Hawkes Processes

Specialeforsvar ved Niklas Nyboe Maltzahn

Titel: Hawkes Processes

Abstract: This thesis addresses Hawkes processes from two perspectives. A dynamic perspective and a branching perspective. The dynamic perspective makes use of thefact that point processes can be constructed through a so-called Poisson embedding. A nice consequence for the constructed process is that it inherit stability properties from the underlying Poisson process. In this framework Hawkes processes (be it linear or non-linear) can be formulated as a solution to a certain Poisson driven stochastic differential equation. The second perspective (the branching perspective) formulates Hawkes processes as a recursive Poisson cluster process. The evolutionary structure of the atoms of such processes can be described by random trees. As a consequence the Hawkes process can be generated from a so-called Hawkes-Poisson forest. Finally an estimation procedure is presented, connecting an equidistant discretization of the Hawkes process to an INAR($\infty$) process

Vejleder:  Thomas Mikosch
Censor:   Søren Asmussen, Aarhus Universitet