Integration Questions Related to Fractional Brownian Motion Yz

Let fB H (u)g u2R be a fractional Brownian motion (fBm) with index H 2 (0; 1) and sp(B H) be the closure in L 2 (() 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 2 sp(B 1=2) can be characterized by a unique function f X 2 L 2 (R), in which case one writes X in an integral form as X = R R f X (u)dB 1=2 (u). From a diierent, 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 sp(B H) can be obtained when H 2 (0; 1=2) or H 2 (1=2; 1). Since it is natural to deene the integral of an elementary function f = P n k=1 f k 1 u k ;u k+1) by P n k=1 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 sp(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 hence incomplete, they form a natural choice and are convenient to work with. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm.