On Stability of Hawkes Process

Abstract

Existence and stability properties are studied for Hawkes process, i.e. point process $S$ that has long-memory and intensity $r(t)=\lambda (g_0(t)+{\sum }_{\tau <t,\tau \in S}h(t-\tau ))$. The approach to Hawkes process presented in this paper allows us to prove the uniqueness of invariant distribution of the process under weaker conditions. New speed of convergence results are also shown. Unlike previous results the function $\lambda$ is not required to be Lipschitz and can be even discontinuous. Some generalizations are also considered.

25 pages, 1 figure

References in corpus (2)

• Limit Theorems for Marked Hawkes Processes with Application to a Risk Model
• Process-Level Large Deviations for Nonlinear Hawkes Point Processes

Cited by in corpus (3)

• Nonlinear Hawkes Processes
• Limit Theorems for Marked Hawkes Processes with Application to a Risk Model
• Hawkes Process Kernel Structure Parametric Search with Renormalization Factors