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Exact simulation of extrinsic stress-release processes

Published online by Cambridge University Press:  14 February 2022

Young Lee*
Affiliation:
Harvard University
Patrick J. Laub*
Affiliation:
University of New South Wales
Thomas Taimre*
Affiliation:
University of Queensland
Hongbiao Zhao*
Affiliation:
Shanghai University of Finance and Economics
Jiancang Zhuang*
Affiliation:
Institute of Statistical Mathematics
*
*Postal address: Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA. Email address:younglee@fas.harvard.edu
**Postal address: School of Risk and Actuarial Studies, UNSW Business School, UNSW Sydney, Sydney, NSW 2052, Australia. Email address: p.laub@unsw.edu.au
***Postal address: School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia. Email address: t.taimre@uq.edu.au
****Postal address: Shanghai University of Finance and Economics, Yangpu District, Shanghai, 200433, China. Email address: h.zhao1@lse.ac.uk
*****Postal address: The Institute of Statistical Mathematics, 10-3 Midori-Cho, Tachikawa-Shi, Tokyo 190-8562, Japan. Email address: zhuangjc@ism.ac.jp

Abstract

We present a new and straightforward algorithm that simulates exact sample paths for a generalized stress-release process. The computation of the exact law of the joint inter-arrival times is detailed and used to derive this algorithm. Furthermore, the martingale generator of the process is derived, and induces theoretical moments which generalize some results of [3] and are used to demonstrate the validity of our simulation algorithm.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bebbington, M. and Harte, D. S. (2001). On the statistics of the linked stress release model. J. Appl. Prob. 38, 176187.10.1239/jap/1085496600CrossRefGoogle Scholar
Bebbington, M. and Harte, D. S. (2003). The linked stress release model for spatio-temporal seismicity: formulations, procedures and applications. Geophys. J. Internat. 154, 925946.CrossRefGoogle Scholar
Borovkov, K. and Vere-Jones, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appl. Prob. 37, 315321.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
Daley, D. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, 2nd edn. Springer, New York.Google Scholar
Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.10.1007/978-1-4613-8643-8CrossRefGoogle Scholar
Isham, V. and Westcott, M. (1979). A self-correcting point process. Stoch. Process. Appl. 8, 335347.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes. Springer.CrossRefGoogle Scholar
Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logistics Quart. 26, 403413.10.1002/nav.3800260304CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes. Springer, New York.CrossRefGoogle Scholar
Liu, J., Vere-Jones, D., Ma, L., Shi, Y.-L. and Zhuang, J. (1998). The principle of coupled stress release model and its application. Acta Seismol. Sinica 11, 273281.CrossRefGoogle Scholar
Lu, C., Harte, D. and Bebbington, M. (1999). A linked stress release model for historical Japanese earthquakes: coupling among major seismic regions. Earth Planets Space 51, 907916.CrossRefGoogle Scholar
Medvegyev, P. (2007). Stochastic Integration Theory (Graduate Texts in Mathematics 14). Oxford University Press.Google Scholar
Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Trans. Inform. Theory 27, 23–31.CrossRefGoogle Scholar
Ogata, Y. (2017). Statistics of earthquake activity: models and methods for earthquake predictability studies. Annu. Rev. Earth Planet. Sci. 45, 497527.10.1146/annurev-earth-063016-015918CrossRefGoogle Scholar
Ogata, Y. and Vere-Jones, D. (1984). Inference for earthquake models: a self-correcting model. Stoch. Process. Appl. 17, 337347.CrossRefGoogle Scholar
Protter, P. (2005). Stochastic Integration and Differential Equations. Springer.10.1007/978-3-662-10061-5CrossRefGoogle Scholar
Shi, Y.-L., Liu, J., Vere-Jones, D., Zhuang, J. and Ma, L. (1998). Application of mechanical and statistical models to the study of seismicity of synthetic earthquakes and the prediction of natural ones. Acta Seismol. Sinica 11, 421430.CrossRefGoogle Scholar
Veen, A. and Schoenberg, F. P. (2008). Estimation of space–time branching process models in seismology using an EM-type algorithm. J. Amer. Statist. Assoc. 103, 614624.10.1198/016214508000000148CrossRefGoogle Scholar
Vere-Jones, D. (1988). On the variance properties of stress release models. Austral. J. Statist. 30A, 123135.10.1111/j.1467-842X.1988.tb00469.xCrossRefGoogle Scholar
Vere-Jones, D. and Ogata, Y. (1984). On the moments of a self-correcting process. J. Appl. Prob. 21, 335342.10.2307/3213644CrossRefGoogle Scholar
Wang, A.-L., Vere-Jones, D. and Zheng, X.-G. (1991). Simulation and estimation procedures for stress release model. Stoch. Process. Appl. 370, 1127.Google Scholar
Zammit-Mangion, A., Dewar, M., Kadirkamanathan, V. and Sanguinetti, G. (2012). Point process modelling of the Afghan War Diary. Proc. Nat. Acad. Sci. USA 109, 1241412419.CrossRefGoogle Scholar
Zheng, X. and Vere-Jones, D. (1991). Applications of stress release models to earthquakes from North China. Pure Appl. Geophys. 135, 559576.10.1007/BF01772406CrossRefGoogle Scholar
Zheng, X. and Vere-Jones, D. (1994). Further applications of the stochastic stress release model to historical earthquake data. Tectonophysics 229, 101121.Google Scholar
Zhuang, J.-C. and Ma, L. (1998). The stress release model and results from modelling features of some seismic regions in China. Acta Seismol. Sinica 11, 5970.CrossRefGoogle Scholar