On fractional Ornstein-Uhlenbeck processes

Abstract

In this paper we study Doob's transform of fractional Brownian motion (FBM).It is well known that Doob's transform of standard Brownian motion is identical in law with the Ornstein-Uhlenbeck diffusion defined as the stationary solution of the (stochastic) Langevin equation where the driving process is a Brownian motion.It is also known that Doob's transform of FBM and the process obtained from the Langevin equation with FBM as the driving process are different.However, also the first one of these can be described as a solution of a Langevin equation but now with some other driving process than FBM.We are mainly interested in the properties of this new driving process denoted Y (1) .We also study the solution of the Langevin equation with Y (1) as the driving process.Moreover, we show that the covariance of Y (1) grows linearly; hence, in this respect Y (1) is more like a standard Brownian motion than a FBM.In fact, it is proved that a properly scaled version of Y (1) converges weakly to Brownian motion.B t := B t , t ≥ 0, B (-) -t , t ≤ 0. It is easily seen that