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Integration questions related to fractional Brownian motion
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  • Published: October 2000

Integration questions related to fractional Brownian motion

  • Vladas Pipiras1 &
  • Murad S. Taqqu1 

Probability Theory and Related Fields volume 118, pages 251–291 (2000)Cite this article

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Abstract.

Let {B H (u)} u ∈ℝ be a fractional Brownian motion (fBm) with index H∈(0, 1) and (B H ) be the closure in L 2(Ω) of the span Sp(B H ) of the increments of fBm B H . It is well-known that, when B H = B 1/2 is the usual Brownian motion (Bm), an element X∈(B 1/2) can be characterized by a unique function f X ∈L 2(ℝ), in which case one writes X in an integral form as X = ∫ℝ f X (u)dB 1/2(u). From a different, though equivalent, perspective, the space L 2(ℝ) forms a class of integrands for the integral on the real line with respect to Bm B 1/2. In this work we explore whether a similar characterization of elements of (B H ) can be obtained when H∈ (0, 1/2) or H∈ (1/2, 1). Since it is natural to define the integral of an elementary function f = ∑ k =1 n f k 1 [uk,uk+1) by ∑ k =1 n f k (B H (u k +1) −B H (u k )), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces. If the space of integrands is not complete, then it characterizes only a strict subset of (B H ). When 0<H<1/2, by using the moving average representation of fBm B H , we construct a complete space of integrands. When 1/2<H<1, however, an analogous construction leads to a space of integrands which is not complete. When 0<H<1/2 or 1/2<H<1, we also consider a number of other spaces of integrands. While smaller and henceincomplete, they form a natural choice and are convenient to workwith. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm.

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Authors and Affiliations

  1. Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215-2411, USA. e-mail: pipiras@math.bu.edu; murad@math.bu.edu, , , , , , US

    Vladas Pipiras & Murad S. Taqqu

Authors
  1. Vladas Pipiras
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  2. Murad S. Taqqu
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Received: 9 August 1999 / Revised version: 10 January 2000 / Published online: 18 September 2000

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Pipiras, V., Taqqu, M. Integration questions related to fractional Brownian motion. Probab Theory Relat Fields 118, 251–291 (2000). https://doi.org/10.1007/s440-000-8016-7

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  • Issue Date: October 2000

  • DOI: https://doi.org/10.1007/s440-000-8016-7

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Keywords

  • Hilbert Space
  • Brownian Motion
  • Real Line
  • Unique Function
  • Elementary Function
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