Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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LIMIT THEOREMS FOR HAWKES PROCESSES WITH UNIFORM IMMIGRANTS

  • Received : 2018.07.19
  • Accepted : 2018.11.21
  • Published : 2019.07.01
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Abstract

Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history. We consider Hawkes processes with uniform immigrants which is a special case of the Hawkes processes with renewal immigrants. We study the limit theorems for Hawkes processes with uniform immigrants. In particular, we obtain a law of large number, a central limit theorem, and a large deviation principle.

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References

  1. E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, Some limit theorems for Hawkes processes and application to financial statistics, Stochastic Process. Appl. 123 (2013), no. 7, 2475-2499. https://doi.org/10.1016/j.spa.2013.04.007
  2. C. Bordenave and G. L. Torrisi, Large deviations of Poisson cluster processes, Stoch. Models 23 (2007), no. 4, 593-625. https://doi.org/10.1080/15326340701645959
  3. P. Bremaud and L. Massoulie, Stability of nonlinear Hawkes processes, Ann. Probab. 24 (1996), no. 3, 1563-1588. https://doi.org/10.1214/aop/1065725193
  4. D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I, second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2003.
  5. A. Dassios and H. Zhao, A dynamic contagion process, Adv. in Appl. Probab. 43 (2011), no. 3, 814-846. https://doi.org/10.1239/aap/1316792671
  6. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, second edition, Applications of Mathematics (New York), 38, Springer-Verlag, New York, 1998.
  7. E. Errais, K. Giesecke, and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM J. Financial Math. 1 (2010), no. 1, 642-665. https://doi.org/10.1137/090771272
  8. R. Fierro, V. Leiva, and J. Moller, The Hawkes process with different exciting functions and its asymptotic behavior, J. Appl. Probab. 52 (2015), no. 1, 37-54. https://doi.org/10.1239/jap/1429282605
  9. X. Gao, X. Zhou, and L. Zhu, Transform analysis for Hawkes processes with applications in dark pool trading, Quant. Finance 18 (2018), no. 2, 265-282. https://doi.org/10.1080/14697688.2017.1403151
  10. A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika 58 (1971), 83-90. https://doi.org/10.1093/biomet/58.1.83
  11. A. G. Hawkes and D. Oakes, A cluster process representation of a self-exciting process, J. Appl. Probability 11 (1974), 493-503. https://doi.org/10.2307/3212693
  12. P. Jagers, Branching Processes with Biological Applications, Wiley-Interscience, London, 1975.
  13. T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes, Ann. Appl. Probab. 25 (2015), no. 2, 600-631. https://doi.org/10.1214/14-AAP1005
  14. T. Jaisson and M. Rosenbaum, Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes, Ann. Appl. Probab. 26 (2016), no. 5, 2860-2882. https://doi.org/10.1214/15-AAP1164
  15. D. Karabash and L. Zhu, Limit theorems for marked Hawkes processes with application to a risk model, Stoch. Models 31 (2015), no. 3, 433-451. https://doi.org/10.1080/15326349.2015.1024868
  16. B. Mehrdad and L. Zhu, On the Hawkes process with different exciting functions, Preprint. arXiv: 1403.0994 (2015).
  17. Y. Seol, Limit theorems for discrete Hawkes processes, Statist. Probab. Lett. 99 (2015), 223-229. https://doi.org/10.1016/j.spl.2015.01.023
  18. Y. Seol, Limit theorems for inverse process $T_n$ of Hawkes process, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 1, 51-60. https://doi.org/10.1007/s10114-016-5470-y
  19. Y. Seol, Moderate deviations for marked Hawkes processes, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 10, 1297-1304. https://doi.org/10.1007/s10114-017-6433-7
  20. Y. Seol, Limit theorems for the compensator of Hawkes processes, Statist. Probab. Lett. 127 (2017), 165-172. https://doi.org/10.1016/j.spl.2017.04.003
  21. S. R. S. Varadhan, Large Deviations and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 46, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984.
  22. S. Wheatley, V. Filimonov, and D. Sorrette, The Hawkes process with renewal immigration & its estimation with an EM algorithm, Computational Statistics & Data Analysis 94, 120-135 (2016). https://doi.org/10.1016/j.csda.2015.08.007
  23. L. Zhu, Central limit theorem for nonlinear Hawkes processes, J. Appl. Probab. 50 (2013), no. 3, 760-771. https://doi.org/10.1239/jap/1378401234
  24. L. Zhu, Moderate deviations for Hawkes processes, Statist. Probab. Lett. 83 (2013), no. 3, 885-890. https://doi.org/10.1016/j.spl.2012.12.011
  25. L. Zhu, Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims, Insurance Math. Econom. 53 (2013), no. 3, 544-550. https://doi.org/10.1016/j.insmatheco.2013.08.008
  26. L. Zhu, Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps, J. Appl. Probab. 51 (2014), no. 3, 699-712. https://doi.org/10.1239/jap/1409932668
  27. L. Zhu, Process-level large deviations for nonlinear Hawkes point processes, Ann. Inst. Henri Poincare Probab. Stat. 50 (2014), no. 3, 845-871. https://doi.org/10.1214/12-AIHP532
  28. L. Zhu, Large deviations for Markovian nonlinear Hawkes processes, Ann. Appl. Probab. 25 (2015), no. 2, 548-581. https://doi.org/10.1214/14-AAP1003