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

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