Ornstein–Uhlenbeck process

Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction.

It is named after Leonard Ornstein and George Eugene Uhlenbeck.

In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.

The Ornstein–Uhlenbeck process is sometimes also written as a Langevin equation of the form where

, also known as white noise, stands in for the supposed derivative

In physics and engineering disciplines, it is a common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations by tacitly assuming that the noise term is a derivative of a differentiable (e.g. Fourier) interpolation of the Wiener process.

This is a linear parabolic partial differential equation which can be solved by a variety of techniques.

In other words, the mean acts as an equilibrium level for the process.

we get whereupon we see From this representation, the first moment (i.e. the mean) is shown to be assuming

Moreover, the Itō isometry can be used to calculate the covariance function by Since the Itô integral of deterministic integrand is normally distributed, it follows that[citation needed] The infinitesimal generator of the process is[8]

, which implies that the mean first passage time for a particle to hit a point on the boundary is on the order of

By using discretely sampled data at time intervals of width

, the maximum likelihood estimators for the parameters of the Ornstein–Uhlenbeck process are asymptotically normal to their true values.

To simulate an OU process numerically with standard deviation

is a normally distributed random number with zero mean and unit variance, sampled independently at every time-step

[10] The Ornstein–Uhlenbeck process can be interpreted as a scaling limit of a discrete process, in the same way that Brownian motion is a scaling limit of random walks.

[12][13] This model has been used to characterize the motion of a Brownian particle in an optical trap.

[13][14] At equilibrium, the spring stores an average energy

[15] The Ornstein–Uhlenbeck process is used in the Vasicek model of the interest rate.

represents the equilibrium or mean value supported by fundamentals;

the rate by which these shocks dissipate and the variable reverts towards the mean.

[17][18][19] A further implementation of the Ornstein–Uhlenbeck process is derived by Marcello Minenna in order to model the stock return under a lognormal distribution dynamics.

This modeling aims at the determination of confidence interval to predict market abuse phenomena.

[22] A Brownian motion model implies that the phenotype can move without limit, whereas for most phenotypes natural selection imposes a cost for moving too far in either direction.

A meta-analysis of 250 fossil phenotype time-series showed that an Ornstein–Uhlenbeck model was the best fit for 115 (46%) of the examined time series, supporting stasis as a common evolutionary pattern.

[23] This said, there are certain challenges to its use: model selection mechanisms are often biased towards preferring an OU process without sufficient support, and misinterpretation is easy to the unsuspecting data scientist.

In addition, in finance, stochastic processes are used where the volatility increases for larger values of

In particular, the CKLS process (Chan–Karolyi–Longstaff–Sanders)[27] with the volatility term replaced by

A multi-dimensional version of the Ornstein–Uhlenbeck process, denoted by the N-dimensional vector

[28] The solution is and the mean is These expressions make use of the matrix exponential.