## Rocky Mountain Journal of Mathematics

### Asymptotics for Hawkes processes with large and small baseline intensities

Youngsoo Seol

#### Abstract

This paper focuses on asymptotic results for linear Hawkes processes with large and small baseline intensities. The intensity process is one of the main tools used to work with the dynamical properties of a general point process. It is of essential interest in credit risk study, in particular. First, we establish a large deviation principle and a moderate deviation principle for the Hawkes process with large baseline intensity. In addition, a law of large numbers and a central limit theorem are also obtained. Second, we observe asymptotic behaviors for the Hawkes process with small baseline intensity. The main idea of the proof relies on the immigration-birth representation and the observations of the moment generating function for the linear Hawkes process.

#### Note

This research was supported from a Dong-A University research grant.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 2 (2019), 661-680.

Dates
Revised: 3 October 2018
First available in Project Euclid: 23 June 2019

https://projecteuclid.org/euclid.rmjm/1561318399

Digital Object Identifier
doi:10.1216/RMJ-2019-49-2-661

Mathematical Reviews number (MathSciNet)
MR3973246

Zentralblatt MATH identifier
07079990

Subjects
Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60F10}

#### Citation

Seol, Youngsoo. Asymptotics for Hawkes processes with large and small baseline intensities. Rocky Mountain J. Math. 49 (2019), no. 2, 661--680. doi:10.1216/RMJ-2019-49-2-661. https://projecteuclid.org/euclid.rmjm/1561318399

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