Are classes of deterministic integrands for fractional Brownian motion on an interval complete?

Abstract

Let B_H be a fractional Brownian motion with self-similarity parameter H∈ (0,1) and a>0 be a fixed real number. Consider the integral ∈t₀^a f(u)\rm dB_H(u), where f belongs to a class of non-random integrands Λ_H,a. The integral will then be defined in the L²(Ω) sense. One would like Λ_H,a to be a complete inner-product space. This corresponds to a desirable situation because then there is an isometry between Λ_H,a and the closure of the span generated by B_H(u), 0≤ u≤ a. We show in this work that, when H∈(½,1), the classes of integrands Λ_H,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 øverline{\mbox{sp}}\{B_H(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 Λ_H,a. We also show, in contrast to the case H∈ (½,1), that there is a class of integrands for fractional Brownian motion B_H with H∈ (0,½) on an interval [0,a] which is a complete inner-product space.