Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion

By Ivan Nourdin. 2007

63 citations recorded by the source corpus.

This note is devoted to a fine study of the convergence of some weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H in (0,1/2). With the help of Malliavin calculus, we show that, correctly renormalized, the weighted quadratic variation of B that we consider converges in L^2 to an explicit limit when H 1/4. In the same spirit, we also show that, correctly renormalized, the weighted cubic variation of B converges in L^2 to an explicit limit when H<1/6.

Subfields: Economics and Econometrics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/07f2d3d4be25acf6bf606583ff534d32817cac7d

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Citations in the corpus

Central Limit Theorems for Multiple Skorokhod Integrals (2010, background): “In [ 13 , 19], the following convergences as n →∞ are shown: • If H< 1 2q , then”
Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion (2008, background): “t index H∈ (0,1). The asymptotic behavior of the q-Hermite power variations of Bwith respect to N, deﬁned as Vn = nX−1 k=0 Hq(Bk+1 −Bk), n≥ 1, (1.1) has recently received a lot of attention, see e.g. [9], [10] and references therein. Here, Hq stands for the Hermite polynomial with degree q, given by”
Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4 (2008, background): “Very recently, the asymptotic behavior of received a lot of attention, see [6, 12, 13, 14, 16] (see also the related works [17, 20, 21, 23]).”