Renewal in Hawkes processes with self-excitation and inhibition

Abstract

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

Maximum likelihood estimation for Hawkes processes with self-excitation or inhibition

“…In this inhibition context, the cluster representation [14] on which is based the construction of a self-exciting Hawkes process, is no longer valid. While the existence and the construction of such nonlinear processes can be found in recent works for the univariate [15] and multivariate [16] cases, statistical estimation of the kernel function has been hardly addressed. A first approach consists in computing an approximation of the likelihood as if the intensity function could take negative values, and optimizing it to get a maximum likelihood estimator [17].…”

Stability for Hawkes processes with inhibition

“…Classical stability results for multivariate nonlinear Hawkes processes found e.g. in [1] or in the recent paper [3], which is devoted to the study of the stabilising effect of inhibitions, are stated in terms of an associated weight function matrix Λ, imposing that the spectral radius of Λ is strictly smaller than one. In this case the process is termed to be subcritical .…”

“…To the best of our knowledge, only few results have been obtained on this natural question in the literature. [1] gives an attempt in this direction but does only deal with the case when c +− and c −+ are of the same sign (see Theorem 6 in [1]), and [3] do only work with the positive part of the weight functions, without profiting from the explicit inhibitory part within the system.…”

Causal Discovery in High-Dimensional Point Process Networks with Hidden Nodes

“…Assumption 2 requires that the intensity rate is strictly bounded, which prevents degenerate processes for all components of the multivariate Hawkes processes. This assumption has been considered in the previous analysis of Hawkes processes [33][34][35]42,48]. Assumption 3.…”