Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2875-2905.

Large deviations and applications for Markovian Hawkes processes with a large initial intensity

Xuefeng Gao and 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, insurance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study linear Hawkes process with an exponential kernel in the asymptotic regime where the initial intensity of the Hawkes process is large. We establish large deviations for Hawkes processes in this regime as well as the regime when both the initial intensity and the time are large. We illustrate the strength of our results by discussing the applications to insurance and queueing systems.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2875-2905.

Dates
Received: April 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051228

Digital Object Identifier
doi:10.3150/17-BEJ948

Mathematical Reviews number (MathSciNet)
MR3779705

Zentralblatt MATH identifier
06853268

Keywords
Hawkes processes insurance large deviations large initial intensity queueing systems

Citation

Gao, Xuefeng; Zhu, Lingjiong. Large deviations and applications for Markovian Hawkes processes with a large initial intensity. Bernoulli 24 (2018), no. 4A, 2875--2905. doi:10.3150/17-BEJ948. https://projecteuclid.org/euclid.bj/1522051228


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References

  • [1] Abergel, F. and Jedidi, A. (2015). Long-time behavior of a Hawkes process-based limit order book. SIAM J. Financial Math. 6 1026–1043.
  • [2] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd ed. Advanced Series on Statistical Science & Applied Probability 14. Hackensack, NJ: World Scientific.
  • [3] 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.
  • [4] Blanchet, J., Chen, X. and Lam, H. (2014). Two-parameter sample path large deviations for infinite-server queues. Stoch. Syst. 4 206–249.
  • [5] Bordenave, C. and Torrisi, G.L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23 593–625.
  • [6] Dassios, A. and Zhao, H. (2012). Ruin by dynamic contagion claims. Insurance Math. Econom. 51 93–106.
  • [7] Dassios, A. and Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Probab. 18 13.
  • [8] Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Probab. 26 216–261.
  • [9] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. New York: Springer.
  • [10] Dragomir, S.S. (2003). Some Gronwall Type Inequalities and Applications. Hauppauge, NY: Nova Science.
  • [11] Errais, E., Giesecke, K. and Goldberg, L.R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1 642–665.
  • [12] Feng, J. and Kurtz, T.G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. Providence, RI: Amer. Math. Soc.
  • [13] Gao, X. and Zhu, L. (2015). Limit theorems for Markovian Hawkes processes with a large initial intensity. Available at arXiv:1512.02155.
  • [14] Gao, X. and Zhu, L. Supplement to “Large deviations and applications for Markovian Hawkes processes with a large initial intensity.” DOI:10.3150/17-BEJ948SUPP.
  • [15] Glynn, P.W. (1995). Large deviations for the infinite server queue in heavy traffic. In Stochastic Networks. IMA Vol. Math. Appl. 71 387–394. New York: Springer.
  • [16] Glynn, P.W. and Whitt, W. (1991). A new view of the heavy-traffic limit theorem for infinite-server queues. Adv. in Appl. Probab. 23 188–209.
  • [17] Hawkes, A.G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83–90.
  • [18] Hawkes, A.G. (1971). Point spectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 33 438–443.
  • [19] Hawkes, A.G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493–503.
  • [20] Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25 600–631.
  • [21] Jaisson, T. and Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Probab. 26 2860–2882.
  • [22] Karabash, D. and Zhu, L. (2015). Limit theorems for marked Hawkes processes with application to a risk model. Stoch. Models 31 433–451.
  • [23] Puhalskii, A. (1995). Large deviation analysis of the single server queue. Queueing Syst. 21 5–66.
  • [24] Stabile, G. and Torrisi, G.L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodol. Comput. Appl. Probab. 12 415–429.
  • [25] Torrisi, G.L. (2016). Gaussian approximation of nonlinear Hawkes processes. Ann. Appl. Probab. 26 2106–2140.
  • [26] Torrisi, G.L. (2017). Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes. Ann. Inst. Henri Poincaré Probab. Stat. 53 679–700.
  • [27] Varadhan, S.R.S. (1984). Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. Philadelphia, PA: SIAM.
  • [28] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. New York: Springer.
  • [29] Zhang, X., Blanchet, J., Giesecke, K. and Glynn, P.W. (2015). Affine point processes: Approximation and efficient simulation. Math. Oper. Res. 40 797–819.
  • [30] Zhu, L. (2013). Nonlinear Hawkes Processes. Ph.D. thesis, New York University.
  • [31] Zhu, L. (2013). Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims. Insurance Math. Econom. 53 544–550.
  • [32] Zhu, L. (2013). Moderate deviations for Hawkes processes. Statist. Probab. Lett. 83 885–890.
  • [33] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 760–771.
  • [34] Zhu, L. (2014). Limit theorems for a Cox–Ingersoll–Ross process with Hawkes jumps. J. Appl. Probab. 51 699–712.
  • [35] Zhu, L. (2014). Process-level large deviations for nonlinear Hawkes point processes. Ann. Inst. Henri Poincaré Probab. Stat. 50 845–871.
  • [36] Zhu, L. (2015). Large deviations for Markovian nonlinear Hawkes processes. Ann. Appl. Probab. 25 548–581.

Supplemental materials

  • Supplement to “Large deviations and applications for Markovian Hawkes processes with a large initial intensity”. We provide proofs for additional results in the paper in the supplemental article [14].