Are Classes of Deterministic Integrands for Fractional Brownian Motion on an Interval Complete?

Abstract

Let BH be a fractional Brownian motion with self-similarity parameter H∈ (0,1) and a>0 be a fixed real number. Consider the integral ∈t0a f(u)\rm dBH(u), where f belongs to a class of non-random integrands ΛH,a. The integral will then be defined in the L2(Ω) sense. One would like ΛH,a to be a complete inner-product space. This corresponds to a desirable situation because then there is an isometry between ΛH,a and the closure of the span generated by BH(u), 0≤ u≤ a. We show in this work that, when H∈(½,1), the classes of integrands ΛH,a one usually considers are not complete inner-product spaces even though they are often assumed in the literature to be complete. Thus, they are isometric not to overline{\mbox{sp}}\{BH(u), 0≤ u≤ a\} but only to a proper subspace. Consequently, there are (random) elements in that closure which cannot be represented by functions f in ΛH,a. We also show, in contrast to the case H∈ (½,1), that there is a class of integrands for fractional Brownian motion BH with H∈ (0,½) on an interval [0,a] which is a complete inner-product space.