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Wikipedia / Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics, the physical sciences, and evolutionary biology. 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.

The Ornstein–Uhlenbeck process is a Gaussian, time-homogeneous Markov process, it admits the invariant Gaussian distribution

The process is stationary when its initial value is distributed according to this invariant distribution. If instead it begins from a fixed value or another nonstationary initial distribution, the process is generally not stationary; its distribution approaches the invariant Gaussian distribution as time increases. gives the process its characteristic mean-reverting behavior.

The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the center. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.

Definition

is defined by the following stochastic differential equation:

denotes the Wiener process.

An additional term is sometimes added:

is a constant called the (long-term) mean. The Ornstein–Uhlenbeck process is sometimes also written as a Langevin equation of the form

, also known as white noise of the Wiener process. does not exist because the Wiener process is nowhere differentiable, and so the Langevin equation only makes sense if interpreted in distributional sense. 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.

Fokker–Planck equation representation

. This function satisfies the Fokker–Planck equation

. This is a linear parabolic partial differential equation which can be solved by a variety of techniques. The transition probability, also known as the Green's function:

.

Mathematical properties

, the mean is

and the covariance is

.

The Ornstein–Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast to the Wiener process; the difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas for the Ornstein–Uhlenbeck process it is dependent on the current value of the process: if the current value of the process is less than the (long-term) mean, the drift will be positive; if the current value of the process is greater than the (long-term) mean, the drift will be negative. In other words, the mean acts as an equilibrium level for the process. This gives the process its informative name, "mean-reverting."

Properties of sample paths

can be represented as a scaled, time-transformed Wiener process:

is the standard Wiener process. This is roughly Theorem 1.2 in this becomes

. For instance, the law of the iterated logarithm becomes

Formal solution

can be formally solved by variation of parameters. Writing

we get

we get

whereupon we see

From this representation, the first moment (i.e. the mean) is shown to be

is constant. Moreover, the Itō isometry can be used to calculate the covariance function by

Kolmogorov equations

The infinitesimal generator of the process iswhich is the defining equation for Hermite polynomials.

Numerical simulation

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

.

Scaling limit interpretation

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 tends to infinity. This was obtained by Mark Kac.

Heuristically one may obtain this as follows.

.

Applications

In physics: noisy relaxation

The Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. A canonical example is a Hookean spring (harmonic oscillator whose dynamics is overdamped. In the presence of thermal fluctuations with temperature; its stochastic dynamics is described by an Ornstein–Uhlenbeck process with

is derived from the Stokes–Einstein equation for the effective diffusion constant. as expected from dimensional analysis.

This model has been used to characterize the motion of a Brownian particle in an optical trap. in accordance with the equipartition theorem.

In financial mathematics

The Ornstein–Uhlenbeck process is used in the Vasicek model of the interest rate. The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates represents the equilibrium or mean value supported by fundamentals the degree of volatility around it caused by shocks the rate by which these shocks dissipate and the variable reverts towards the mean. One application of the process is a trading strategy known as pairs trade.

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.

In evolutionary biology

The Ornstein–Uhlenbeck process has been proposed as an improvement over a Brownian motion model for modeling the change in organismal phenotypes over time. 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. 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.

Generalizations

It is possible to define a Lévy-driven Ornstein–Uhlenbeck process, in which the background driving process is a Lévy process instead of a Wiener process:

.

. In particular, the CKLS process (Chan–Karolyi–Longstaff–Sanders), which corresponds to the Cox–Ingersoll–Ross model (CIR-model).

Higher dimensions

A multi-dimensional version of the Ornstein–Uhlenbeck process, denoted by the N, can be defined from

is an N are constant N×N matrices. The solution is

and the mean is

These expressions make use of the matrix exponential.

, which satisfies the Fokker–Planck equation

moreover exists, given by

is determined from the Lyapunov equation.

See also

Notes

References