Diffusion Approximations for Self-excited Systems with Applications to General Branching Processes
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866913214041686016 |
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| author | Xu, Wei |
| author_facet | Xu, Wei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_01288 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Diffusion Approximations for Self-excited Systems with Applications to General Branching Processes Xu, Wei Probability 60F17, 60G55, 60J80, 62P10 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. |
| title | Diffusion Approximations for Self-excited Systems with Applications to General Branching Processes |
| topic | Probability 60F17, 60G55, 60J80, 62P10 |
| url | https://arxiv.org/abs/2101.01288 |