Expectations of Linear and Nonlinear Hawkes Processes Using a Field-Theoretical Approach

Abstract

Moments play a crucial role for understanding the mathematical properties and practical applications of Hawkes processes. Here, we derive expectations of Hawkes processes and their intensity functions using a recently introduced Markovian embedding of (generally non-Markovian) linear and nonlinear Hawkes processes via a field-theoretical approach. The necessary and sufficient conditions for the stability of the Hawkes processes are also given by using the expectations of intensity functions directly via some matrix manipulations. Two kinds of Hawkes processes are considered, the standard linear Hawkes process with non-Markovian memory function expressed as a finite sum of exponentials, and the nonlinear Hawkes process with an intensity function that is quadratic as a function of an internal variable (“tension”) itself expressed as the sum over all past events with memory function given as a finite sum of exponentials and with zero mean random marks. All results obtained for the quadratic Hawkes processes are new contributions to the literature. The results obtained for linear Hawkes processes recover already known conclusions, while providing a novel alternative approach to existing methods. The matrix method presented in this paper gives a new way for finding the necessary and sufficient conditions for the stability of Hawkes processes.