Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales
Mihai Gradinaru, Institut de Math'ematiques 'Elie Cartan, Universit'e Henri Poincar'e
Ivan Nourdin, Institut de Math'ematiques 'Elie Cartan, Universit'e Henri Poincar'e
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=1/2).
If Y is a continuous process and if m is a positive integer, we study the
existence of the limit, as epsilon tends to 0, of the approximations
Iepsilon(Y,X)
:={int_0^t Ys
((Xs+epsilon-Xs)/(epsilon)alpha)mds, t>=0}
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 epsilon tends to 0, of
(epsilon)-½Iepsilon(1,X)
is studied. We prove that the limit is a Brownian motion when
X is the fractional Brownian motion of index H in (0,1/2], and it is in
term of a two dimensional standard Brownian motion when X is a semimartingale.
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