Expectations of Linear and Nonlinear Hawkes Processes by Employing a Field-theoretical Approach

AbstractMoments are some important characteristics for understanding the features and applications of Hawkes processes. In the paper, a field theoretical approach is used to present expectations of Hawkes processes and their intensity functions, and the necessary and sufficient conditions of 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, one is the linear Hawkes process with intensity function in the sum of exponentials, and the other is the non-linear Hawkes process with a quadratic form intensity function in an internal variable in the sum of exponentials and zero mean marks. All results for non-linear Hawkes processes are new conclusions in literature, although the results for linear Hawkes processes recover already known conclusions. The matrix method presented in this paper gives a new way for finding the necessary and sufficient conditions of stability of Hawkes processes.