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dc.contributor.authorMuzy, Jean-François
HAL ID: 8572
dc.contributor.authorDelattre, Sylvain
dc.contributor.authorHoffmann, Marc
dc.contributor.authorBacry, Emmanuel
HAL ID: 735850
ORCID: 0000-0001-5997-1942
dc.subjectStatistics of random processesen
dc.subjectDiscretisation of stochastic processesen
dc.subjectLimit theoremsen
dc.subjectHawkes processesen
dc.subjectPoint processesen
dc.titleSome limit theorems for Hawkes processes and application to financial statisticsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherSPE CNRS-UMR 6134 and Université de Corte;France
dc.contributor.editoruniversityotherUniversité Paris Diderot and LPMA CNRS-UMR 7599;France
dc.contributor.editoruniversityotherCMAP CNRS-UMR 7641 and École Polytechnique;France
dc.description.abstractenIn the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0,T] when T→∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh Δ over [0,T] up to some further time shift τ. The behaviour of this functional depends on the relative size of Δ and τ with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce important empirical stylised fact such as the Epps effect and the lead-lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.en
dc.relation.isversionofjnlnameStochastic Processes and their Applications
dc.subject.ddclabelEconomie financièreen

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