Some limit theorems for Hawkes processes and application to financial statistics
Muzy, Jean-François; Delattre, Sylvain; Hoffmann, Marc; Bacry, Emmanuel (2013), Some limit theorems for Hawkes processes and application to financial statistics, Stochastic Processes and their Applications, 123, 7, p. 2475–2499. http://dx.doi.org/10.1016/j.spa.2013.04.007
TypeArticle accepté pour publication ou publié
Journal nameStochastic Processes and their Applications
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Abstract (EN)In 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)  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.
Subjects / KeywordsStatistics of random processes; Discretisation of stochastic processes; Limit theorems; Hawkes processes; Point processes
Showing items related by title and author.
Achab, Massil; Bacry, Emmanuel; Gaïffas, Stéphane; Mastromatteo, Iacopo; Muzy, Jean-François (2017) Article accepté pour publication ou publié