Integration questions related to fractional Brownian motion

Abstract

Let {BH(u)}U∈ℝ be a fractional Brownian motion (fBm) with index H ∈ (0, 1) and Sp̄(BH) be the closure in L2(Ω) of the span Sp(BH) of the increments of fBm BH. It is well-known that, when BH = B1/2 is the usual Brownian motion (Bm), an element X ∈ Sp̄(B1/2) can be characterized by a unique function fx ∈ L2(ℝ), in which case one writes X in an integral form as X = ∫ℝ fx(u)d B1/2 (u). From a different, though equivalent, perspective, the space L2 (ℝ) forms a class of integrands for the integral on the real line with respect to Bm B1/2. In this work we explore whether a similar characterization of elements of Sp̄(BH) 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 = Σnk=1 fk 1 [uk, uk+1) by Σnk=1, fk (BH (uk+1) - BH (uk)), 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 Sp̄(BH). When 0 < H < 1/2, by using the moving average representation of fBm BH, 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.