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Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case \(H=1/4\). (English) Zbl 1200.60023

The authors complete the recent analysis of weighted quadratic variations of fractional Brownian motion by studying the missing case \(H=1/4\). More precisely, they analyse \[ V(f)= \sum_{k=0}^{n-1} f(B^{1/4}_{k/n}) \big [ \sqrt{n} (B^{1/4}_{(k+1)/n} - B^{1/4}_{k/n})^2 -1\big] \] where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a sufficiently smooth function and \(B^{1/4}\) is a fractional Brownian motion with Hurst parameter \(H=1/4\). The authors show that \[ \frac{1}{\sqrt{n}} V(f) \, \overset{Law}{\longrightarrow} \, C_{1/4} \int_0^1 f(B^{1/4}) d W_s + \frac{1}{4} \int_0^1 f''(B_s^{1/4}) \,d s \] for \(n \rightarrow \infty\), where \(W\) is a Brownian motion independent of \(B^{1/4}\) and \(C_{1/4}\) is an explicitly known constant.
Moreover, they show that the limit of the midpoint rule, i.e. \[ I(f):= \lim_{n \rightarrow \infty} \sum_{k=1}^{ \lfloor n/2 \rfloor} f(B_{(2k-1)/n}^{1/4}) (B_{2k/n}^{1/4}- B_{(2k-2)/n}^{1/4}) \] exists in probability, and one has the non-classical change of variable rule \[ I(f') \overset{Law}{=}f(B_1^{1/4})-f(0)- \frac{\kappa}{2} \int_0^1 f''(B^{1/4}) d W_s, \] where \(\kappa\) is a universal constant and \(W\) is again a Brownian motion, independent of \(B^{1/4}\).
The proofs of both results are based on Malliavin calculus tools.
Reviewer: Andreas Neuenkirch

MSC:

60F05 Central limit and other weak theorems
60H05 Stochastic integrals
60G22 Fractional processes, including fractional Brownian motion
60H07 Stochastic calculus of variations and the Malliavin calculus
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