Multipower variation for Brownian semistationary processes

作者：Barndorff Nielsen Ole E^*; Corcuera Jose Manuel; Podolskij Mark

来源：Bernoulli, 2011, 17(4): 1159-1194.

DOI：10.3150/10-BEJ316

• central limit theorem
• Gaussian processes
• intermittency
• non-semimartingales
• turbulence
• volatility
• Wiener chaos

In this paper we study the asymptotic behaviour of power and multipower variations of processes Y:
Y-t =integral(t)(-infinity) g(t - s)sigma W-s(ds)+ Z(t),
where g: (0, infinity) -> R is deterministic, sigma > 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 sigma. 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 sigma. 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.

• 出版日期2011-11