Delayed Hawkes birth-death processes

arXiv (Cornell University)

Abstract

We introduce, and formally establish, a variant of the Hawkes-fed birth-death process -- the delayed Hawkes birth-death process -- in which the conditional intensity does not increase at arrivals but at departures from the system. In a scaling limit where sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$, the delayed Hawkes process behaves markedly differently from its classical counterpart. We design a family of models admitting a cluster representation and containing the Hawkes and delayed Hawkes processes as special cases. The cluster representation allows for transform characterizations by a fixed-point equation and for analysis of heavy-tailed asymptotics. We compare the delayed Hawkes process to the classical Hawkes process using stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behavior. In the Markovian network case, a recursive procedure is presented to calculate the $d$th-order moments analytically.

Authors 3

1. Justin Baars Aachen Dept. of Quantitative Economics with CentER

   Leiden University · Eindhoven University of Technology · Tilburg University

   Affiliation as printed

   Dept. of Quantitative Economics , University of Amster- dam. RL is also with Eurandom , Eindhoven University of Technology , and with CentER , Tilburg University. MM is with the Mathematical Institute , Leiden University ,

2. Roger J. A. Laeven Aachen Dept. of Quantitative Economics with CentER

   Leiden University · University of Amsterdam · Eindhoven University of Technology · Tilburg University

   Affiliation as printed

   Dept. of Quantitative Economics , University of Amster- dam. RL is also with Eurandom , Eindhoven University of Technology , and with CentER , Tilburg University. MM is with the Mathematical Institute , Leiden University ,

   Korteweg-de Vries Institute for Mathematics , University of Amsterdam ; Eurandom , Eindhoven University of Technology , Eindhoven ;

3. Michel Mandjes Aachen Dept. of Quantitative Economics with CentER

   Leiden University · University of Amsterdam · Eindhoven University of Technology · Tilburg University · Amsterdam University of Applied Sciences · Amsterdam University College

   Affiliation as printed

   Amsterdam Business School , University of Amsterdam

   Dept. of Quantitative Economics , University of Amster- dam. RL is also with Eurandom , Eindhoven University of Technology , and with CentER , Tilburg University. MM is with the Mathematical Institute , Leiden University ,