Integration questions related to fractional Brownian motion

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. Received: 9 August 1999 / Revised version: 10 January 2000 / Published online: 18 September 2000