Properties and Applications of Hawkes Processes

Abstract

Point processes in probability theory are special stochastic processes which model the occurrence

of events (points) in some space and parameterized by their frequency. Hawkes

processes are general point processes where the frequency can be self-exciting or mutually

exciting. They occur in various areas such as geological sciences, epidemiology and financial

mathematics. The focus of this thesis lies in discussing different types of Hawkes processes

and their properties as well as presenting financial applications. The (linear) Hawkes process

or self-exciting Hawkes process is a process with a single counting process such that that the

occurrence of an event increases the probability of the occurrence of another event. In the

case of the (linear) Hawkes process, we prove the Law of Large Numbers and the Central

Limit Theorem. Moreover, we study the Hawkes likelihood function and the Hawkes loglikelihood

function. Besides a self-exciting Hawkes process there exists a mutually exciting

Hawkes process, which is a process with multiple counting processes that are depended on

each other. Meaning that the occurrence of an event in one of these counting processes

also lead to an increased probability of an event occurring in the other counting processes.

We study the Hawkes likelihood function and the Hawkes log-likelihood function for the

mutually exciting Hawkes process. The marked Hawkes process is a (linear) Hawkes process

with added random variables called the random marks. We prove the Hawkes likelihood

function and the Hawkes log-likelihood function as well as the Central Limit Theorem.

Furthermore, we derive the dynamics for the Hawkes jump-diffusion model given by three

stochastic differential equations and prove the Law of Large Numbers and Central Limit

Theorem for this particular model.