Asymptotic results for a class of Markovian self-exciting processes

Journal of Inequalities and Applications

Research

Youngsoo Seol¹ 

[1] Department of Mathematics, Dong-A University, Saha-gu, Nakdong-daero 550, 37, Busan, Republic of Korea;

关键词: Hawkes process;    Inverse Markovian;    Self-exciting point processes;    Central limit theorems;    Law of large numbers;    60G55;    60F05;    60F10;   

DOI  :  10.1186/s13660-023-02989-z

 received in 2022-03-30, accepted in 2023-05-17,  发布年份 2023

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY   
© The Author(s) 2023

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