# Ornstein-Uhlenbeck process

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## Ornstein-Uhlenbeck process

[¦ȯrn‚stīn ′ü·lən‚bek ‚prä‚ses]
(statistics)
A stochastic process used as a theoretical model for Brownian motion.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
TABLE 3 The parameter setting of Ornstein-Uhlenbeck process in case 1.
Wylomanska, "Time-changed Ornstein-Uhlenbeck process," Journal of Physics A: Mathematical and Theoretical, vol.
The mean-reverting process [Y.sub.t] evolves as an Ornstein-Uhlenbeck process with a positive mean-reverting rate [alpha], an equilibrium level m, and the volatility of the volatility [beta].
For the Ornstein-Uhlenbeck process, the infinitesimal generator is G = -[mu]x(d/dx) + ([[sigma].sup.2]/2)([d.sup.2]/d[x.sup.2]).
As we have been working with continuous processes throughout this work, we choose an Ornstein-Uhlenbeck process to model the daily maximum.
We assume that the asset price process {[X.sub.t]; t [greater than or equal to] 0} is conditionally lognormal and the volatility process {[Y.sub.t]; t [greater than or equal to] 0} is a fractional Ornstein-Uhlenbeck process. {[X.sub.t]; t [greater than or equal to] 0} and {[Y.sub.t]; t [greater than or equal to] 0} satisfy the following equations:
The Ornstein-Uhlenbeck process satisfies the part of Condition I2 on the regressor's density in [21, page 136].
(a) The Case of an Ornstein-Uhlenbeck Process (OU) Introduce the function
yielding the Ornstein-Uhlenbeck process, where m, q, and v are non-negative constants interpretable as the long-range mean to which [r.sub.t] tends to revert, the speed of adjustment, and the volatility, respectively.
The Hull-White  model uses a lognormal process for the volatility, the Scott  and Stein-Stein  models use a mean-reverting Ornstein-Uhlenbeck process, and the Heston model  uses the Feller (Cox-Ingersoll-Ross) process.
A large volume of empirical researches in financial market which indicates the assumption that these variables are stochastic volatile and follow a certain stochastic process (e.g., Ornstein-Uhlenbeck process) is more realistic [3, 4].
The solution is often called an Ornstein-Uhlenbeck process. In fact,
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