@article{4a22a0b0e23c411f81beba0f9b982bf7,

title = "Statistical inference for the doubly stochastic self-exciting process",

abstract = "We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter T −10T θt∗ dt, where θt∗ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it na{\"i}vely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-na{\"i}ve estimator which is constructed as the na{\"i}ve one when applying a first-order bias reduction to the local MLE. We show the associated central limit theorem. Monte Carlo simulations show the importance of the bias correction and that the method performs well in finite sample, whereas the empirical study discusses the implementation in practice and documents the stochastic behavior of the parameters.",

keywords = "Hawkes process, High-frequency data, Integrated parameter, Self-exciting process, Stochastic, Time-varying parameter",

author = "Simon Clinet and Yoann Potiron",

note = "Funding Information: The research of Simon Clinet is in part supported by CREST Japan Science and Technology Agency. The research of Yoann Potiron is supported by the National Science Foundation [DMS 14-07812], Japanese Society for the Promotion of Science Grant-in-Aid for Young Scientists (60781119) and a private grant from Keio University. All financial data is provided by the Chair of Quantitative Finance of the Ecole Centrale Paris. We would like to thank Per Myk-land, Nakahiro Yoshida, Frederic Abergel, Feng Chen, Holger Dette (the Editor), an anonymous Associate Editor and two anonymous referees for helpful discussions and advice. Funding Information: The research of Simon Clinet is in part supported by CREST Japan Science and Technology Agency. The research of Yoann Potiron is supported by the National Science Foundation [DMS 14-07812], Japanese Society for the Promotion of Science Grant-in-Aid for Young Scientists (60781119) and a private grant from Keio University. All financial data is provided by the Chair of Quantitative Finance of the Ecole Centrale Paris. We would like to thank Per Mykland, Nakahiro Yoshida, Frederic Abergel, Feng Chen, Holger Dette (the Editor), an anonymous Associate Editor and two anonymous referees for helpful discussions and advice. Publisher Copyright: {\textcopyright} 2018 ISI/BS.",

year = "2018",

month = nov,

doi = "10.3150/17-BEJ966",

language = "English",

volume = "24",

pages = "3469--3493",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "International Statistical Institute",

number = "4B",

}