Time-changed spectrally positive L\'evy processes starting from infinity

Abstract

Consider a spectrally positive L\'evy 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 $\eta(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. 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 L\'evy 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 $(x_t,t\geq 0)$ solution to ${\rm d} x_t=-\gamma R(x_t) {\rm d} t$, with $x_0=\infty$. When $\Psi(\lambda)\sim c\lambda^{\alpha}$, with $\lambda \rightarrow 0$, $\alpha\in (1,2]$, $c>0$ 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.