## Bernoulli

- Bernoulli
- Volume 7, Number 6 (2001), 873-897.

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

Vladas Pipiras and Murad S. Taqqu

#### 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 ∈t_{0}^{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 L^{2}(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.*

*Article information*

*Article information*

**Source**

Bernoulli, Volume 7, Number 6 (2001), 873-897.

**Dates**

First available in Project Euclid: 10 March 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1078951127

**Mathematical Reviews number (MathSciNet)**

MR1873833

**Zentralblatt MATH identifier**

1003.60055

**Keywords**

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

*Citation*

*Citation*

*Pipiras, Vladas; Taqqu, Murad S. Are classes of deterministic integrands for fractional Brownian motion on an interval complete?.
Bernoulli 7 (2001), no. 6, 873--897. https://projecteuclid.org/euclid.bj/1078951127*