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http://dx.doi.org/10.4134/JKMS.j200251

ASYMPTOTICS FOR AN EXTENDED INVERSE MARKOVIAN HAWKES PROCESS  

Seol, Youngsoo (Department of Mathematics Dong-A University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.4, 2021 , pp. 819-833 More about this Journal
Abstract
Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history and has been widely applied in insurance, finance, queueing theory, statistics, and many other fields. Seol [27] proposed the inverse Markovian Hawkes processes and studied some asymptotic behaviors. In this paper, we consider an extended inverse Markovian Hawkes process which combines a Markovian Hawkes process and inverse Markovian Hawkes process with features of several existing models of self-exciting processes. We study the limit theorems for an extended inverse Markovian Hawkes process. In particular, we obtain a law of large number and central limit theorems.
Keywords
Hawkes process; inverse Markovian; self-exciting point processes; central limit theorems; law of large numbers;
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1 F. Gao and L. Zhu, Some asymptotic results for nonlinear Hawkes processes, Stochastic Process. Appl. 128 (2018), no. 12, 4051-4077. https://doi.org/10.1016/j.spa.2018.01.007   DOI
2 X. Gao and L. Zhu, Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues, Queueing Syst. 90 (2018), no. 1-2, 161-206. https://doi.org/10.1007/s11134-018-9570-5   DOI
3 J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften, 288, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-662-02514-7
4 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   DOI
5 L. Massouli'e, Stability results for a general class of interacting point processes dynamics, and applications, Stochastic Process. Appl. 75 (1998), no. 1, 1-30. https://doi.org/10.1016/S0304-4149(98)00006-4   DOI
6 B. Mehrdad and L. Zhu, On the Hawkes process with different exciting functions, Preprint. arXiv: 1403.0994 2017.
7 Y. Seol, Limit theorems for inverse process Tn of Hawkes process, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 1, 51-60. https://doi.org/10.1007/s10114-016-5470-y   DOI
8 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   DOI
9 Y. Seol, Limit theorems for an inverse Markovian Hawkes processes, Statist. Probab. Lett. 155 (2019).
10 R. Fierro, V. Leiva, and J. Moller, The Hawkes process with different exciting functions and its asymptotic behavior, J. Appl. Prob 52 (2015), 37-54.   DOI
11 E. Errais, K. Giesecke, and L. Goldberg, Affine point processes and portfolio credit risk, SIAM J. Financial Math. Vol. 1 (2010), 642-665.   DOI
12 C. Bordenave and G. L. Torrisi, Large deviations of Poisson cluster processes, Stoch.Models 23 (2007), 593-625.   DOI
13 P. Br'emaud and L. Massouli'e, Stability of nonlinear Hawkes processes, Ann. Probab. 24 (1996), 1563-1588.   DOI
14 A. Dassios and H. Zhao, A dynamic contagion process, Adv. in Appl. Probab. 43 (2011), 814-846.   DOI
15 P. Jagers, Branching Processes with Biological Applications, Wiley-Interscience, London, 1975.
16 F. G. Foster, On the stochastic matrices associated with certain queuing processes, Ann. Math. Statistics 24 (1953), 355-360. https://doi.org/10.1214/aoms/1177728976   DOI
17 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   DOI
18 X. Gao and L. Zhu, Limit theorems for Markovian Hawkes processes with a large initial intensity, Stochastic Process. Appl. 128 (2018), no. 11, 3807-3839. https://doi.org/10.1016/j.spa.2017.12.001   DOI
19 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   DOI
20 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   DOI
21 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   DOI
22 L. Zhu, Process-level large deviations for nonlinear Hawkes point processes, Ann. Inst. Henri Poincar'e Probab. Stat. 50 (2014), no. 3, 845-871. https://doi.org/10.1214/12-AIHP532   DOI
23 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   DOI
24 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   DOI
25 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   DOI
26 X. Gao and L. Zhu, Large deviations and applications for Markovian Hawkes processes with a large initial intensity, Bernoulli 24 (2018), no. 4A, 2875-2905. https://doi.org/10.3150/17-BEJ948   DOI
27 E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, Scaling limits for Hawkes processes and application to financial statistics, Stochastic Process. Appl. 123 (2012), 2475-2499.
28 D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume I and II, Springer, Second Edition 2003.
29 S. Wheatley, V. Filimonov, and D. Sorrette, The Hawkes process with renewal immigration & its estimation with an EM algorithm, Computational Statistics & Data Analysis 94 (2016), 120-135.   DOI
30 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   DOI
31 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   DOI
32 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   DOI
33 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   DOI
34 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   DOI