Ornstein–Uhlenbeck process explained
Ornstein–Uhlenbeck process should not be confused with Ornstein–Uhlenbeck operator.
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.
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. 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.
Definition
The Ornstein–Uhlenbeck process
is defined by the following
stochastic differential equation:
dxt=-\thetaxtdt+\sigmadWt
where
and
are parameters and
denotes the
Wiener process.
An additional drift term is sometimes added:
dxt=\theta(\mu-xt)dt+\sigmadWt
where
is a constant.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. However,
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
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
. 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, also known as the
Green's function,
is a Gaussian with mean
and variance
:
} \exp \left[-\frac{\theta}{2D} \frac{(x - x' e^{-\theta (t-t')})^2}{1 - e^{-2\theta (t-t')}}\right]
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 properties
Conditioned on a particular value of
, the mean is
\operatornameE(xt\midx0)=x0e-\theta+\mu(1-e-\theta)
and the
covariance is
\operatorname{cov}(xs,xt)=
\left(e-\theta|t-s|-e-\theta(t+s)\right).
For the stationary (unconditioned) process, the mean of
is
, and the covariance of
and
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
A temporally homogeneous Ornstein–Uhlenbeck process can be represented as a scaled, time-transformed Wiener process:
} e^ W_where
is the standard Wiener process. This is roughly Theorem 1.2 in . Equivalently, with the change of variable
this becomes
} s^ x_, \qquad s > 0
Using this mapping, one can translate known properties of
into corresponding statements for
. For instance, the law of the iterated logarithm for
becomes
\limsupt
| xt |
\sqrt{(\sigma2/\theta)lnt |
} = 1, \quad \text
Formal solution
The stochastic differential equation for
can be formally solved by
variation of parameters. Writing
we get
\begin{align}
df(xt,t)&=
dt+e\thetadxt\\[6pt]
&=e\theta\theta\mudt+\sigmae\thetadWt.
\end{align}
Integrating from
to
we get
xte\theta=x0+
e\theta\theta\muds+
\sigmae\thetadWs
whereupon we see
xt=
+\mu(1-e-\theta)+\sigma
e-\thetadWs.
From this representation, the first moment (i.e. the mean) is shown to be
\operatornameE(xt)=x0e-\theta+\mu(1-e-\theta)
assuming
is constant. Moreover, the
Itō isometry can be used to calculate the
covariance function by
\begin{align}
\operatorname{cov}(xs,xt)&=\operatornameE[(xs-\operatornameE[xs])(xt-\operatornameE[xt])]\\[5pt]
&=\operatornameE\left[
\sigmae\thetadWu
\sigmae\thetadWv\right]\\[5pt]
&=\sigma2e-\theta\operatornameE\left[
e\thetadWu
e\thetadWv\right]\\[5pt]
&=
e-\theta(e2\theta-1)\\[5pt]
&=
\left(e-\theta|t-s|-e-\theta(t+s)\right).
\end{align}
Since the Itô integral of deterministic integrand is normally distributed, it follows that
xt=x0e-\theta+\mu(1-e-\theta)+\tfrac{\sigma}{\sqrt{2\theta}}
Kolmogorov equations
The infinitesimal generator of the process is[1] If we let
}, then the eigenvalue equation simplifies to:
which is the defining equation for
Hermite polynomials. Its solutions are
, with
, which implies that the mean first passage time for a particle to hit a point on the boundary is on the order of
.
Numerical simulation
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. More precisely,
To simulate an OU process numerically with standard deviation
and correlation time
, one method is to apply the finite-difference formula
where
is a normally distributed random number with zero mean and unit variance, sampled independently at every time-step
.
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. 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. This was obtained by
Mark Kac.
Heuristically one may obtain this as follows.
Let
}, and we will obtain the stochastic differential equation at the
limit. First deduce
With this, we can calculate the mean and variance of
, which turns out to be
and
. Thus at the
limit, we have
, with solution (assuming
distribution is standard normal)
.
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) with spring constant
whose dynamics is
overdampedwith friction coefficient
. In the presence of thermal fluctuations with
temperature
, the length
of the spring fluctuates around the spring rest length
; its stochastic dynamics is described by an Ornstein–Uhlenbeck process with
\begin{align}
\theta&=k/\gamma,\\
\mu&=x0,\\
\sigma&=\sqrt{2kBT/\gamma},
\end{align}
where
is derived from the
Stokes–Einstein equation
for the effective diffusion constant. This model has been used to characterize the motion of a Brownian particle in an
optical trap.
At equilibrium, the spring stores an average energy
\langleE\rangle=k\langle
\rangle/2=kBT/2
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, 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.
[2] [3] 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.[4] [5]
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:
dxt=-\thetaxtdt+\sigmadLt
Here, the differential of the Wiener process
has been replaced with the differential of a Lévy process
.
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) 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 dimensions
A multi-dimensional version of the Ornstein–Uhlenbeck process, denoted by the N-dimensional vector
, can be defined from
dxt=-\boldsymbol{\beta}xtdt+\boldsymbol{\sigma}dWt.
where
is an
N-dimensional Wiener process, and
and
are constant
N×
N matrices. The solution is
xt=e-\boldsymbol{\betat}x0+
e-\boldsymbol{\beta(t-t')}\boldsymbol{\sigma}dWt'
and the mean is
\operatornameE(xt)=e-\boldsymbol{\betat}\operatornameE(x0).
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
=\sumi,j\betaij
(xjP)+\sumi,jDij
| \partial2P |
\partialxi\partialxj |
.
where the matrix
with components
is defined by
\boldsymbol{D}=\boldsymbol{\sigma}\boldsymbol{\sigma}T/2
. 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
Pst(x)=(2\pi)-N/2(\det\boldsymbol{\omega})-1/2\exp\left(-
xT\boldsymbol{\omega}-1x\right)
where the matrix
is determined from the
Lyapunov equation \boldsymbol{\beta}\boldsymbol{\omega}+\boldsymbol{\omega}\boldsymbol{\beta}T=2\boldsymbol{D}
.
See also
References
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- Hunt . G. . The relative importance of directional change, random walks, and stasis in the evolution of fossil lineages . Proceedings of the National Academy of Sciences . 104 . 47 . 2007-11-14 . 0027-8424 . 10.1073/pnas.0704088104 . 18404–18408. 18003931 . 2141789 . free .
- Iglehart . Donald L. . June 1968 . Limit Theorems for the Multi-urn Ehrenfest Model . The Annals of Mathematical Statistics . 39 . 3 . 864–876 . 10.1214/aoms/1177698318 . 0003-4851. free .
- Goerlich . Rémi . Li . Minghao . Albert . Samuel . Manfredi . Giovanni . Hervieux . Paul-Antoine . Genet . Cyriaque . Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process . Physical Review E . 103 . 3 . 2021-03-19 . 032132 . 2470-0045 . 10.1103/physreve.103.032132 . 33862817 . 2007.12246. 2021PhRvE.103c2132G . 220768666 .
- Jespersen . Sune . Metzler . Ralf . Fogedby . Hans C. . Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions . Physical Review E . 59 . 3 . 1999-03-01 . 1063-651X . 10.1103/physreve.59.2736 . 2736–2745 . cond-mat/9810176. 1999PhRvE..59.2736J . 51944991 .
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External links
Notes and References
- Web site: Holmes-Cerfon . Miranda . 2022 . Lecture 12: Detailed balance and Eigenfunction methods .
- http://www.cs.sunysb.edu/~skiena/691/lectures/lecture23.pdf Advantages of Pair Trading: Market Neutrality
- http://www.ms.unimelb.edu.au/publications/RampertshammerStefan.pdf An Ornstein–Uhlenbeck Framework for Pairs Trading
- Web site: Detecting Market Abuse . 2 November 2004 . Risk Magazine.
- Web site: The detection of Market Abuse on financial markets: a quantitative approach . Consob – The Italian Securities and Exchange Commission.