Scaling limits for Hawkes processes and application to financial statistics

TL;DR

This paper establishes large-scale limit theorems for multivariate Hawkes processes and applies these results to model and analyze complex financial phenomena like the Epps and lead-lag effects across different time scales.

Contribution

It provides the first law of large numbers and functional central limit theorem for multivariate Hawkes processes, with applications to financial statistics and modeling of asset variations.

Findings

Derived the macroscopic diffusion limit of a Hawkes-based financial model

Characterized the asymptotic behavior of covariation in Hawkes processes

Reproduced stylized facts like the Epps and lead-lag effects across scales

Abstract

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.