Moderate deviations for marked Hawkes processes

Seol, Youngsoo

doi:10.1007/s10114-017-6433-7

Cited by 11 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Zhu [Zhu3b] proved the moderate deviation principle for a univariate linear Hawkes process, which fills in a gap between central limit theorem and large deviations. The moderate deviation principle for expansion of Hawkes processes like marked linear Hawkes processes and multivariate unmarked linear Hawkes processes were studied in [Seo17] and [Yao18].…”

Section: Large Deviations For Hawkes Processesmentioning

confidence: 99%

Large and moderate deviations for a discrete-time marked Hawkes process

Wang¹

2020

Preprint

View full text Add to dashboard Cite

The Hawkes process is a simple point process with wide applications in finance, social networks, criminology, seismology, and many other fields. The Hawkes process is defined for continuous time setting. However, data is often recorded in a discrete-time scheme. In this paper, we study a discrete-time marked Hawkes process, which is appealing for applications. In particular, we study large deviations for the model.

show abstract

Section: Large Deviations For Hawkes Processesmentioning

confidence: 99%

Large and moderate deviations for a discrete-time marked Hawkes process

Wang¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, Wang [31,32] studied limit behaviors of a discrete-time Hawkes process with random marks and proved large and moderate deviations for a discrete-time Hawkes process with marks. Seol [27,28] studied the moderate deviation principle of marked Hawkes processes and also studied asymptotic behaviors for the compensator processes of Hawkes models. Furthermore, Gao and Zhu [10,[12][13][14] made some progress in the direction of limit behaviors other than the large time scale limits.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic results for a class of Markovian self-exciting processes

Seol

2023

J Inequal Appl

Self Cite

View full text Add to dashboard Cite

Hawkes process is a class of self-exciting point processes with clustering effect whose jump rate relies on their entire past history. This process is usually defined as a continuous-time setting and has been widely applied in several fields, including insurance, finance, queueing theory, and statistics. The Hawkes model is generally non-Markovian because the future development of a self-exciting point process is determined by the timing of past events. However, it can be Markovian in special cases such as when the exciting function is an exponential function or a sum of exponential functions. Difficulty arises when the exciting function is not an exponential function or a sum of exponentials, in which case the process can be non-Markovian. The inverse Markovian case for Hawkes processes was introduced by Seol (Stat. Probab. Lett. 155:108580, 2019) who studied some asymptotic behaviors. An extended version of the inverse Markovian Hawkes process was also studied by Seol (J. Korean Math. Soc. 58(4):819–833, 2021). In the current work, we propose a class of Markovian self-exciting processes that interpolates between the Hawkes process and the inverse Hawkes process. We derived limit theorems for the newly considered class of Markovian self-exciting processes. In particular, we established both the law of large numbers (LLN) and central limit theorems (CLT) with some key results.

show abstract

“…Recently, Wang [29,30] studied limit behaviors of a discrete-time Hawkes process with random marks, and furthermore, proved the large and moderate deviations for a discrete-time Hawkes process with marks. Seol [31,32] has studied the moderate deviation principle of marked Hawkes processes and also studied asymptotic behaviors for the compensator processes of Hawkes models. Furthermore, Gao and Zhu [33][34][35][36] made some progress in the direction of limit behaviors other than the large time scale limits.…”

Section: Introductionmentioning

confidence: 99%

Non-Markovian Inverse Hawkes Processes

Seol

2022

Mathematics

Self Cite

View full text Add to dashboard Cite

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process.

show abstract

Moderate deviations for marked Hawkes processes

Cited by 11 publications

References 18 publications

Large and moderate deviations for a discrete-time marked Hawkes process

Large and moderate deviations for a discrete-time marked Hawkes process

Asymptotic results for a class of Markovian self-exciting processes

Non-Markovian Inverse Hawkes Processes

Contact Info

Product

Resources

About