Loglinear Hawkes processes

TL;DR

This paper introduces loglinear Hawkes processes with exponential rate functions, providing conditions for explosion, nonexplosion, and stability, thereby establishing a theoretical foundation for their further study and applications.

Contribution

It offers the first comprehensive theoretical analysis of loglinear Hawkes processes, including conditions for explosion, nonexplosion, and stability.

Findings

Conditions for explosion and nonexplosion are established.

Loglinear Hawkes processes are stable with nonpositive memory functions.

Theoretical basis for future research and applications is provided.

Abstract

This paper discusses a special class of nonlinear Hawkes processes, where the rate function is the exponential function. We call these processes loglinear Hawkes processes. In the main theorem, we give sufficient conditions for explosion and nonexplosion that cover a large class of practically relevant memory functions. We also investigate stability. In particular, we show that for nonpositive memory functions the loglinear Hawkes process is stable. The paper aims at providing a theoretical basis for further research and applications of these processes.