Instructor: Dr. Wei Xu
This lecture gives a systematic introduction into the theory of branching particle systems and self-exciting point processes. This lecture consists of four parts. In the first part, we give an introduction on time-inhomogeneous random point processes, including Cox processes, cluster processes, self-exciting point processes and marked point processes. We not only study their basic properties but also explore their spectral representations and limit theorems. In the second part, we first introduce some elementary properties of continuous-time Galton-Watson branching processes including their generating function, criticality and asymptotic properties. Subsequently, we give a short but detailed introduction into the general theory of Non-Markovian branching processes including their generating functions and extinction properties. In the third part, we study marked point processes associated to the general branching processes. Conversely, for any self-exciting process we also give a cluster representation in term of some general branching process. Based on this one-to-one correspondence, we study the inner branching mechanism of self-exciting processes and its asymptotic properties via the theory of branching particle systems. Moreover, applying the convergence theorems of classic branching processes, we give several diffusion approximations for self-exciting processes including functional central limit theorems and scaling limit theorems. In the last part of this lecture, we offer a method to explore the general branching processes via the related self-exciting processes. We close with a brief overview on recent own research.
Prerequisites (1) Stochastics I and II
(2) Undergraduate Real Analysis and Measure Theory
(3) Some familiarity with stochastic integrals would be helpful
Lecture notes will be available as the course progresses. Other good books can be referred.
Athreya, K.B. and Ney, P.E. (1972): Branching Processes. Springer, Berlin Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley & Sons, London and New York.
Daley, D. J. and Vere-Jones, D. (2003). An introduction to the theory of point processes. Vol. I. Probability and its Applications. Springer Science & Business Media.
Daley, D. J. and Vere-Jones, D. (2007). An introduction to the theory of point processes: volume II: general theory and structure. Springer Science & Business Media.