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April 2015 Large deviations for Markovian nonlinear Hawkes processes
Lingjiong Zhu
Ann. Appl. Probab. 25(2): 548-581 (April 2015). DOI: 10.1214/14-AAP1003

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.

Citation

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Lingjiong Zhu. "Large deviations for Markovian nonlinear Hawkes processes." Ann. Appl. Probab. 25 (2) 548 - 581, April 2015. https://doi.org/10.1214/14-AAP1003

Information

Published: April 2015
First available in Project Euclid: 19 February 2015

zbMATH: 1312.60019
MathSciNet: MR3313748
Digital Object Identifier: 10.1214/14-AAP1003

Subjects:
Primary: 60F10, 60G55

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 2 • April 2015
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