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Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales

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1.	Title	Title of document	Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales
2.	Creator	Author's name, affiliation, country	Mihai Gradinaru; Institut de Math'ematiques 'Elie Cartan, Universit'e Henri Poincar'e
2.	Creator	Author's name, affiliation, country	Ivan Nourdin; Institut de Math'ematiques 'Elie Cartan, Universit'e Henri Poincar'e
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	Let $X$ be the fractional Brownian motion of any Hurst index $H\in (0,1)$ (resp. a semimartingale) and set $\alpha=H$ (resp. $\alpha=\frac{1}{2}$). If $Y$ is a continuous process and if $m$ is a positive integer, we study the existence of the limit, as $\varepsilon\rightarrow 0$, of the approximations $$ I_{\varepsilon}(Y,X) :=\left\{\int_{0}^{t}Y_{s}\left(\frac{X_{s+\varepsilon}-X_{s}}{\varepsilon^{\alpha}}\right)^{m}ds,\,t\geq 0\right\} $$ of $m$-order integral of $Y$ with respect to $X$. For these two choices of $X$, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the $m$-th moment of the Gaussian standard random variable. In particular, if $m$ is an odd integer, the limit equals to zero. In this case, the convergence in distribution, as $\varepsilon\rightarrow 0$, of $\varepsilon^{-\frac{1}{2}} I_{\varepsilon}(1,X)$ is studied. We prove that the limit is a Brownian motion when $X$ is the fractional Brownian motion of index $H\in (0,\frac{1}{2}]$, and it is in term of a two dimensional standard Brownian motion when $X$ is a semimartingale.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2003-10-30
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/166
10.	Identifier	Digital Object Identifier	10.1214/EJP.v8-166
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 8
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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