An extension of Hawkes processes with ephemeral nearest effects

Abstract

Hawkes processes have been widely studied in both theory and applications because of their self-exciting effects, but there are still some questions and room left for exploring the self-exciting point processes. In this paper, an ephemeral self-exciting Hawkes process is developed, which contains 1-memory and 0-memory self-exciting point processes. After giving the definition of the ephemeral self-exciting Hawkes process, some explanations on this new point process, properties, declustering of the occurrences of events, positive and negative self-exciting effects are presented. Moments of this point process are derived as well in the paper by using the Laplace transforms and the elementary method. Some special cases are considered and the computation methods of moments are also presented by the difference method and inverse Laplace transforms, respectively. Combining dynamics of the self-exciting point process and the delayed renewal process is an important feature of the proposed ephemeral self-exciting Hawkes process, which may shed light on the related studies in the future.

Keywords:

• Delayed renewal process
• Hawkes process
• Laplace transform
• moments
• self-exciting effect

Acknowledgements

The authors also thank two anonymous referees for their suggestions for improving the paper.

Additional information

Funding

This work is supported by the NSF of China under grants 71871021 and 71631001.