Limit theorems for Hawkes processes with uniform immigrants

다국어 초록 (Multilingual 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.

참고문헌 (Reference)

1 X. Gao, "Transform analysis for Hawkes processes with applications in dark pool trading" 18 (18): 265-282, 2018

2 S. Wheatley, "The Hawkes process with renewal immigra-tion & its estimation with an EM algorithm" 94 : 120-135, 2016

3 R. Fierro, "The Hawkes process with different exciting functions and its asymptotic behavior" 52 (52): 37-54, 2015

4 P. Bremaud, "Stability of nonlinear Hawkes processes" 24 (24): 1563-1588, 1996

5 A. G. Hawkes, "Spectra of some self-exciting and mutually exciting point processes" 58 : 83-90, 1971

6 E. Bacry, "Some limit theorems for Hawkes processes and application to financial statistics" 123 (123): 2475-2499, 2013

7 L. Zhu, "Ruin probabilities for risk processes with non-stationary arrivals and subexpo-nential claims" 53 (53): 544-550, 2013

8 T. Jaisson, "Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes" 26 (26): 2860-2882, 2016

9 L. Zhu, "Process-level large deviations for nonlinear Hawkes point processes" 50 (50): 845-871, 2014

10 B. Mehrdad, "On the Hawkes process with different exciting functions"

11 Y. Seol, "Moderate deviations for marked Hawkes processes" 33 (33): 1297-1304, 2017

12 L. Zhu, "Moderate deviations for Hawkes processes" 83 (83): 885-890, 2013

13 Y. Seol, "Limit theorems for the compensator of Hawkes processes" 127 : 165-172, 2017

14 T. Jaisson, "Limit theorems for nearly unstable Hawkes processes" 25 (25): 600-631, 2015

15 D. Karabash, "Limit theorems for marked Hawkes processes with application to a risk model" 31 (31): 433-451, 2015

16 Y. Seol, "Limit theorems for inverse process Tn of Hawkes process" 33 (33): 51-60, 2017

17 Y. Seol, "Limit theorems for discrete Hawkes processes" 99 : 223-229, 2015

18 L. Zhu, "Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps" 51 (51): 699-712, 2014

19 C. Bordenave, "Large deviations of Poisson cluster processes" 23 (23): 593-625, 2007

20 L. Zhu, "Large deviations for Markovian nonlinear Hawkes processes" 25 (25): 548-581, 2015

21 S. R. S. Varadhan, "Large Deviations and Applications" Society for Industrial and Applied Mathematics (SIAM) 1984

22 A. Dembo, "Large Deviations Techniques and Applications" Springer-Verlag 1998

23 L. Zhu, "Central limit theorem for nonlinear Hawkes processes" 50 (50): 760-771, 2013

24 P. Jagers, "Branching Processes with Biological Applications" Wiley-Interscience 1975

25 D. J. Daley, "An Introduction to the Theory of Point Processes. Vol. I" Springer-Verlag 2003

26 E. Errais, "Affine point processes and portfolio credit risk" 1 (1): 642-665, 2010

27 A. Dassios, "A dynamic contagion process" 43 (43): 814-846, 2011

28 A. G. Hawkes, "A cluster process representation of a self-exciting process" 11 : 493-503, 1974