CONTINUOUS-STATE BRANCHING PROCESSES WITH IMMIGRATION
Zenghu Li
These notes were used
in a short graduate course on branching processes the author gave in Beijing
Normal University. They provide a brief introduction to continuous-state
branching processes with or without immigration. The processes are constructed
by taking rescaling limits of classical discrete-state branching models. We
give quick developments of the martingale problems and stochastic equations of
the continuous-state processes. The proofs 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. Using the stochastic equations, we give characterizations of the
local and global maximal jumps of the processes. Under suitable conditions,
their strong Feller property and exponential ergodicity
are studied by a coupling method based on one of the stochastic equations.
The notes were
published as a chapter in: From Probability to Finance (2020), pp. 1-69, edited
by Y. Jiao, Mathematical Lectures from Peking University, Springer.
Download the pdf file of the notes.