Limit theorems for an inverse Markovian Hawkes process

Seol, Youngsoo

doi:10.1016/j.spl.2019.108580

Cited by 6 publications

(7 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In a recent paper by Seol [39], an inverse version of the Markovianvian Hawkes process was developed and studied. In contrast to the general Hawkes processes, the new model has some specific features.…”

Section: Inverse Markovian Hawkes Processmentioning

confidence: 99%

“…in distribution as t → ∞. Furthermore, Seol [40] introduced a model combining the Hawkes process and inverse Hawkes process, which is an extended version of the inverse Markovian Hawkes process. The extended model can be defined as…”

Section: Inverse Markovian Hawkes Processmentioning

confidence: 99%

“…This is known as the renewal Hawkes process (see [38]). Recently, Seol [39] showed that the inverse case of Markovian Hawkes processes can be represented by several existing models of self-exciting processes, and studied the asymptotic behaviors of the inverse Markovian Hawkes processes. Seol [40] further studied the extended version of the inverse case of the Markovian Hawkes model.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Seol [39] showed that the inverse case of Markovian Hawkes processes can be represented by several existing models of self-exciting processes, and studied the asymptotic behaviors of the inverse Markovian Hawkes processes. Seol [40] further studied the extended version of the inverse case of the Markovian Hawkes model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and important applications to several financial models, such as a shot-noise process and a jump-diffusion process with no diffusions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Section: Inverse Markovian Hawkes Processmentioning

confidence: 99%

Section: Inverse Markovian Hawkes Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-Markovian Inverse Hawkes Processes

Seol

2022

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…That is known as the renewal Hawkes process (see [33]). Recently, Seol [29] introduced the inverse case of Markovian Hawkes processes represented as several existing models of self-exciting point processes and proved some asymptotic behaviors of the inverse Marko-vian Hawkes processes. Seol [30] further studied an extended version of the inverse case of Markovian Hawkes model.…”

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