Journal of the
Korean Mathematical Society JKMS

Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history and has been widely applied in insurance, finance, queueing theory, statistics, and many other fields. Seol~\cite{Seol5} proposed the inverse Markovian Hawkes processes and studied some asymptotic behaviors. In this paper, we consider an extended inverse Markovian Hawkes process which combines a Markovian Hawkes process and inverse Markovian Hawkes process with features of several existing models of self-exciting processes. We study the limit theorems for an extended inverse Markovian Hawkes process. In particular, we obtain a law of large number and central limit theorems.

Keywords: Hawkes process, inverse Markovian, self-exciting point processes, central limit theorems, law of large numbers

MSC numbers: Primary 60G55, 60F05, 60F10

Supported by: This research is supported by the Dong-A University research grant.