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DISTRIBUTIONS OF QUADRATIC FUNCTIONALS OF THE FRACTIONAL BROWNIAN MOTION BASED ON A MARTINGALE APPROXIMATION

Published online by Cambridge University Press:  02 April 2014

Katsuto Tanaka
Katsuto Tanaka*
Affiliation:
Gakushuin University, Tokyo, Japan
*
*Address correspondence to Katsuto Tanaka, Department of Economics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan; e-mail: katsuto.tanaka@gakushuin.ac.jp.
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Abstract

The present paper deals with the distributions related to the fractional Brownian motion (fBm). In particular, we try to compute the distributions of (ratios of) its quadratic functionals, not by simulations, but by numerically inverting the associated characteristic functions (c.f.s). Among them is the fractional unit root distribution. It turns out that the derivation of the c.f.s based on the standard approaches used for the ordinary Bm is inapplicable. Here the martingale approximation to the fBm suggested in the literature is used to compute an approximation to the distributions of such functionals. The associated c.f. is obtained via the Fredholm determinant. Comparison of the first two moments of the approximate with true distributions is made, and simulations are conducted to examine the performance of the approximation. We also find an interesting moment property of the approximate fractional unit root distribution, and a conjecture is given that the same property will hold for the true fractional unit root distribution.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 5 , October 2014 , pp. 1078 - 1109
Copyright
Copyright © Cambridge University Press 2014 

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