This repository provides an overview of papers on Hawkes processes, tailored primarily for researchers. Our aim is to categorize papers into different applications, provide links to official and unofficial code resources, and explore the general properties of Hawkes processes.

Currently, the repository is incomplete. For those with limited knowledge about stochastic processes and Hawkes processes, we highly recommend starting with Laub et al. [4] or Rizoiu et al. [5] to familiarize yourself with the basics before delving into the original papers [1, 2, 3].

• [1] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83-90.
• [2] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society Series B: Statistical Methodology, 33(3), 438-443.
• [3] Hawkes, A. G., & Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3), 493-503.

Fascinating insights into the origins of Hawkes processes:

Hawkes, A., & Chen, J. (2021). A personal history of Hawkes process. Proceedings of the Institute of Statistical Mathematics (統計数理), 69(2), 123-143.

• [4] Laub, P. J., Lee, Y., & Taimre, T. (2021). The elements of Hawkes processes. Springer Cham, Switzerland. [Code]
• [5] Rizoiu, M. A., Lee, Y., Mishra, S., & Xie, L. (2017). Hawkes processes for events in social media. In Frontiers of multimedia research (pp. 191-218).

For a general introduction into stochastic processes refer to:

• [6] Daley, D. J., & Vere-Jones, D. (2003). An introduction to the theory of point processes: Volume I: Elementary theory and methods. Springer New York, NY.
• [7] Daley, D. J., & Vere-Jones, D. (2008). An introduction to the theory of point processes: Volume II: General theory and structure. Springer New York, NY.

If you're searching for a reference and not interested in reading a book, here are a couple of literature reviews, with most focusing on a selected topic:

• Overview of Hawkes processes:
  • [8] Lima, R. (2023). Hawkes processes modeling, inference, and control: An overview. SIAM Review, 65(2), 331-374.
• Review with a focus on spatio-temporal processes:
  • [9] Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33(3), 299-318.
• Reviews with a focus on financial applications:
  • [10] Bacry, E., Mastromatteo, I., & Muzy, J. F. (2015). Hawkes processes in finance. Market Microstructure and Liquidity, 1(01), 1550005.
  • [11] Hawkes, A. G. (2018). Hawkes processes and their applications to finance: A review. Quantitative Finance, 18(2), 193-198.
  • [12] Hawkes, A. G. (2022). Hawkes jump-diffusions and finance: A brief history and review. European Journal of Finance, 28(7), 627-641.
• Review on neural Hawkes processes:
  • [13] Shchur, O., T ̈urkmen, A. C., Januschowski, T., & G ̈unnemann, S. (2021). Neural temporal point processes: A review. In Proceedings of the 30th International Joint Conference on Artificial Intelligence (pp. 4585–4593).

• [14] Oakes, D. (1975). The Markovian self-exciting process. Journal of Applied Probability, 12(1), 69-77.
• [15] Brémaud, P., & Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Annals of Probability, 24(3), 1563-1588.
• [16] Brémaud, P., Nappo, G., & Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. Journal of Applied Probability, 39(1), 123-136.
• [17] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. Journal of Applied Probability, 50(3), 760-771.
• [18] Jovanović, S., Hertz, J., & Rotter, S. (2015). Cumulants of Hawkes point processes. Physical Review E, 91(4), 042802.
• [19] Cui, L., Hawkes, A., & Yi, H. (2020). An elementary derivation of moments of Hawkes processes. Advances in Applied Probability, 52(1), 102-137.
• [20] Daw, A. (2024). Conditional uniformity and Hawkes processes. Mathematics of Operations Research, 49(1), 40-57.

• [21] Ogata, Y. (1981). On Lewis' simulation method for point processes. IEEE Transactions on Information Theory, 27(1), 23-31.
• [22] Dassios, A., & Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Probab., 18, 1-13

Fitting Hawkes processes is a notoriously difficult task. The selection of a statistical inference method should always depend on the amount of data at hand, the specific application domain, and the task to be accomplished. General statistical inference methods include the following:

• Maximimum-likelihood estimation (MLE)
  • [23] Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics, 30, 243-261
  • [24] Ozaki, T. (1979). Maximum likelihood estimation of Hawkes' self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31, 145-155.
• Expectation maximization (EM)
  • ...
  • [25] Mark, M., & Weber, T. A. (2020). Robust identification of controlled Hawkes processes. Physical Review E, 101(4), 043305.
• Bayesian
  • [26] Rasmussen, J. G. (2013). Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 15, 623-642.
• Non-parametric estimation:
  • [27] Lewis, E., & Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Technical Report.
  • [28] Bacry, E., & Muzy, J. F. (2016). First-and second-order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Transactions on Information Theory, 62(4), 2184-2202.
  • [29] Kirchner, M. (2017). An estimation procedure for the Hawkes process. Quantitative Finance, 17(4), 571-595.
  • [30] Achab, M., Bacry, E., Gaïffas, S., Mastromatteo, I., & Muzy, J. F. (2017). Uncovering causality from multivariate Hawkes integrated cumulants. In Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1-10. [Code]
• ...

• [31] Mei, H., & Eisner, J. M. (2017). The neural Hawkes process: A neurally self-modulating multivariate point process. Advances in Neural Information Processing Systems, 30.
• [32] Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M., & Song, L. (2016). Recurrent marked temporal point processes: Embedding event history to vector. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1555-1564). [Code]
• [33] Upadhyay, U., De, A., & Gomez Rodriguez, M. (2018). Deep reinforcement learning of marked temporal point processes. Advances in Neural Information Processing Systems, 31. [Code]
• [34] Zuo, S., Jiang, H., Li, Z., Zhao, T., & Zha, H. (2020). Transformer Hawkes process. In Proceedings of the 37th International Conference on Machine Learning, PMLR 119:11692-11702. [Code]

In the following, we aim to highlight the general applicability of Hawkes processes to a plethora of applications. The provided references represent just a hand-selected subset of publications. In the subdirectories, we strive to provide a more comprehensive overview of the significant contributions towards adopting Hawkes processes.

• Earthquake modeling:
  • [35] Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9-27.
  • [36] Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50, 379-402.
• Social media:
  • [37] Zhao, Q., Erdogdu, M. A., He, H. Y., Rajaraman, A., & Leskovec, J. (2015). SEISMIC: A self-exciting point process model for predicting tweet popularity. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1513-1522). [Code]
  • [38] Rizoiu, M. A., Xie, L., Sanner, S., Cebrian, M., Yu, H., & Van Hentenryck, P. (2017). Expecting to be hip: Hawkes intensity processes for social media popularity. In Proceedings of the 26th International Conference on World Wide Web (pp. 735-744).[Code]
  • [39] Schneider, P. J., & Rizoiu, M. A. (2023). The effectiveness of moderating harmful online content. Proceedings of the National Academy of Sciences, 120(34), e2307360120. [Code]
• Criminology:
  • [40] Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P., & Tita, G. E. (2011). Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100-108.
• Finance:
  • [41] Aït-Sahalia, Y., Cacho-Diaz, J., & Laeven, R. J. (2015). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics, 117(3), 585-606.
  • [42] Rambaldi, M., Pennesi, P., & Lillo, F. (2015). Modeling foreign exchange market activity around macroeconomic news: Hawkes-process approach. Physical Review E, 91(1), 012819.
  • [43] Mark, M., Sila, J., & Weber, T. A. (2022). Quantifying endogeneity of cryptocurrency markets. European Journal of Finance, 28(7), 784-799.
• Epidemiology:
  • [44] Rizoiu, M. A., Mishra, S., Kong, Q., Carman, M., & Xie, L. (2018). SIR-Hawkes: Linking epidemic models and Hawkes processes to model diffusions in finite populations. In Proceedings of the 2018 World Wide Web Conference (pp. 419-428). [Code]
  • [45] Kim, M., Paini, D., & Jurdak, R. (2019). Modeling stochastic processes in disease spread across a heterogeneous social system. Proceedings of the National Academy of Sciences, 116(2), 401-406.
  • [46] Bertozzi, A. L., Franco, E., Mohler, G., Short, M. B., & Sledge, D. (2020). The challenges of modeling and forecasting the spread of COVID-19. Proceedings of the National Academy of Sciences, 117(29), 16732-16738. [Code]