Probabilistic and statistical study of conditionally heteroskedastic nonlinear models.

Summary We study a class of nonlinear conditionally heteroskedastic models (ARCH). The volatility of the variable at date t depends on the relative position of the past variables. We show that a family of such models admits a Markovian representation allowing to study its stability. Using Lyapunov criteria, we establish conditions for the existence of moments. The asymptotic properties (strong and law convergence) of three types of parameter estimators are established. These results are illustrated at finite distance using simulated methods and on real series.