PhD Defence Mads Bonde Raad

Title: Stability of Age Dependent Hawkes Processes

Abstract:

The primary subject of this dissertation is Hawkes processes with a focus on stability and with neuroscience as a primary application. The Hawkes process has received much attention in the last decade for modeling events that exhibit self excitation -or inhibition. There are many examples of phenomena of interest which exhibit such behavior, including nance that propagate through a market giving rise to volatility clustering observations [2], interactions on social media [42], and pattern dependencies in DNA [39].

Hawkes processes are point processes where the intensity function is stochastic and allowed to depend on the past history, introducing memory in the temporal evolution of the stochastic process. We work with an extension to the ordinary Hawkes model called an age dependent Hawkes process, where the intensity of a unit may also depend on the time since its last jump.

The classical results concerning stability for Hawkes processes found in [4] assume bounds on the strength of the functional connectivity between units in the system. However, by taking inhibition into account, we will be able to produce new stability results for Hawkes processes. We shall also discuss stability for Hawkes processes from a regeneration point of view. We will give a constructive proof of a random time such that the incremented Hawkes process is a Hawkes process in itself independent of the past. Standard Markov chain techniques allow us to prove various asymptotic results for the Hawkes process. Finally, we end by studying the meaneld limit of age dependent Hawkes processes.

 Super Advisor: Professor Susanne Ditlevsen, University of Copenhagen, Denmark

Co-supervisor: Eva Löcherbach, Université de Paris 1 Panthéon-Sorbonne Paris, Frankrig

Assessment committee:

Professor Thomas Valentin Mikosch (Chairman), University of Copenhagen, Denmark

Professor Jesper Møller, Aalborg University, Denmark

Professor Carl Graham, Ecole Polytechnique, Palaiseau