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December 2001 Are classes of deterministic integrands for fractional Brownian motion on an interval complete?
Vladas Pipiras, Murad S. Taqqu
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Bernoulli 7(6): 873-897 (December 2001).

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 øverline{\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.

Citation

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Vladas Pipiras. Murad S. Taqqu. "Are classes of deterministic integrands for fractional Brownian motion on an interval complete?." Bernoulli 7 (6) 873 - 897, December 2001.

Information

Published: December 2001
First available in Project Euclid: 10 March 2004

zbMATH: 1003.60055
MathSciNet: MR1873833

Keywords: completeness , fractional Brownian motion , fractional integrals and derivatives , inner-product spaces , integration in the L ²sense

Rights: Copyright © 2001 Bernoulli Society for Mathematical Statistics and Probability

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Vol.7 • No. 6 • December 2001
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