The Annals of Applied Probability

Large deviations for Markovian nonlinear Hawkes processes

Lingjiong Zhu

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Abstract

Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalize it to get the result for sum of exponentials exciting functions. We then provide an alternative proof for the large deviation principle for a linear Hawkes process. Finally, we use an approximation approach to prove the large deviation principle for a special class of nonlinear Hawkes processes with general exciting functions.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 548-581.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355123

Digital Object Identifier
doi:10.1214/14-AAP1003

Mathematical Reviews number (MathSciNet)
MR3313748

Zentralblatt MATH identifier
1312.60019

Subjects
Primary: 60G55: Point processes 60F10: Large deviations

Keywords
Large deviations rare events point processes Hawkes processes self-exciting processes

Citation

Zhu, Lingjiong. Large deviations for Markovian nonlinear Hawkes processes. Ann. Appl. Probab. 25 (2015), no. 2, 548--581. doi:10.1214/14-AAP1003. https://projecteuclid.org/euclid.aoap/1424355123


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