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How Heavy are the Tails of a Stationary HARCH(k) Process? A Study of the Moments

  • Paul Embrechts
  • Gennady Samorodnitsky
  • Michel M. Dacorogna
  • Ulrich A. Müller
Part of the Trends in Mathematics book series (TM)

Abstract

Probabilistic properties of HARCH(k) processes as special stochastic volatility models are investigated. We present necessary and sufficient conditions for the existence of a strongly stationary version of a HARCH(k) process with finite (2m)th moments, m ⩾ 1. Our approach is based on the general Markov chain techniques of (Meyn and Tweedie, 1993). The conditions are explicit in the case of second moments, and also in the case of 4th moments of the HARCH(2) process. We also deduce explicit necessary and explicit sufficient conditions for higher order moments of general HARCH(k) models. We start by studying the HARCH(2) process (in which case our results are the most explicit) and then generalize the results to a general HARCH(k) process.

Keywords and phrases

HARCH processes heavy tails Markov chains heteroscedasticity Harris recurrence random recursion stochastic volatility. 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Paul Embrechts
    • 1
  • Gennady Samorodnitsky
    • 2
  • Michel M. Dacorogna
    • 3
  • Ulrich A. Müller
    • 3
  1. 1.Department of MathematicsETH ZentrumZürichSwitzerland
  2. 2.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA
  3. 3.Research Institute for Applied EconomicsOlsen & AssociatesZürichSwitzerland

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