12. Continuous-State Branching Processes

See also Silverstein (1968).

As usual, for x≥0, the measure P _x satisfies the property P _x (Y ₀=x)=1.

Recall that our definition of spectrally positive processes excludes subordinators. See the discussion following Lemma 2.14.

As usual, we understand the process X killed at rate q to mean that it is sent to +∞ after an independent and exponentially distributed time with parameter q. Further q=0 means there is no killing.

Here and later on, we make an abuse of notation in working with the independent and exponentially distributed random variables e _q . In taking limits as q↓0 in, for example, (12.21), we appear to be working with an uncountable sequence of independent exponential random variables on the same probability space. This is possible, on account of the fact that, for each q>0, we may write e _q =q ⁻¹ e ₁.

Examples of such processes when ϕ(λ)=cλ ^α for α∈(0,1) and c>0 are considered by Etheridge and Williams (2003).

This exercise is due to Prof. A.G. Pakes.

In Theorem 4.7, the Kella–Whitt martingale was only introduced for Lévy processes with bounded variation paths. Thereafter it was noted that, in fact, the conclusion of this theorem is still valid when the Lévy process has paths of unbounded variation.

There is a minor error in Grey (1974). In the current setting, Theorem 3 (ii) of this paper states that P ₁(Ξ=0)=P ₁(ζ<∞), which cannot be true for all supercritical continuous-state branching processes. Indeed, suppose that ∫^∞1/ψ(ξ)dξ=∞, so that P ₁(ζ<∞)=0. In that case, Theorem 12.7 tells us that P ₁(lim _t↑∞ Y _t =0)=exp{−Φ(0)}. However, since λ∈(0,Φ(0)), η _t (λ)→0 as t↑∞ and we see that η _t (λ)Y _t →0 on {lim _t↑∞ Y _t =0}. This also implies that exp{−Φ(0)}≤P ₁(Ξ=0), which is a contradiction. The error occurs on line 11 of p. 675 where it is claimed that “ϕ(θ)→−logq (which may be +∞) as θ→∞”.