Renewal Hawkes Processes: Expectations and Applications

TL;DR

This paper introduces three new classes of renewal Hawkes processes incorporating renewal factors, derives their expectations through integral equations, and applies them to optimize systems with cascading failures.

Contribution

The paper presents novel renewal Hawkes process models with exogenous and endogenous factors, along with their expectations and applications to system optimization.

Findings

Derived expectations via integral equations.

Established a general renewal equation for these processes.

Applied models to optimize periodic replacement policies.

Abstract

Hawkes processes are point processes with self-exciting and clustering properties that are popular in applications. In recent years, renewal Hawkes processes have gained attention, due to their versatility such as the capability of capturing dependence between clusters. In this paper, three classes of novel renewal Hawkes processes are introduced after incorporating exogenous and endogenous renewal factors. The relationship among five related Hawkes processes is studied. The expectations of the three classes of renewal Hawkes processes are derived by establishing a set of integral equations. A general common renewal equation for these expectations is derived and further discussions are provided. The special case of constant exogenous factor is discussed as well for the three proposed Hawkes processes. Finally, we apply our proposed models to study the optimization problems that arise in a periodic replacement policy for systems with cascading failures and provide numerical illustrations. The numerical solutions rely on computing the expectations of the proposed renewal Hawkes processes that can be efficiently obtained by using the direct Riemann integration method.