The Annals of Applied Probability

Large deviations for Markovian nonlinear Hawkes processes

Lingjiong Zhu

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Abstract

Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalize it to get the result for sum of exponentials exciting functions. We then provide an alternative proof for the large deviation principle for a linear Hawkes process. Finally, we use an approximation approach to prove the large deviation principle for a special class of nonlinear Hawkes processes with general exciting functions.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 548-581.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355123

Digital Object Identifier
doi:10.1214/14-AAP1003

Mathematical Reviews number (MathSciNet)
MR3313748

Zentralblatt MATH identifier
1312.60019

Subjects
Primary: 60G55: Point processes 60F10: Large deviations

Keywords
Large deviations rare events point processes Hawkes processes self-exciting processes

Citation

Zhu, Lingjiong. Large deviations for Markovian nonlinear Hawkes processes. Ann. Appl. Probab. 25 (2015), no. 2, 548--581. doi:10.1214/14-AAP1003. https://projecteuclid.org/euclid.aoap/1424355123


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References

  • [1] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Process. Appl. 123 2475–2499.
  • [2] Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23 593–625.
  • [3] Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Probab. 24 1563–1588.
  • [4] Cox, D. R. and Isham, V. (1980). Point Processes. Chapman & Hall, London.
  • [5] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, 2nd ed. Springer, New York.
  • [6] Davis, M. H. A. (1993). Markov Models and Optimization. Chapman & Hall, London.
  • [7] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • [8] Echeverría, P. (1982). A criterion for invariant measures of Markov processes. Probab. Theory Related Fields 61 1–16.
  • [9] Fan, K. (1953). Minimax theorems. Proc. Natl. Acad. Sci. USA 39 42–47.
  • [10] Frenk, J. B. G. and Kassay, G. (2003). The level set method of Joó and its use in minimax theory. Technical Report E.I 2003-03, Econometric Institute, Erasmus Univ., Rotterdam.
  • [11] Hairer, M. (2010). Convergence of Markov processes. Lecture Notes. Univ. Warwick. Available at http://www.hairer.org/notes/Convergence.pdf.
  • [12] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83–90.
  • [13] Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493–503.
  • [14] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.
  • [15] Joó, I. (1984). Note on my paper: “A simple proof for von Neumann’s minimax theorem” [Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 91–94; MR0576940 (81i:49008)]. Acta Math. Hungar. 44 363–365.
  • [16] Karabash, D. and Zhu, L. (2012). Limit theorems for marked Hawkes processes with application to a risk model. Preprint. Available at arXiv:1211.4039.
  • [17] Koralov, L. B. and Sinai, Y. G. (2007). Theory of Probability and Random Processes, 2nd ed. Springer, Berlin.
  • [18] Liniger, T. (2009). Multivariate Hawkes processes. Ph.D. thesis, ETH, Zürich.
  • [19] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes. II, 2nd ed. Springer, Berlin.
  • [20] Oakes, D. (1975). The Markovian self-exciting process. J. Appl. Probab. 12 69–77.
  • [21] Stabile, G. and Torrisi, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodol. Comput. Appl. Probab. 12 415–429.
  • [22] Varadhan, S. R. S. (2001). Probability Theory. Amer. Math. Soc., Providence, RI.
  • [23] Varadhan, S. R. S. (2008). Large deviations. Ann. Probab. 36 397–419.
  • [24] Zhu, L. (2014). Limit theorems for a Cox–Ingersoll–Ross process with Hawkes jumps. J. Appl. Probab. To appear.
  • [25] Zhu, L. (2014). Process-level large deviations for nonlinear Hawkes point processes. Ann. Inst. Henri Poincaré Probab. Stat. 50 845–871.
  • [26] Zhu, L. (2013). Nonlinear Hawkes processes. Ph.D. thesis, New York Univ.
  • [27] Zhu, L. (2013). Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims. Insurance Math. Econom. 53 544–550.
  • [28] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 760–771.
  • [29] Zhu, L. (2013). Moderate deviations for Hawkes processes. Statist. Probab. Lett. 83 885–890.