Distributions of Quadratic Functionals of the Fractional Brownian Motion Based on a Martingale Approximation

Abstract

We discuss some computational problems associated with distributions of statistics arising from the fractional Brownian motion (fBm). In particular, we deal with (ratios of) its quadratic functionals. While it is easy in principle to deal with the standard Bm, the fBm is difficult to analyze because of its non-semimartingale nature. Here we suggest how to derive and compute the distributions of such functionals by using a martingale approximation. For this purpose we employ the Fredholm theory concerning the integral equations, illustrating how to compute the characteristic function via the Fredholm determinant. We also apply the present methodology to compute the fractional unit root distribution, and demonstrate some interesting moment properties.