First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation
| Autor: | Emmanuel Bacry, Jean-François Muzy |
|---|---|
| Přispěvatelé: | Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Sciences pour l'environnement (SPE), Centre National de la Recherche Scientifique (CNRS)-Université Pascal Paoli (UPP) |
| Rok vydání: | 2016 |
| Předmět: |
Multivariate statistics
Computer science high-frequency trading events 01 natural sciences Power law 010104 statistics & probability Mathematical model [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] earthquakes occurrence dynamics second-order statistics characterization ComputingMilieux_MISCELLANEOUS correlation methods 050208 finance inverse problems Covariance matrix 05 social sciences Shape circular dependence Inverse problem discrete-event systems Correlation Computer Science Applications Kernel power-law estimation error numerical inversion [PHYS.PHYS.PHYS-DATA-AN]Physics [physics]/Physics [physics]/Data Analysis Statistics and Probability [physics.data-an] Information Systems integral equations first-order statistics characterization nonparametric estimation procedure three-variate processes microstructure Library and Information Sciences Kernel (linear algebra) nonpositive kernels statistical analysis Stochastic processes 0502 economics and business Wiener-Hopf integral equations Applied mathematics financial markets 0101 mathematics earthquakes [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph] Hawkes kernel matrix estimation Stochastic process multivariate point processes Nonparametric statistics covariance matrices matrix algebra Integral equation multivariate Hawkes process [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] monovariate processes correlation matrix |
| Zdroj: | IEEE Transactions on Information Theory IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2016, 62 (4), pp.2184-2202. ⟨10.1109/TIT.2016.2533397⟩ |
| ISSN: | 1557-9654 0018-9448 |
| DOI: | 10.1109/tit.2016.2533397 |
| Popis: | We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix through a system of Wiener–Hopf integral equations. A Wiener–Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequently that the first- and second-order properties fully characterize a Hawkes process. The numerical inversion of this system of integral equations allows us to propose a fast and efficient method, which main principles were initially sketched by Bacry and Muzy, to perform a non-parametric estimation of the Hawkes kernel matrix. In this paper, we perform a systematic study of this non-parametric estimation procedure in the general framework of marked Hawkes processes. We precisely describe this procedure step by step. We discuss the estimation error and explain how the values for the main parameters should be chosen. Various numerical examples are given in order to illustrate the broad possibilities of this estimation procedure ranging from monovariate (power-law or non-positive kernels) up to three-variate (circular dependence) processes. A comparison with other non-parametric estimation procedures is made. Applications to high-frequency trading events in financial markets and to earthquakes occurrence dynamics are finally considered. |
| Databáze: | OpenAIRE |
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