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April 2015 Limit theorems for nearly unstable Hawkes processes
Thibault Jaisson, Mathieu Rosenbaum
Ann. Appl. Probab. 25(2): 600-631 (April 2015). DOI: 10.1214/14-AAP1005


Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^{1}$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox–Ingersoll–Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65–77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.


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Thibault Jaisson. Mathieu Rosenbaum. "Limit theorems for nearly unstable Hawkes processes." Ann. Appl. Probab. 25 (2) 600 - 631, April 2015.


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

zbMATH: 1319.60101
MathSciNet: MR3313750
Digital Object Identifier: 10.1214/14-AAP1005

Primary: 60F05, 60F17, 60G55, 62P05

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 2015
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