  We introduce the notion of continuous invertibility on a compact set for
volatility models driven by a Stochastic Recurrence Equation (SRE). We prove
the strong consistency of the Quasi Maximum Likelihood Estimator (QMLE) when
the optimization procedure is done on a continuously invertible domain. This
approach gives for the first time the strong consistency of the QMLE used by
Nelson in \cite{nelson:1991} for the EGARCH(1,1) model under explicit but non
observable conditions. In practice, we propose to stabilize the QMLE by
constraining the optimization procedure to an empirical continuously invertible
domain. The new method, called Stable QMLE (SQMLE), is strongly consistent when
the observations follow an invertible EGARCH(1,1) model. We also give the
asymptotic normality of the SQMLE under additional minimal assumptions.
