Asymptotic behavior of weighted quadratic variation of bi-fractional Brownian motion

Rachid Belfadli

doi:10.5802/ambp.281

Abstract

We prove, by means of Malliavin calculus, the convergence in L 2 of some properly renormalized weighted quadratic variations of bi-fractional Brownian motion (biFBM) with parameters H and K, when H<1/4 and K∈(0,1].

Highlights

There has been recently a lot of interests in the literature to the study of weighted power variations
For a given integer p > 1, a smooth enough function h : R → R and a process X, the analysis of the asymptotic behavior, as n tends to infinity, of quantities such as n−1
We recall the definition of the bi-fractional Brownian motion and present the elements of Malliavin calculus that will be needed in the sequel

Summary

Introduction

There has been recently a lot of interests in the literature to the study of weighted power variations. This study origins in the work [5] by Nourdin, in the case where X is a fractional Brownian motion (f.B.m, in short). (1.1), in the particular case p = 2 and X the fractional Brownian motion of Hurst parameter 1/4, allowed the authors of [7] to derive a new type of change of variable formula for X, with a correction term that is an ordinary Itô integral with respect to a Wiener process that is independent of X. A complete description of the nature of the convergence of weighted p-power variation of the form (1.1) in the case where X is the fractional Brownian motion with Hurst parameter H ∈ (0, 1) has been given in [6, 5, 7].

Results

Conclusion