Some limit theorems for Hawkes processes and application to financial statistics |
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Authors: | E Bacry S Delattre M Hoffmann JF Muzy |
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Institution: | 1. CMAP CNRS-UMR 7641 and École Polytechnique, 91128 Palaiseau, France;2. Université Paris Diderot and LPMA CNRS-UMR 7599, Boîte courrier 7012, 75251 Paris Cedex 16, France;3. Université Paris-Dauphine and CEREMADE CNRS-UMR 7534, Place du maréchal de Lattre de Tassigny, 75775 Paris, France;4. SPE CNRS-UMR 6134 and Université de Corte, 20250 Corte, France |
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Abstract: | 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 et al. (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 the 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. |
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Keywords: | 60F05 60G55 62M10 |
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