Diffusion Approximations for Self-excited Systems with Applications to General Branching Processes

TL;DR

This paper establishes convergence and diffusion approximation results for nearly critical self-excited systems modeled by multivariate Hawkes processes, linking them to multi-type branching processes and revealing phenomena like state space collapse.

Contribution

It introduces new diffusion approximation results for multivariate Hawkes processes and their applications to branching processes and queueing theory, highlighting the state space collapse phenomenon.

Findings

Rescaled density processes behave like multi-type continuous-state branching processes.

Diffusion approximations for multi-type Crump-Mode-Jagers branching processes are established.

Observation of state space collapse in queueing systems at large time scales.

Abstract

In this work, several convergence results are established for nearly critical self-excited systems in which event arrivals are described by multivariate marked Hawkes point processes. Under some mild high-frequency assumptions, the rescaled density process behaves asymptotically like a multi-type continuous-state branching process with immigration, which is the unique solution to a multi-dimensional stochastic differential equation with dynamical mechanism similar to that of multivariate Hawkes processes. To illustrate the strength of these limit results, we further establish diffusion approximations for multi-type Crump-Mode-Jagers branching processes counted with various characteristics by linking them to marked Hawkes shot noise processes. In particular, an interesting phenomenon in queueing theory, well-known as state space collapse, is observed in the behavior of the population structure at a large time scale. This phenomenon reveals that the rescaled complex biological system can be recovered from its population process by a lifting map.