COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1391

Fractional Brownian Motion as a Differentiable Generalized Gaussian Process

Victoria Zinde-Walsh
Department of Economics, McGill University & CIREQ

Peter C.B.Phillips
Cowles Foundation, Yale University, University of Auckland & University of York

October 15, 2002

Brownian motion can be characterized as a generalized random process and, as such, has a generalized derivative whose covariance functional is the delta function. In a similar fashion, fractional Brownian motion can be interpreted as a generalized random process and shown to possess a generalized derivative. The resulting process is a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the delta-function.

Keywords: Brownian motion, fractional Brownian motion, fractional derivative, covariance functional, delta function, generalized derivative, generalized Gaussian process

JEL Classification: C32 Time Series Models