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Some limit theorems for Hawkes processes and application to financial statistics. (English) Zbl 1292.60032
Summary: 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 $$T\to \infty$$. 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 $$\varDelta$$ over $$[0,T]$$ up to some further time shift $$\tau$$. The behaviour of this functional depends on the relative size of $$\varDelta$$ and $$\tau$$ 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 E. Bacry et al. [Quant. Finance 13, No. 1, 65–77 (2013; Zbl 1280.91073)] 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.

##### MSC:
 60F05 Central limit and other weak theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and financial mathematics
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