Fractional Brownian Motion

3 Approximation of fBm by Various Classes of Stochastic Processes

In Chapters 1 and 2 we studied the projections of fractional Brownian motion (fBm) on various subspaces, the distance from which to fBm was non-zero. So, in such subspaces, we cannot approximate fBm. However, a very interesting problem is still to approximate fBm using stochastic processes of comparatively simple structure or alternatively using the series of specially selected structure. The chapter is organized as follows. In section 3.1, fBm is represented as uniformly convergent series of Lebesgue integrals. Section 3.2 is devoted to the semimartingale approximation of fBm and to the approximation of the pathwise integral with respect to fBm by the integrals with respect to semimartingales. In section 3.3, we construct smooth processes that converge to fBm in the certain Besov-type space. This allows us to approximate the stochastic integral with respect to fBm by the integrals with respect to absolutely continuous processes. Section 3.4 contains a construction of absolute continuous approximations for the so-called multifractional Brownian motion, which is a generalization of fBm for the case of a time-dependent Hurst index.

3.1. Approximation of fBm by uniformly convergent series of Lebesgue integrals

In this section, we consider representation of fBm via the uniformly convergent series of special Lebesgue integrals. Let be an fBm with H ∈ (, 1). Consider the following supplementary Gaussian process: ...