Antoine Lotz - Some Limit Theorems For Locally Stationary Hawkes Processes, With Statistical Applications

Hawkes processes are a class of path-dependent point processes in which past events influence later occurrences. The dependence structure of the process is characterised in terms of regulating or excitating interactions between its coordinates k ∈ {1, · · · ,K}, encoded by a matrix kernel function φ = (φkl) : R → RK × RK. The intensity at coordinate k then depends upon the φkl via an activation function Φk : R 7→ [0,∞). In the particular case Φk(x) ∝ x, the process is endowed with a population representation, in which kernel norms ∥φkl∥L1 =R∞0 |φkl(t)| dt correspond to fertility rates. The spectral radius α ∈ [0, 1) of the matrix with coefficients ∥φkl∥L1 , hence referred to as a reproduction or endogeneity rate, is a key characteristic of the process. We consider an extension of the Hawkes process wherein the reproduction rate is allowed to fluctuate in time, corresponding to a subclass of the so-called locally stationary Hawkes processes introduced by Roueff et al. In the linear case Φk(x) ∝ x we prove a functional law of large numbers and central limit theorem, characterising the behaviour of the process as the coarse scale T → ∞. In the general case
of Lipschitz-continuous activation function Φk, our asymptotic theory yields important implications for parametric inference. We obtain the asymptotic distribution of the likelihood ratio test for two interaction selection problems. Firstly, we consider whether the reproduction rate does vary in time, against the alternative that it remains constant. Secondly, whether some selected set of interactions between two coordinates do exist, against the alternative that they are null. Finally, our model is applied on high-frequency financial data from German intraday power markets, revealing a recurring pattern in the rate of market participation.