Abstract
The classical stationary Ornstein-Uhlenbeck process can be obtained in two different ways. On the one hand, it is a stationary solution of the Langevin equation with Brownian motion noise. On the other hand, it can be obtained from Brownian motion by the so called Lamperti transformation. We show that the Langevin equation with fractional Brownian motion noise also has a stationary solution and that the decay of its auto-covariance function is like that of a power function. Contrary to that, the stationary process obtained from fractional Brownian motion by the Lamperti transformation has an auto-covariance function that decays exponentially.
Highlights
IntroductionDefinition 1.1 A fractional Brownian motion with Hurst parameter H ∈ (0, 1], is an almost surely continuous, centered Gaussian process (BtH )t∈R with
Let (Ω, F, P ) be a probability space.Definition 1.1 A fractional Brownian motion with Hurst parameter H ∈ (0, 1], is an almost surely continuous, centered Gaussian process (BtH )t∈R with Cov BtH, BsH =|t|2H + |s|2H − |t − s|2H, t, s ∈ R . (1.1)For an in-depth introduction to fractional Brownian motions we refer the reader to Section 7.2 of Samorodnitsky and Taqqu (1994) or Chapter 4 of Embrechts and Maejima (2002)
Zt1 := e−λtBα1 exp(λt) = αη, t ∈ R, if α. This leads us to the question whether for Langevin equation with noise processt≥0 has a stationary solution, if its distribution is unique and if it is equal in some sense to the Lamperti transform
Summary
Definition 1.1 A fractional Brownian motion with Hurst parameter H ∈ (0, 1], is an almost surely continuous, centered Gaussian process (BtH )t∈R with. 1), the Langevin equation with noise process (σBtH )t≥0 has a stationary solution, if its distribution is unique and if it is equal in some sense to the Lamperti transform. There exists a stationary, almost surely continuous, centered Gaussian process (YtH )t∈R such that (YtH )t≥0 solves the Langevin equation with fractional Brownian motion noise, and every other stationary solution is equal to (YtH )t≥0 in distribution.
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