(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

Abstract We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process (Xₙ) ₍₀ with parameters a₁, , aₚ R, p, and 0, such that, for all n p, conditioned on X₀, , X₍-₁, Xₙ is Poisson distributed with parameter (a₁ X₍-₁ + + aₚ X₍-₏ +) _+. This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case p = 2, for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation xₙ = a₁x₍-₁ + a₂x₍-₂ + remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in a₁ and a₂, in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.