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Exponential control of the trajectories of iterated function systems with application to semi-strong GARCH $\boldsymbol{{(P, Q)}}$ models

Part of: Inference from stochastic processes Stochastic analysis Applications

Published online by Cambridge University Press:  15 May 2023

Baye Matar Kandji Open the ORCID record for Baye Matar Kandji [Opens in a new window]
Baye Matar Kandji*
Affiliation:
CREST, ENSAE, Institut Polytechnique de Paris
*
*Postal address: 5 Avenue Henri Le Chatelier, 91120 Palaiseau, France. Email: bayematar.kandji@ensae.fr
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Abstract

We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.

Keywords

Inference without momentsquasi-maximum likelihoodsemi-strong GARCHstochastic recurrence equation

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 62M10: Time series, auto-correlation, regression, etc. 62P05: Applications to actuarial sciences and financial mathematics
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1501 - 1515
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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