Continuous-state branching processes (CB-processes) and continuous-state branching processes with immigration (CBI-processes) constitute important classes of Markov processes taking values in the positive real line. They were introduced as probabilistic models describing the evolution of large populations with small individuals. The study of CB-processes was initiated by Feller (1951), who noticed that a branching diffusion process may arise in a limit theorem of Galton–Watson discrete branching processes. A characterization of CB-processes by random time changes of Levy processes was given by Lamperti (1967). The convergence of rescaled discrete branching processes with immigration to CBI-processes has also been studied by a number of authors. From a mathematical point of view, the continuous-state processes are usually easier to deal with because both their time and state spaces are smooth, and the distributions that appear are infinitely divisible.
A continuous CBI-process with subcritical branching mechanism was used by Cox, Ingersoll and Rose (1985) to describe the evolution of interest rates and it has been known in mathematical finance as the Cox–Ingersoll–Ross model (CIR-model). Compared with other financial models introduced before, the CIR-model is more appealing as it is positive and mean-reverting. A natural generalization of the CBI-process is the affine Markov process, which has also been used a lot in mathematical finance.
The approach of stochastic equations has been proved useful in the recent developments in the theory and applications of CB- and CBI-processes. A flow of CB-processes was constructed in Bertoin and Le Gall (2006) by weak solutions to a jump-type stochastic equation. The strong existence and uniqueness for a stochastic equation of general CBI-processes were established in Dawson and Li (2006). The results of Bertoin and Le Gall (2006) were extended to flows of CBI-processes in Dawson and Li (2012) using strong solutions to stochastic equations driven by time-space noises. For the stable branching CBI-process, a strong stochastic differential equation was established in Fu and Li (2010).
The purpose of this course is to provide an introduction to CB- and CBI-processes including a quick development of their stochastic equations. The proofs given here are more elementary than those appearing in the literature before. We hope the treatments are understandable to the audience with reasonable background in probability theory and stochastic processes. Main contents: Construction of CB-processes; Some basic properties; Positive integral functionals; Construction of CBI-processes; Martingale problems for CBI-processes; Stochastic equations for CBI-processes; Local and global maximal jumps; Reconstructions from excursions.