Fractional Ornstein–Uhlenbeck processes. Joseph effect in models with infinite variance

Abstract

We consider five fractional generalizations of the Markovian α -stable Ornstein–Uhlenbeck process and explore the dependence structure of these stochastic models. Since the variance of α -stable distributed random variables is infinite, we describe the dependence structure of the introduced processes in the language of the function called codifference. We present exact formulas for the asymptotic behavior of codifference and answer the question of long-range dependence property (Joseph effect) for the discussed fractional α -stable models. We show that the fractional Ornstein–Uhlenbeck processes can display both Noah and Joseph effect.