Abstract
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 $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator.
Highlights
Introduction and motivationThe quadratic variation, or the pathwise volatility, of stochastic processes is of paramount importance in mathematical finance
It was the major discovery of the celebrated article by Black and Scholes [8] that the prices of financial derivatives depend only on the volatility of the underlying asset
This article investigates the asymptotic normality of the randomized periodogram estimator for the mixed Brownian– fractional Brownian model
Summary
The quadratic variation, or the pathwise volatility, of stochastic processes is of paramount importance in mathematical finance It was the major discovery of the celebrated article by Black and Scholes [8] that the prices of financial derivatives depend only on the volatility of the underlying asset. There are numerous articles that study the asymptotic behavior of realized quadratic variation; see [4, 3, 16, 14, 15] and references therein Another approach, suggested by Dzhaparidze and Spreij [12], is to use the randomized periodogram estimator. This article investigates the asymptotic normality of the randomized periodogram estimator for the mixed Brownian– fractional Brownian model. Some technical calculations are deferred into Appendix A.1 and Appendix A.2
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