigma^2 s^2/2)\big\}$$ for all real $t\ge0$ and all real $s$, where $c\in\R$ and $\sigma\in(0,\infty)$.

(There is a typo in the expression of $N(u)$ in the cited theorem: there must be $-\al$ there in place of $\al$.)