Time-changed spectrally positive Lévy processes started from infinity

Abstract

Consider a spectrally positive Lévy process Z with log-Laplace exponent Ψ and a positive continuous function R on $(0,\infty )$. We investigate the entrance from infinity of the process X obtained by changing time in Z with the inverse of the additive functional $\mathit{\eta }(\mathit{t})={\int }_0^{\mathit{t}}\frac{\mathrm{d}\mathit{s}}{\mathit{R}({\mathit{Z}}_{\mathit{s}})}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process X. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift $\mathit{\delta }:=-\mathit{\gamma }<0$, sufficient conditions over R and Ψ are found for the process to come down from infinity along the deterministic function $({\mathit{x}}_{\mathit{t}},\mathit{t}\ge 0)$ solution to $\mathrm{d}{\mathit{x}}_{\mathit{t}}=-\mathit{\gamma }\mathit{R}({\mathit{x}}_{\mathit{t}})\mathrm{d}\mathit{t}$ with ${\mathit{x}}_0=\infty$. If $\mathrm{\Psi }(\mathit{\lambda })\sim \mathit{c}{\mathit{\lambda }}^{\mathit{\alpha }}$ as $\mathit{\lambda }\to 0$, $\mathit{\alpha }\in (1,2]$, $\mathit{c}>0$ and R is regularly varying at ∞ with index $\mathit{\theta }>\mathit{\alpha }$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.