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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

Type
Article accepté pour publication ou publié
Date
2013
Journal name
Stochastic Processes and their Applications
Volume
123
Number
7
Publisher
Elsevier
Pages
2475–2499
Publication identifier
http://dx.doi.org/10.1016/j.spa.2013.04.007
Metadata
Show full item record
Author(s)
Muzy, Jean-François
Delattre, Sylvain
Hoffmann, Marc
Bacry, Emmanuel cc
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) [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.
Subjects / Keywords
Statistics of random processes; Discretisation of stochastic processes; Limit theorems; Hawkes processes; Point processes
JEL
C1 - Econometric and Statistical Methods and Methodology: General

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