Ornstein–Uhlenbeck process

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

A simulation with θ = 1.0, σ = 3 and μ = (0, 0). Initially at the position (10, 10), the particle tends to move to the central point μ.
A 3D simulation with θ = 1.0, σ = 3, μ = (0, 0, 0) and the initial position (10, 10, 10).

The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.[1] Over time, the process tends to drift towards its mean function: such a process is called mean-reverting.

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.

DefinitionEdit

The Ornstein–Uhlenbeck process   is defined by the following stochastic differential equation:

 

where   and   are parameters and   denotes the Wiener process.[2][3][4]

An additional drift term is sometimes added:

 

where   is a constant. In financial mathematics, this is also known as the Vasicek model.[5]

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   of the Wiener process.[6] However,   does not exist because the Wiener process is nowhere differentiable, and so the Langevin equation is, strictly speaking, only heuristic.[7] 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 representationEdit

The Ornstein–Uhlenbeck process can also be described in terms of a probability density function,  , which specifies the probability of finding the process in the state   at time  .[8] This function satisfies the Fokker–Planck equation

 

where  . This is a linear parabolic partial differential equation which can be solved by a variety of techniques. The transition probability   is a Gaussian with mean   and variance  :

 

This gives the probability of the state   occurring at time   given initial state   at time  . Equivalently,   is the solution of the Fokker-Planck equation with initial condition  .

Mathematical propertiesEdit

Assuming   is constant, 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 pathsEdit

A temporally homogeneous Ornstein–Uhlenbeck process can be represented as a scaled, time-transformed Wiener process:

 

where   is the standard Wiener process.[1] Equivalently, with the change of variable   this becomes

 

Using this mapping, one can translate known properties of   into corresponding statements for  . For instance, the law of the iterated logarithm for   becomes[1]

 

Formal solutionEdit

The stochastic differential equation for   can be formally solved by variation of parameters.[9] Writing

 

we get

 

Integrating from   to   we get

 

whereupon we see

 

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

 

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

 

Numerical samplingEdit

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.[10] More precisely,[failed verification]

 

 
three sample paths of different OU-processes with θ = 1, μ = 1.2, σ = 0.3:
blue: initial value a = 0 (a.s.)
green: initial value a = 2 (a.s.)
red: initial value normally distributed so that the process has invariant measure

Scaling limit interpretationEdit

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. Consider an urn containing   blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let   be the number of blue balls in the urn after   steps. Then   converges in law to an Ornstein–Uhlenbeck process as   tends to infinity.

ApplicationsEdit

In physical sciencesEdit

The Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. Consider for example a Hookean spring with spring constant   whose dynamics is highly overdamped with friction coefficient  . In the presence of thermal fluctuations with temperature  , the length   of the spring will fluctuate stochastically around the spring rest length  ; its stochastic dynamic is described by an Ornstein–Uhlenbeck process with:

 

where   is derived from the Stokes–Einstein equation   for the effective diffusion constant.

In physical sciences, the stochastic differential equation of an Ornstein–Uhlenbeck process is rewritten as a Langevin equation

 

where   is white Gaussian noise with   Fluctuations are correlated as

 

with correlation time  .

At equilibrium, the spring stores an average energy   in accordance with the equipartition theorem.

In financial mathematicsEdit

The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter   represents the equilibrium or mean value supported by fundamentals;   the degree of volatility around it caused by shocks, and   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.[11][12][13]

In evolutionary biologyEdit

The Ornstein–Uhlenbeck process has been proposed as an improvement over a Brownian motion model for modeling the change in organismal phenotypes over time.[14] 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.

GeneralizationsEdit

It is possible to extend Ornstein–Uhlenbeck processes to processes where the background driving process is a Lévy process (instead of a simple Brownian motion).[clarification needed]

In addition, in finance, stochastic processes are used where the volatility increases for larger values of  . In particular, the CKLS (Chan–Karolyi–Longstaff–Sanders) process[15] with the volatility term replaced by   can be solved in closed form for  , as well as for  , which corresponds to the conventional OU process. Another special case is  , which corresponds to the Cox–Ingersoll–Ross model (CIR-model).

Higher dimensionsEdit

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

 

where   is an N-dimensional Wiener process, and   and   are constant N×N matrices.[16] The solution is

 

and the mean is

 

Note that these expressions make use of the matrix exponential.

The process can also be described in terms of the probability density function  , which satisfies the Fokker–Planck equation[17]

 

where the matrix   with components   is defined by  . As for the 1d case, the process is a linear transformation of Gaussian random variables, and therefore itself must be Gaussian. Because of this, the transition probability   is a Gaussian which can be written down explicitly. If the real parts of the eigenvalues of   are larger than zero, a stationary solution   moreover exists, given by

 

where the matrix   is determined from the Lyapunov equation  .[18]

See alsoEdit

NotesEdit

  1. ^ a b c Doob, J.L. (April 1942). "The Brownian Movement and Stochastic Equations". Annals of Mathematics. 43 (2): 351–369. doi:10.2307/1968873. JSTOR 1968873.
  2. ^ Karatzas, Ioannis; Shreve, Steven E. (1991), Brownian Motion and Stochastic Calculus (2nd ed.), Springer-Verlag, p. 358, ISBN 978-0-387-97655-6
  3. ^ Gard, Thomas C. (1988), Introduction to Stochastic Differential Equations, Marcel Dekker, p. 115, ISBN 978-0-8247-7776-0
  4. ^ Gardiner, C.W. (1985), Handbook of Stochastic Methods (2nd ed.), Springer-Verlag, p. 106, ISBN 978-0-387-15607-1
  5. ^ Björk, Tomas (2009). Arbitrage Theory in Continuous Time (3rd ed.). Oxford University Press. pp. 375, 381. ISBN 978-0-19-957474-2.
  6. ^ Risken (1984)
  7. ^ Lawler, Gregory F. (2006). Introduction to Stochastic Processes (2nd ed.). Chapman & Hall/CRC. ISBN 978-1584886518.
  8. ^ Risken, H. (1984), The Fokker-Planck Equation: Methods of Solution and Application, Springer-Verlag, pp. 99–100, ISBN 978-0-387-13098-9
  9. ^ Gardiner (1985) p. 106
  10. ^ Aït-Sahalia, Y. (April 2002). "Maximum Likelihood Estimation of Discretely Sampled Diffusion: A Closed-Form Approximation Approach". Econometrica. 70 (1): 223–262. doi:10.1111/1468-0262.00274.
  11. ^ Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications. World Scientific Publishing Co. 2016. ISBN 978-9814725910.
  12. ^ Advantages of Pair Trading: Market Neutrality
  13. ^ An Ornstein-Uhlenbeck Framework for Pairs Trading
  14. ^ Martins, E.P. (1994). "Estimating the Rate of Phenotypic Evolution from Comparative Data". Amer. Nat. 144 (2): 193–209. doi:10.1086/285670.
  15. ^ Chan et al. (1992)
  16. ^ Gardiner (1985), p. 109
  17. ^ Gardiner (1985), p. 97
  18. ^ Risken (1984), p. 156

ReferencesEdit

External linksEdit