Fractional Ornstein-Uhlenbeck processes

By Patrick Cheridito, H. Kawaguchi, M. Maejima. 2003

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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.

Subfields: Economics and Econometrics, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4ad5c32283e7f9ae90ac48cc2325335645f48435

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Citations in the corpus

ASYMPTOTIC THEORY FOR ESTIMATING DRIFT PARAMETERS IN THE FRACTIONAL VASICEK MODEL (2018, background): “(A.18) Substituting (A.13)-(A.18) into (A.10), we obtain (2).”
Estimation and pricing under long-memory stochastic volatility (2012, background): “See Cheridito et al. [6] for more details.”
Parameter estimation for fractional Ornstein–Uhlenbeck processes (2009, background): “For H = 12 this is well-known and for H > 1 2 this is proved in [2].”
Analysis of the Rosenblatt process (2006, methodology): “This has been done in [25] for general Hermite processes of order k following the argument in [8] for the fBm case.”
Statistical aspects of the fractional stochastic calculus (2006, background): “In [9] it is shown that, for any H ∈ (0,1), there is a unique almost surely continuous process X satisfying (7), and it can be represented as Xt = ∫ t”
Representation formulae for the fractional Brownian motion (2009, background): “ if TH is the time inversion operator deﬁned in (39), then one has the equality in law THΞ≃ Ξ. Notice that this holds even when H is not positive. Remark 3.1. The Lamperti transform (see for instance [5]) Ξ(t); t >0 → e−HtΞ(et); t ∈R (46) maps H-self-similar processesΞt into stationary processes ”
Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem (2007, methodology): “As an application, we discuss the existence and the properties of the Hermite Ornstein-Uhlenbeck process (the correspondent of the fractional Ornstein-Uhlenbeck process, see [6]) that appears as the unique solution of a Langevin type equation.”
Pricing under rough volatility (2015, unknown): “…μt is a suitable drift term, Zt is a standard Brownian motion, Xt is a fractional Ornstein–Uhlenbeck process (fOU process for short) satisfying dXt = ν dW Ht − α (Xt − m)dt, where m ∈ R, and ν and α are positive parameters, see Cheridito et al. (2003), with Z and W H correlated in general.”
Out of sample forecasts of quadratic variation (2008, background): “Let us start with a brief description of fractional OU processes; for details, we refer the reader to Cheridito et al. (2003).”
Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter (2017, methodology): “3 is to prove the asymptotic consistency of θ̂T by using a new method, different from that in Hu and Nualart (2010), which is valid for all H ∈ (0, 1).”