= Lévy measure of a symmetric Laplace time-one law
{c}
{title2=$K(dy)=e^{-\alpha|y|}|y|^{-1}dy$}
For $\alpha>0$, the <Lévy process> with <characteristic function> $\mathbb E e^{iuX_t}=(1+u^2/\alpha^2)^{-t}$ has zero Gaussian coefficient, zero uncompensated drift, and the displayed <Lévy measure>. Indeed, the symmetric jump integral is $2\int_0^\infty(\cos(uy)-1)e^{-\alpha y}dy/y$. Differentiating in $u$ gives $-2u/(\alpha^2+u^2)$, and integrating back from zero gives $-\log(1+u^2/\alpha^2)$. The time-one law is the <Laplace distribution> of rate $\alpha$. The jump activity is infinite, since $K$ has infinite mass near zero, but the jump variation is finite because $\int(1\wedge|y|)K(dy)<\infty$.
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