Scaling limits for Hawkes processes and application to financial statistics

By Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, Jean-François Muzy. 2012

84 citations recorded by the source corpus.

We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval $[0,T]$ in the limit $T \rightarrow \infty$. 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 $\Delta$ over $[0,T]$ up to some further time shift $\tau$. The behaviour of this functional depends on the relative size of $\Delta$ 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 a previous work 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 enable to track these effects across scales in rigorous mathematical terms.

Subfields: Applied Mathematics, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/4bc9644edc2d0fdfc528111e021071c48a41dd3b

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Citations in the corpus

Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations (2012, methodology): “In the case of the Hawkes model, Assumption 1 was shown to hold in Bacry et al. (2010). Also, these assumptions are quite plausible for high frequency quotes for liquid US stocks since, as argued in Section 3:”
Limit theorems for nearly unstable Hawkes processes (2013, methodology): “Moreover, it is shown in [5] that when one considers this price at large time scales, the stability condition implies that after suitable renormalization, it converges towards a Brownian motion (with a given volatility).”
Limit Theorems for Marked Hawkes Processes with Application to a Risk Model (2012, background): “[1] proved a functional central limit theorem for the linear multivariate Hawkes process under certain assumptions which includes the linear Hawkes process as a special case and they proved that N·t − ·μt √ t → σB(·), as t→ ∞, (1.”
Hawkes model for price and trades high-frequency dynamics (2013, methodology): “In the large time limit, one can show that these processes converge to correlated Brownian motions (see Bacry et al. (2013a) or section 3.3†).”
Ergodicity and scaling limit of a constrained multivariate Hawkes process (2013, unknown): “use marked Hawkes processes to reproduce empirically microstructure noise and the stylized facts such as Epps effect and lead-lag in [5] and reach to derive and characterize the exact macroscopic diffusion limit of this model in [6].”