Multipower Variation for Brownian Semistationary Processes

Abstract

In this paper we study the asymptotic behaviour of power and multipower variations of processes Y :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.

A Weak Limit Theorem for Numerical Approximation of Brownian Semi-stationary Processes

“…In the setting of the previous remark the overall Riemann approximation error is max (N −1 , N /c N ). Recalling that c N /N → ∞, the obtained rate is definitely slower than the one associated with Fourier approximation proposed at (6).…”

“…According to the estimate (11) and the upper bound for the Fourier coefficient of Remark 2 applied for n = 1, we readily deduce the rate N −1 for the L 2 -error approximation connected to (6). On the other hand, the effective sample size of the Riemann approximation at (16) is c N .…”

“…In the past years stochastic analysis, probabilistic properties and statistical inference for Lévy semi-stationary processes have been studied in numerous papers. We refer to [2,3,6,7,11,12,15,17,20,25] for the mathematical theory as well as to [5,26] for a recent survey on theory of ambit fields and their applications. For practical applications in sciences numerical approximation of Lévy (Brownian) semi-stationary processes, or, more generally, of ambit fields, is an important issue.…”

“…Obviously, the above approximation is an L 2 -projection onto the linear subspace generated by orthogonal functions {cos(kπ x/τ ), sin(kπ x/τ )} N k=0 , hence we deal with a classical Fourier expansion of the function h (recall that the function h is even by definition, thus the sinus terms do not appear at (6)). Now, the basic idea of the numerical approximation method proposed in [14,18] is based upon the following relationship:…”

“…In other words, the stochastic field X λ,u (t, y) is simulated via a simple recursive scheme without using the knowledge of g, while the kernel g is approximated via the Fourier transform at (6). This is in contrast to a straightforward discretization scheme…”