Shot noise on cluster processes with cluster marks, and studies of long range dependence

Abstract

With the aim of providing greater flexibility in developing and applying shot noise models, this paper studies shot noise on cluster point processes with both pointwise and cluster marks. For example, in financial modelling, responses to events in the financial market may occur in clusters, with random amplitudes including a ‘cluster component’ reflecting a degree of commonness among responses within a cluster. For such shot noise models, general formulae for the characteristic functional are developed and specialized to the case of Neyman-Scott clustering with cluster marks. For several general forms of response function, long range dependence of the corresponding equilibrium shot noise models is investigated. It is shown, for example, that long range dependence holds when the ‘structure component’ of the response function decays slowly enough, or when the response function has a finite random duration with a heavy tailed distribution.