Continuous-state branching processes with immigration

TL;DR

This paper introduces continuous-state branching processes with immigration, providing accessible proofs, explicit calculations, and new stochastic equations, enhancing understanding and analysis of these stochastic processes for graduate students.

Contribution

It offers elementary proofs, explicit functional calculations, and new stochastic equations for CB- and CBI-processes, making the theory more accessible and detailed.

Findings

Explicit Laplace transform calculations for functionals

Construction of processes as limits of Galton--Watson models

Proof of strong Feller property and exponential ergodicity

Abstract

This work provides a brief introduction to continuous-state branching processes (CB-processes) and continuous-state branching processes with immigration (CBI-processes) accessible to graduate students with reasonable background in probability theory and stochastic processes. In particular, we give a quick development of the stochastic equations of the processes and some immediate applications. The proofs given here are more elementary than those appearing in the literature before. We have made them readable without requiring too much preliminary knowledge on branching processes and stochastic analysis. In Section 1, we review some properties of Laplace transforms of finite measures on the positive half line. In Section 2, a construction of CB-processes is given as rescaling limits of Galton--Watson branching processes. This approach also gives the physical interpretation of the CB-processes. Some basic properties of the processes are developed in Section 3. The Laplace transforms of some positive integral functionals are calculated explicitly in Section 4. In Section 5, the CBI-processes are constructed as rescaling limits of Galton--Watson branching processes with immigration. In Section 6, we present reconstructions of the CB- and CBI-processes by Poisson random measures determined by entrance laws, which reveal the structures of the trajectories of the processes. Several equivalent formulations of martingale problems for CBI-processes are given in Section 7. From those we derive the stochastic equations of the processes in Section 8. Using the stochastic equations, some characterizations of local and global maximal jumps of the CB- and CBI-processes are given in Section 9. In Section 10, we prove the strong Feller property and the exponential ergodicity of the CBI-process under suitable conditions using a coupling based on one of the stochastic equations.