Two-type continuous-state branching processes in varying environments
Zenghu Li, Junyan Zhang
TL;DR
The work advances the theory of two-type continuous-state branching processes in varying environments by formulating and solving a backward cumulant semigroup equation under time-dependent, càdlàg parameters. Through a two-dimensional Gronwall framework and a nonlinear h-transform, the authors prove existence and uniqueness for both simplified and full backward equations, while carefully handling bottlenecks with a finite set and a moment condition. The resulting inhomogeneous transition semigroup is characterized by a Laplace-transform representation, with a Lévy–Khintchine type structure that aligns with a detailed branching mechanism. This provides a rigorous basis for constructing and analyzing multi-type CBVE processes and sets the stage for broader generalizations.
Abstract
A basic class of two-type continuous-state branching processes in varying environments are constructed by solving the backward equation determining the cumulant semigroup. The parameters of the process are allowed to be càdlàg in time and the difficulty brought about by the bottlenecks are overcome by introducing a suitable moment condition.