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An ephemerally self-exciting point process

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  14 March 2022

Andrew Daw Open the ORCID record for Andrew Daw [Opens in a new window]  and
Jamol Pender Open the ORCID record for Jamol Pender [Opens in a new window]
Andrew Daw*
Affiliation:
University of Southern California
Jamol Pender*
Affiliation:
Cornell University
*
*Postal address: Marshall School of Business, University of Southern California, 401B Bridge Hall, Los Angeles, CA 90089. Email address: andrew.daw@usc.edu
**Postal address: School of Operations Research and Information Engineering, Cornell University, 228 Rhodes Hall, Ithaca, NY 14853.
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Abstract

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

Keywords

Self-excitementHawkes processpoint processcontagioninteraction models

MSC classification

Primary: 60G55: Point processes
Secondary: 60K25: Queueing theory 60J75: Jump processes
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 340 - 403
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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