Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T21:45:56.698Z Has data issue: false hasContentIssue false
  • English
  • Français

An ephemerally self-exciting point process

Part of: Stochastic processes Markov processes Special 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.
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmed, A. and Xing, E. (2008). Dynamic non-parametric mixture models and the recurrent Chinese restaurant process: with applications to evolutionary clustering. In Proc. 2008 SIAM International Conference on Data Mining, Society for Industrial and Applied Mathematics, Philadelphia, pp. 219–230.CrossRefGoogle Scholar
At-Sahalia, Y., Cacho-Diaz, J. and Laeven, R. J. (2015). Modeling financial contagion using mutually exciting jump processes. J. Financial Econom. 117, 585–606.CrossRefGoogle Scholar
Aldous, D. J. (1985). Exchangeability and related topics. In École d’ÉtÉ de ProbabilitÉs de Saint-Flour XIII—1983, Springer, Berlin, Heidelberg, pp. 1–198.CrossRefGoogle Scholar
Azizpour, S., Giesecke, K. and Schwenkler, G. (2016). Exploring the sources of default clustering. J. Financial Econom. 129, 154–183.CrossRefGoogle Scholar
Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.CrossRefGoogle Scholar
Bacry, E., Jaisson, T. and Muzy, J.-F. (2016). Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics. Quant. Finance 16, 1179–1201.CrossRefGoogle Scholar
Bacry, E. and Muzy, J.-F. (2014). Hawkes model for price and trades high-frequency dynamics. Quant. Finance 14, 1147–1166.CrossRefGoogle Scholar
Ball, F. (1983). The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Blackwell, D. (1948). A renewal theorem. Duke Math. J. 15, 145–150.CrossRefGoogle Scholar
Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1, 353–355.CrossRefGoogle Scholar
Blanc, P., Donier, J. and Bouchaud, J.-P. (2017). Quadratic Hawkes processes for financial prices. Quant. Finance 17, 171–188.CrossRefGoogle Scholar
Blei, D. M. and Frazier, P. I. (2011). Distance dependent Chinese restaurant processes. J. Machine Learning Res. 12, 2461–2488.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.CrossRefGoogle Scholar
Brockmeyer, E., Halstrom, H. and Jensen, A. (1948). The Life and Works of A. K. Erlang. Copenhagen Telephone Company, Copenhagen.Google Scholar
Da Fonseca, J. and Zaatour, R. (2014). Hawkes process: fast calibration, application to trade clustering, and diffusive limit. J. Futures Markets 34, 548–579.Google Scholar
Daley, D. J. and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure. Springer, New York.Google Scholar
Dassios, A. and Zhao, H. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814–846.CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2017). Efficient simulation of clustering jumps with CIR intensity. Operat. Res. 65, 1494–1515.CrossRefGoogle Scholar
Davis, M. H. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Daw, A. and Pender, J. (2018). Queues driven by Hawkes processes. Stoch. Systems 8, 192–229.CrossRefGoogle Scholar
Daw, A. and Pender, J. (2019). Matrix calculations for moments of Markov processes. Preprint. Available at https://arxiv.org/abs/1909.03320.Google Scholar
Du, N. et al. (2015). Dirichlet–Hawkes processes with applications to clustering continuous-time document streams. In KDD ’15: Proc. 21st ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Association for Computing Machinery, New York, pp. 219–228.CrossRefGoogle Scholar
Eick, S. G., Massey, W. A. and Whitt, W. (1993). The physics of the $\textrm{M}_t$ /G/ $\infty$ queue. Operat. Res. 41, 731–742.CrossRefGoogle Scholar
Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1, 642–665.CrossRefGoogle Scholar
Ertekin, Ş., Rudin, C. and McCormick, T. H. (2015). Reactive point processes: a new approach to predicting power failures in underground electrical systems. Ann. Appl. Statist. 9, 122–144.CrossRefGoogle Scholar
Farajtabar, M. et al. (2017). Fake news mitigation via point process based intervention. In Proc. 34th International Conference on Machine Learning (PMLR 70), pp. 1097–1106.Google Scholar
Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Gao, X., Zhou, X. and Zhu, L. (2018). Transform analysis for Hawkes processes with applications in dark pool trading. Quant. Finance 18, 265–282.CrossRefGoogle Scholar
Gao, X. and Zhu, L. (2018). Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues. Queueing Systems 90, 161–206.CrossRefGoogle Scholar
Gao, X. and Zhu, L. (2018). Large deviations and applications for Markovian Hawkes processes with a large initial intensity. Bernoulli 24, 2875–2905.CrossRefGoogle Scholar
Guo, X., Ruan, Z. and Zhu, L. (2015). Dynamics of order positions and related queues in a limit order book. Preprint. Available at https://arxiv.org/abs/1505.04810.Google Scholar
Gupta, A., Farajtabar, M., Dilkina, B. and Zha, H. (2018). Discrete interventions in Hawkes processes with applications in invasive species management. In IJCAI ’18: Proc. 27th International Joint Conference on Artificial Intelligence, AAAI Press, Palo Alto, CA, pp. 3385–3392.CrossRefGoogle Scholar
Hale, J. K. and Lunel, S. M. V. (2013). Introduction to Functional Differential Equations. Springer, New York.Google Scholar
Halpin, P. F. and De Boeck, P. (2013). Modelling dyadic interaction with Hawkes processes. Psychometrika 78, 793–814.CrossRefGoogle ScholarPubMed
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438–443.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Ibrahim, R., Ye, H., L’Ecuyer, P. and Shen, H. (2016). Modeling and forecasting call center arrivals: a literature survey and a case study. Internat. J. Forecasting 32, 865–874.CrossRefGoogle Scholar
Karlis, D. and Xekalaki, E. (2005). Mixed Poisson distributions. Internat. Statist. Rev. 73, 35–58.CrossRefGoogle Scholar
Kelly, F. P. (2011). Reversibility and Stochastic Networks. Cambridge University Press.Google Scholar
Koops, D., Saxena, M., Boxma, O. and Mandjes, M. (2018). Infinite-server queues with Hawkes input. J. Appl. Prob. 55, 920–943.CrossRefGoogle Scholar
Krumin, M., Reutsky, I. and Shoham, S. (2010). Correlation-based analysis and generation of multiple spike trains using Hawkes models with an exogenous input. Frontiers Comput. Neurosci. 4, 147.CrossRefGoogle ScholarPubMed
Lewis, E. and Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Preprint. Available at http://paleo.sscnet.ucla.edu/Lewis-Molher-EM_Preprint.pdf.Google Scholar
Li, L. and Zha, H. (2018). Energy usage behavior modeling in energy disaggregation via Hawkes processes. ACM Trans. Intellig. Systems Technol. 9, 36.CrossRefGoogle Scholar
Lindvall, T. (1977). A probabilistic proof of Blackwell’s renewal theorem. Ann. Prob. 5, 482–485.CrossRefGoogle Scholar
Lum, K., Swarup, S., Eubank, S. and Hawdon, J. (2014). The contagious nature of imprisonment: an agent-based model to explain racial disparities in incarceration rates. J. R. Soc. Interface 11, 20140409.CrossRefGoogle ScholarPubMed
Malmgren, R. D., Stouffer, D. B., Motter, A. E. and Amaral, L. A. (2008). A Poissonian explanation for heavy tails in e-mail communication. Proc. Nat. Acad. Sci. USA 105, 18153–18158.CrossRefGoogle Scholar
Massey, W. A. and Pender, J. (2013). Gaussian skewness approximation for dynamic rate multi-server queues with abandonment. Queueing Systems 75, 243–277.CrossRefGoogle Scholar
Masuda, N., Takaguchi, T., Sato, N. and Yano, K. (2013). Self-exciting point process modeling of conversation event sequences. In Temporal Networks, Springer, Berlin, Heidelberg, pp. 245–264.CrossRefGoogle ScholarPubMed
Mei, H. and Eisner, J. M. (2017). The neural Hawkes process: A neurally self-modulating multivariate point process. In Advances in Neural Information Processing Systems 30 (NIPS 2017), Neural Information Processing Systems, San Diego, CA, pp. 6754–6764.Google Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.CrossRefGoogle Scholar
Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20, 206–224.CrossRefGoogle Scholar
Rambaldi, M., Bacry, E. and Lillo, F. (2017). The role of volume in order book dynamics: a multivariate Hawkes process analysis. Quant. Finance 17, 999–1020.CrossRefGoogle Scholar
Rizoiu, M.-A. et al. (2018). SIR–Hawkes: linking epidemic models and Hawkes processes to model diffusions in finite populations. In WWW ’18: Proc. 2018 World Wide Web Conference, Association for Computing Machinery, New York, pp. 419–428.CrossRefGoogle Scholar
Rizoiu, M.-A. et al. (2017). Expecting to be hip: Hawkes intensity processes for social media popularity. In WWW ’17: Proc. 26th International Conference on World Wide Web, Association for Computing Machinery, New York, pp. 735–744.CrossRefGoogle Scholar
Singh, S. and Myers, C. R. (2014). Outbreak statistics and scaling laws for externally driven epidemics. Phys. Rev. E 89, 042108.CrossRefGoogle ScholarPubMed
Truccolo, W. et al. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. J. Neurophysiology 93, 1074–1089.CrossRefGoogle ScholarPubMed
Van der Hofstad, R. and Keane, M. (2008). An elementary proof of the hitting time theorem. Amer. Math. Monthly 115, 753–756.CrossRefGoogle Scholar
Veen, A. and Schoenberg, F. P. (2008). Estimation of space–time branching process models in seismology using an EM–type algorithm. J. Amer. Statist. Assoc. 103, 614–624.CrossRefGoogle Scholar
Willmot, G. (1986). Mixed compound Poisson distributions. ASTIN Bull. 16, S59–S79.CrossRefGoogle Scholar
Wolff, R. W. (1982). Poisson arrivals see time averages. Operat. Res. 30, 223–231.CrossRefGoogle Scholar
Wu, P., Rambaldi, M., Muzy, J.-F. and Bacry, E. (2019). Queue-reactive Hawkes models for the order flow. Preprint. Available at https://arxiv.org/abs/1901.08938.Google Scholar
Xu, H., Luo, D. and Zha, H. (2017). Learning Hawkes processes from short doubly-censored event sequences. In Proc. 34th International Conference on Machine Learning (PMLR 70), pp. 3831–3840.Google Scholar
Xu, H., Zhen, Y. and Zha, H. (2015). Trailer generation via a point process-based visual attractiveness model. In Proc. Twenty-Fourth International Joint Conference on Artificial Intelligence, AAAI Press/International Joint Conferences on Artificial Intelligence, Palo Alto, CA, pp. 2198–2204.Google Scholar
Yan, J. et al. (2013). Towards effective prioritizing water pipe replacement and rehabilitation. In Proc. Twenty-Third International Joint Conference on Artificial Intelligence, AAAI Press/International Joint Conferences on Artificial Intelligence, Menlo Park, CA, pp. 2931–2937.Google Scholar
Zhang, X., Blanchet, J., Giesecke, K. and Glynn, P. W. (2015). Affine point processes: approximation and efficient simulation. Math. Operat. Res. 40, 797–819.CrossRefGoogle Scholar
Zhang, X.-W., Glynn, P. W., Giesecke, K. and Blanchet, J. (2009). Rare event simulation for a generalized Hawkes process. In Proc. 2009 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 1291–1298.CrossRefGoogle Scholar
Zino, L., Rizzo, A. and Porfiri, M. (2018). Modeling memory effects in activity-driven networks. SIAM J. Appl. Dynam. Systems 17, 2830–2854.CrossRefGoogle Scholar