Time-changed spectrally positive Lévy processes starting from infinity

Abstract

Consider a spectrally positive Lévy process $Z$ with log-Laplace exponent $\Psi$ and a positive continuous function $R$ on $(0,\infty )$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $\eta (t)={\int }_0^t\frac{\mathrm{d}s}{R(Z_s)}$. 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 $\delta :=-\gamma <0$, sufficient conditions over $R$ and $\Psi$ are found for the process to come down from infinity along the deterministic function $(x_t,t\ge 0)$ solution to $\mathrm{d}{x}_t=-\gamma R(x_t)\mathrm{d}t$, with $x_0=\infty$. When $\Psi (\lambda )\sim c{\lambda }^{\alpha }$, with $\lambda \to 0$, $\alpha \in (1,2]$, $c>0$ and $R$ is regularly varying at $\infty$ with index $\theta >\alpha$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.