We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian–fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter

The quadratic variation, or the pathwise volatility, of stochastic processes is of paramount importance in mathematical finance. Indeed, it was the major discovery of the celebrated article by Black and Scholes [

An important class of pricing models is the mixed Brownian–fractional Brownian model. This is a model where the quadratic variation is determined by the Brownian part and the correlation structure is determined by the fractional Brownian part. Thus, this is a pricing model that captures the long-range dependence while leaving the Black–Scholes pricing formulas intact. The mixed Brownian–fractional Brownian model has been studied in the pricing context, for example, in [

By the hedging paradigm the prices and hedges of financial derivative depend only on the pathwise quadratic variation of the underlying process. Consequently, the statistical estimation of the quadratic variation is an important problem. One way to estimate the quadratic variation is to use directly its definition by the so-called

The rest of the paper is organized as follows. In Section

It is well known that (see [

Dzhaparidze and Spreij [

Recently, the convergence (

It is known that (see [

Given

Let

Föllmer [

The rest of the section contains the essential elements of Gaussian analysis and Malliavin calculus that are used in this paper. See, for instance, Refs. [

Let

By iteration, the

For every

To obtain Berry–Esseen-type estimate, we shall use the following result from [

In this subsection, we briefly describe how two Gaussian processes can be embedded into an isonormal Gaussian process. Let

Let

Now, for any

The fractional Brownian motion

The following proposition establishes the link between pathwise integral and Skorokhod integral in Malliavin calculus associated to fractional Brownian motion and will play an important role in our analysis.

For further use, we also need the following ancillary facts related to the isonormal Gaussian process derived from the covariance function of the mixed Brownian–fractional Brownian motion. Assume that

Throughout this section, we assume that

We start with the following fact, which is one of our key ingredients.

Our first aim is to transform the pathwise integral in (

First, note that

We will also pose the following assumption for characteristic function

Note that Assumption

We continue with the following technical lemma, which in fact provides a correct normalization for our central limit theorems.

We point it out that the variance

Throughout the proof,

We also apply the following proposition. The proof is rather technical and is postponed to Appendix A.

Our main theorem is the following.

First, by applying Lemmas

Note that the proof of Theorem

As a consequence of the proof of Theorem

By proof of Theorem

Now, by using the proof of Proposition

In many cases of interest, the leading term in

Consider the case

Azmoodeh is supported by research project F1R-MTH-PUL-12PAMP from University of Luxembourg, and Lauri Viitasaari was partially funded by Emil Aaltonen Foundation. The authors are grateful to Christian Bender for useful discussions.

Denote

We have