Approximation at First and Second Order of m-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales
نویسندگان
چکیده
Let X be the fractional Brownian motion of any Hurst index H ∈ (0, 1) (resp. a semimartingale) and set α = H (resp. α = 12). If Y is a continuous process and if m is a positive integer, we study the existence of the limit, as ε→ 0, of the approximations Iε(Y,X) := {∫ t 0 Ys ( Xs+ε −Xs εα )m ds, 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 ε → 0, of ε− 1 2 Iε(1, X) is studied. We prove that the limit is a Brownian motion when X is the fractional Brownian motion of index H ∈ (0, 1 2 ], and it is in term of a two dimensional standard Brownian motion when X is a semimartingale.
منابع مشابه
Integration in a Normal World: Fractional Brownian Motion and Beyond
Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Lauri Viitasaari Name of the doctoral dissertation Integration in a Normal World: Fractional Brownian Motion and Beyond Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 14/2014 Field of research Mathematics Manuscript submitted 12 ...
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