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

Wei Xu

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.

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

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.

Paper Structure

This paper contains 33 sections, 37 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Branching representation
  4. Stochastic Volterra representation
  5. Examples
  6. Limit theorems for self-excited dynamical systems
  7. Scaling limit for intensity processes
  8. Asymptotic assumptions
  9. Asymptotic analysis in intuition
  10. Weak convergence of rescaled intensity processes
  11. Examples
  12. Scaling limits for marked Hawkes shot noise processes
  13. Cumulative response function
  14. Instantaneous response function
  15. Limits theorems for multi-type CMJI-processes
  16. ...and 18 more sections

Key Result

Proposition 2.1

The random point measure $N'_\mathcal{H}(dt,du)$ is a realization of the multivariate MHPI-measure defined by (eqn1.01)-(eqn1.04).

Theorems & Definitions (62)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4: Martingale representation
  • Example : Exponential type
  • Example : Completely monotone type
  • Example : Convolution type
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 52 more