Some limit theorems for Hawkes processes and application to financial statistics

Abstract

A Special Issue on the Occasion of the 2013 International Year of StatisticsInternational audienceAbstract 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 ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? 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

Fractional Skellam processes with applications to finance

“…where N (1) (t), t ≥ 0 and N (2) (t), t ≥ 0 are two independent homogeneous Poisson processes with intensities λ (1) > 0 and λ (2) > 0, respectively.…”

“…A fractional Sekellam process of type I X(t) has marginal laws of fractional Skellam type I denoted by X(t) ∼ f Sk(k, t; λ (1) , α (1) , λ (2) , α (2) ), which is a new four parameter distribution.…”

“…(1) i and T (2) j are waiting times between events from (2.9). Since these random variables follow MittagLeffler distribution, the distribution of their sum has a density, and the probabilities of the events summed above all have probability zero.…”

“…This can be quantified through the 95% confidence interval for α (2) computed as (0.9557, 0.9597), concluding that α (2) = 1 as would be the case if inter-arrival times where exponential in law.…”

“…Recent literature on the subject includes [1,2,3,10] where models based on the difference of two point processes are proposed. Difference of Poisson processes is considered in [3,10], and Hawkes processes are discussed in [1,2]. This paper extends the models where the forward price of a risky asset c 2014 Diogenes Co., Sofia pp.…”