How heavy are the tails of a stationary HARCH(k) process? A study of the moments

Probabilistic properties of HARCH(k) processes, as special stochastic volatility models, are investigated. We present necessary and sufficient conditions for existence of a 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, 1990). 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