Time-changed spectrally positive Lévy processes starting from infinity

Abstract

Consider a spectrally positive Lévy process Z with log-Laplace exponent Ψ and a positive continuous function R on (0, ∞). We investigate the entrance from ∞ of the process X obtained by changing time in Z with the inverse of the additive functional, for any t ≥ 0. This process can be viewed as a continuous-state branching process with non-linear branching rate defined recently in Li et al. [28]. We provide a necessary and sufficient condition for ∞ to be an entrance boundary. 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 δ := −γ < 0, sufficient conditions over R and Ψ are found for the process to come down from infinity along the deterministic function (x t , t ≥ 0) solution to dx t = −γR(x t )dt, with x 0 = ∞. When the Lévy process oscillates, the process X may come down from infinity for certain functions R. We find a renormalisation in law of its running infimum at small times, when Ψ(λ) ∼ λ α , as λ → 0, for α ∈ (1, 2] and R is regularly varying at ∞ with index θ > α.

On the explosion of a class of continuous-state nonlinear branching processes

“…The behaviors of extinction, explosion and coming down from infinity for such a process are discussed in Li (2019). A more general class of continuous-state branching processes is proposed in Foucart et al (2019) via Lamperti type time change of stopped spectrally positive Lévy processes using rate functions R defined on (0, ∞), where the classical continuous-state branching process corresponds to the linear rate function of R(x) = x and the model in Li (2019) corresponds to the rate function of R(x) = x θ . The above continuous-state nonlinear branching processes are further generalized in as solutions to more general versions of the Dawson-Li equation.…”

“…For the continuous-state nonlinear branching processes, on one hand, the nonadditive branching mechanism allows richer boundary behaviors such as coming down from infinity; on the other hand, many classical techniques based on the additive branching property fail to work. Criteria for extinction, explosion and coming down from infinity are developed in Li (2019), and Foucart et al (2019) for the respective continuousstate nonlinear branching processes via a martingale approach and fluctuation theory for spectrally positive Lévy processes.…”

“…The speed of coming down from infinity for such processes is studied in Foucart et al (2019) by analyzing the asymptotic behaviors of weighted occupation times for the associated spectrally positive Lévy process. Sufficient conditions are found under which the continuous-state nonlinear branching process comes down from infinity along a deterministic curve.…”

“…For the continuous-state nonlinear branching processes introduced in Foucart et al (2019), explosion occurs when the process X has a positive drift and the rate function increases fast enough near infinity. In this paper we study the explosion behaviors for such a continuous-state branching process X.…”

“…In particular, in the fast regime the speed of explosion is asymptotically proportional to − log t as time t → 0+. Some parts of our approach resemble those in Foucart et al (2019) and in Bansaye et al (2016) for studying the coming down from infinity behaviors of the respective processes. But an additional difficulty emerges in our work due to the overshoot when the nonlinear branching process first upcrosses a level x at time T +…”