Abstract
In this article we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d \geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a suitable comparison with one-dimensional CBI processes and an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for transience of multi-type CBI processes.
Highlights
Continuous-state branching processes with immigration form a class of time-homogeneous Markov processes with state spaceRd+ = {x ∈ Rd | x1, . . . , xd ≥ 0}, d ∈ N, whose Laplace transform is an exponentially affine function of the initial state variable
We provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction
The same technique is used to provide a sufficient condition for the transience of multi-type CBI processes
Summary
Continuous-state branching processes with immigration (abbreviated as CBI processes) form a class of time-homogeneous Markov processes with state space. In order to study the multidimensional case, we establish first in Section 2 a general comparison principle for multi-type CBI processes Such comparison allows us to relate two CBI processes with different admissible parameters with respect to the classical order on R+. Assuming in addition mild additional conditions such that this CBI process has a unique invariant measure (see [22]), this method can be combined with the coupling argument from [16] to show that its transition probabilities converge in the total variation distance to the unique invariant measure Such an approach was first established in [13] for the anisotropic stable JCIR process, see Example 3.8 for its definition. Let us mention that convergence in total variation for extensions of CBI processes have been studied by different techniques in [32, 18]
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