Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Zhu, Lingjiong

doi:10.1017/s002190020001161x

Cited by 22 publications

(12 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Gao

Zhu

2018

Queueing Syst

Self Cite

View full text Add to dashboard Cite

show abstract

“…Assumptions are readily verified as linear functions that are differentiable. Boundedness of the first and second moment of the stationary solution, which is proven in Zhu (), can be used to verify Assumptions and . Assumption holds thanks to the linearity of λ .…”

Section: Lan For Continuous‐time Observationsmentioning

confidence: 99%

“…In fact, by inspection of the proofs in the appendix, it is clear that given the linearity of μ , σ 2 and λ , Assumption can be replaced by

{double-struckE}_{bold-italicθ, bold-italicη} ([], | ξ |^{2}) < \infty

, as long as f satisfies proper conditions. The boundedness of the second moment of ξ is proven in Zhu (). As far as f is concerned, it is sufficient, for example, that

\partial_{x} \frac{\nabla_{bold-italicθ} f^{1 / 2} (bold-italicθ, x)}{f^{1 / 2} (bold-italicθ, x)} \leq M < \infty

.…”

Section: Equivalence Of the Discrete‐time Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

Becheri

Drost

Werker

2015

Scandinavian J Statistics

View full text Add to dashboard Cite

show abstract

“…We apply functional central limit theorem to obtain approximations and use that to study finite time ruin probability. The limit theorems have also been studied for an extension of linear Hawkes processes and Cox-Ingersoll-Ross processes in Zhu [33], which has applications in short interest rate models in finance.…”

Section: Introductionmentioning

confidence: 99%

Diffusion Approximation of a Risk Model with Non-Stationary Hawkes Arrivals of Claims

Cheng

Seol

2019

Methodol Comput Appl Probab

View full text Add to dashboard Cite

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given. MSC2010: primary 91B30; secondary 60F17, 60G55.

show abstract

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Cited by 22 publications

References 25 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

Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

Diffusion Approximation of a Risk Model with Non-Stationary Hawkes Arrivals of Claims

Contact Info

Product

Resources

About