The Annals of Applied Probability

Limit theorems for nearly unstable Hawkes processes

Thibault Jaisson and Mathieu Rosenbaum

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 600-631.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355125

Digital Object Identifier
doi:10.1214/14-AAP1005

Mathematical Reviews number (MathSciNet)
MR3313750

Zentralblatt MATH identifier
1319.60101

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G55: Point processes 62P05: Applications to actuarial sciences and financial mathematics

Keywords
Point processes Hawkes processes limit theorems microstructure modeling high-frequency data order flows Cox–Ingersoll–Ross model Heston model

Citation

Jaisson, Thibault; Rosenbaum, Mathieu. Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25 (2015), no. 2, 600--631. doi:10.1214/14-AAP1005. https://projecteuclid.org/euclid.aoap/1424355125


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