Multipower variation for Brownian semistationary processes

Abstract

In this paper we study the asymptotic behaviour of power and multipower variations of processes Y: Yt = ∫t-∞ g(t - s)σsW(ds) + Zt, where g : (0,∞)→R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency σ. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of Y as a basis for studying properties of the intermittency process σ. Notably the processes Y are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given. © 2011 ISI/BS.