Affine Point Processes: Approximation and Efficient Simulation

Zhang, Xiaowei; Blanchet, José; Giesecke, Kay

doi:10.1287/moor.2014.0696

Cited by 27 publications

(39 citation statements)

References 41 publications

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“…Note that most of the existing literature on limit theorems for Hawkes processes are for large-time asymptotics, where one scales both time and space. See [1,6,57] for large-time asymptotics of linear Hawkes processes, [38,59] for large-time asymptotics for extensions of linear Hawkes processes, [34,35] for the nearly unstable case where h L 1 ≈ 1, [55] for the generalized Markovian Hawkes processes (or affine point processes), and [56] for large-time asymptotics of nonlinear Hawkes processes.…”

Section: Introductionmentioning

confidence: 99%

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Gao

Zhu

2018

Queueing Syst

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A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infiniteserver queues with multivariate Hawkes traffic.Organization of this paper. The rest of the paper is organized as follows. In Section 2, we formally introduce stationary linear Hawkes processes and review some of their properties. In Section 3, we state the main result on the functional central limit theorem for univariate stationary Hawkes processes with large baseline intensity µ and describe the properties of the limiting Gaussian process. In Section 4, we develop heavytraffic approximations for infinite-server queues with univariate Hawkes traffic. We also discuss in detail the special case when service times are exponentially distributed. In Section 5, we extend our results to multivariate stationary Hawkes processes and study infinite-server queues with multivariate Hawkes traffic. The proofs of all the results are collected in the Appendix.

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Section: Introductionmentioning

confidence: 99%

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Gao

Zhu

2018

Queueing Syst

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“…The large deviations for extensions of Hawkes processes have also been studied in the literature, see e.g. Karabash and Zhu [24] for the linear marked Hawkes process, and Zhu [35] for the Cox-Ingersoll-Ross process with Hawkes jumps and also Zhang et al [31] for affine point processes. Other than the large deviations, the central limit theorems for linear, nonlinear and extensions of Hawkes processes have been considered in, e.g., [4,36,35].…”

Section: Introductionmentioning

confidence: 92%

Large deviations and applications for Markovian Hawkes processes with a large initial intensity

Gao¹,

Zhu²

2018

Bernoulli

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Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, insurance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study linear Hawkes process with an exponential kernel in the asymptotic regime where the initial intensity of the Hawkes process is large. We establish large deviations for Hawkes processes in this regime as well as the regime when both the initial intensity and the time are large. We illustrate the strength of our results by discussing the applications to insurance and queueing systems.where λ(·) : R + → R + is locally integrable, left continuous, φ(·) : R + → R + and we always assume that, where τ are the occurrences of the points before time t. In the literature, 1imsart-bj ver.

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“…In order to fulfil Assumption , additional conditions on the drift, the volatility, the jump size and the intensity function are needed. Such conditions for affine‐jump diffusions can be found in Zhang et al (). Masuda () provides conditions for jump diffusions of the Lévy type, which can be used when the intensity does not depend on X .…”

Section: Lan For Continuous‐time Observationsmentioning

confidence: 97%

Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

Becheri

Drost

Werker

2015

Scandinavian J Statistics

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Affine Point Processes: Approximation and Efficient Simulation

Abstract: If it is the author's pre-published version, changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this work, please refer to the published version.

Cited by 27 publications

References 41 publications

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Large deviations and applications for Markovian Hawkes processes with a large initial intensity

Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

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