Ornstein–Uhlenbeck process

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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).

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.^[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.

Definition[]

Simplified formula for the Ornstein–Uhlenbeck process from the mural shown below.

Dutch artist collective De Strakke Hand: Leonard Ornstein mural, showing Ornstein as a cofounder of the Dutch Physical Society (Netherlands Physical Society) at his desk in 1921, and illustrating twice the random walk of a drunkard with a simplified formula for the Ornstein–Uhlenbeck process. Oosterkade, Utrecht, The Netherlands, not far from Ornstein's laboratory. Translated text: Prof. Ornstein researches random motion 1930.

The Ornstein–Uhlenbeck process ${\displaystyle x_{t}}$ is defined by the following stochastic differential equation:

${\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dW_{t}}$

where ${\displaystyle \theta >0}$ and ${\displaystyle \sigma >0}$ are parameters and ${\displaystyle W_{t}}$ denotes the Wiener process.^[2][3][4]

An additional drift term is sometimes added:

${\displaystyle dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}}$

where ${\displaystyle \mu }$ 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

${\displaystyle {\frac {dx_{t}}{dt}}=-\theta \,x_{t}+\sigma \,\eta (t)}$

where ${\displaystyle \eta (t)}$, also known as white noise, stands in for the supposed derivative ${\displaystyle dW_{t}/dt}$ of the Wiener process.^[6] However, ${\displaystyle dW_{t}/dt}$ 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 representation[]

The Ornstein–Uhlenbeck process can also be described in terms of a probability density function, ${\displaystyle P(x,t)}$, which specifies the probability of finding the process in the state ${\displaystyle x}$ at time ${\displaystyle t}$.^[6] This function satisfies the Fokker–Planck equation

${\displaystyle {\frac {\partial P}{\partial t}}=\theta {\frac {\partial }{\partial x}}(xP)+D{\frac {\partial ^{2}P}{\partial x^{2}}}}$

where ${\displaystyle D=\sigma ^{2}/2}$. 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, ${\displaystyle P(x,t\mid x',t')}$ is a Gaussian with mean ${\displaystyle x'e^{-\theta (t-t')}}$ and variance ${\displaystyle {\frac {D}{\theta }}\left(1-e^{-2\theta (t-t')}\right)}$:

${\displaystyle P(x,t\mid x',t')={\sqrt {\frac {\theta }{2\pi D(1-e^{-2\theta (t-t')})}}}\exp \left[-{\frac {\theta }{2D}}{\frac {(x-x'e^{-\theta (t-t')})^{2}}{1-e^{-2\theta (t-t')}}}\right]}$

This gives the probability of the state ${\displaystyle x}$ occurring at time ${\displaystyle t}$ given initial state ${\displaystyle x'}$ at time ${\displaystyle t'<t}$. Equivalently, ${\displaystyle P(x,t\mid x',t')}$ is the solution of the Fokker-Planck equation with initial condition ${\displaystyle P(x,t')=\delta (x-x')}$.

Mathematical properties[]

Assuming ${\displaystyle x_{0}}$ is constant, the mean is

${\displaystyle \operatorname {\mathbb {E} } (x_{t})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})}$

and the covariance is

${\displaystyle \operatorname {cov} (x_{s},x_{t})={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right)}$

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:

${\displaystyle x_{t}={\frac {\sigma }{\sqrt {2\theta }}}e^{-\theta t}W_{e^{2\theta t}}}$

where ${\displaystyle W_{t}}$ is the standard Wiener process. This is roughly Theorem 1.2 in Doob 1942. Equivalently, with the change of variable ${\displaystyle s=e^{2\theta t}}$ this becomes

${\displaystyle W_{s}={\frac {\sqrt {2\theta }}{\sigma }}s^{1/2}x_{(\ln s)/(2\theta )},\qquad s>0}$

Using this mapping, one can translate known properties of ${\displaystyle W_{t}}$ into corresponding statements for ${\displaystyle x_{t}}$. For instance, the law of the iterated logarithm for ${\displaystyle W_{t}}$ becomes^[1]

${\displaystyle \limsup _{t\rightarrow \infty }{\frac {x_{t}}{\sqrt {(\sigma ^{2}/\theta )\ln t}}}=1,\quad {\text{with probability 1.}}}$

Formal solution[]

The stochastic differential equation for ${\displaystyle x_{t}}$ can be formally solved by variation of parameters.^[8] Writing

${\displaystyle f(x_{t},t)=x_{t}e^{\theta t}\,}$

we get

${\displaystyle {\begin{aligned}df(x_{t},t)&=\theta \,x_{t}\,e^{\theta t}\,dt+e^{\theta t}\,dx_{t}\\[6pt]&=e^{\theta t}\theta \,\mu \,dt+\sigma \,e^{\theta t}\,dW_{t}.\end{aligned}}}$

Integrating from ${\displaystyle 0}$ to ${\displaystyle t}$ we get

${\displaystyle x_{t}e^{\theta t}=x_{0}+\int _{0}^{t}e^{\theta s}\theta \,\mu \,ds+\int _{0}^{t}\sigma \,e^{\theta s}\,dW_{s}\,}$

whereupon we see

${\displaystyle x_{t}=x_{0}\,e^{-\theta t}+\mu \,(1-e^{-\theta t})+\sigma \int _{0}^{t}e^{-\theta (t-s)}\,dW_{s}.\,}$

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

${\displaystyle \operatorname {E} (x_{t})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})\!\ }$

assuming ${\displaystyle x_{0}}$ is constant. Moreover, the Itō isometry can be used to calculate the covariance function by

${\displaystyle {\begin{aligned}\operatorname {cov} (x_{s},x_{t})&=\operatorname {E} [(x_{s}-\operatorname {E} [x_{s}])(x_{t}-\operatorname {E} [x_{t}])]\\[5pt]&=\operatorname {E} \left[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}\right]\\[5pt]&=\sigma ^{2}e^{-\theta (s+t)}\operatorname {E} \left[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}\right]\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\,e^{-\theta (s+t)}(e^{2\theta \min(s,t)}-1)\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right).\end{aligned}}}$

Since the Itô integral of deterministic integrand is normally distributed, it follow that

${\textstyle x_{t}=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})+{\frac {\sigma }{\sqrt {2\theta }}}W_{1-e^{-2\theta t}}}$

Numerical sampling[]

By using discretely sampled data at time intervals of width ${\displaystyle t}$, the maximum likelihood estimators for the parameters of the Ornstein–Uhlenbeck process are asymptotically normal to their true values.^[9] More precisely,^{[failed verification]}

${\displaystyle {\sqrt {n}}\left({\begin{pmatrix}{\widehat {\theta }}_{n}\\{\widehat {\mu }}_{n}\\{\widehat {\sigma }}_{n}^{2}\end{pmatrix}}-{\begin{pmatrix}\theta \\\mu \\\sigma ^{2}\end{pmatrix}}\right)\xrightarrow {d} \ {\mathcal {N}}\left({\begin{pmatrix}0\\0\\0\end{pmatrix}},{\begin{pmatrix}{\frac {e^{2t\theta }-1}{t^{2}}}&0&{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}\\0&{\frac {\sigma ^{2}\left(e^{t\theta }+1\right)}{2\left(e^{t\theta }-1\right)\theta }}&0\\{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}&0&{\frac {\sigma ^{4}\left[\left(e^{2t\theta }-1\right)^{2}+2t^{2}\theta ^{2}\left(e^{2t\theta }+1\right)+4t\theta \left(e^{2t\theta }-1\right)\right]}{t^{2}\left(e^{2t\theta }-1\right)\theta ^{2}}}\end{pmatrix}}\right)}$

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 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 ${\displaystyle n}$ blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let ${\displaystyle X_{n}}$ be the number of blue balls in the urn after ${\displaystyle n}$ steps. Then ${\displaystyle {\frac {X_{[nt]}-n/2}{\sqrt {n}}}}$ converges in law to an Ornstein–Uhlenbeck process as ${\displaystyle n}$ tends to infinity.

Applications[]

In physical sciences[]

The Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. A canonical example is a Hookean spring with spring constant ${\displaystyle k}$ whose dynamics is highly overdamped with friction coefficient ${\displaystyle \gamma }$. In the presence of thermal fluctuations with temperature ${\displaystyle T}$, the length ${\displaystyle x(t)}$ of the spring will fluctuate stochastically around the spring rest length ${\displaystyle x_{0}}$; its stochastic dynamics is described by an Ornstein–Uhlenbeck process with

${\displaystyle {\begin{aligned}\theta &=k/\gamma ,\\\mu &=x_{0},\\\sigma &={\sqrt {2k_{B}T/\gamma }},\end{aligned}}}$

where ${\displaystyle \sigma }$ is derived from the Stokes–Einstein equation ${\displaystyle D=\sigma ^{2}/2=k_{B}T/\gamma }$ for the effective diffusion constant.

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

${\displaystyle {\dot {x}}(t)=-{\frac {k}{\gamma }}(x(t)-x_{0})+\xi (t)}$

where ${\displaystyle \xi (t)}$ is white Gaussian noise with ${\displaystyle \langle \xi (t_{1})\xi (t_{2})\rangle =2k_{B}T/\gamma \,\delta (t_{1}-t_{2}).}$ Fluctuations are correlated as

${\displaystyle \langle (x(t_{0})-x_{0})(x(t_{0}+t)-x_{0})\rangle ={\frac {k_{B}T}{k}}\exp(-|t|/\tau )}$

with correlation time ${\displaystyle \tau =\gamma /k}$.

At equilibrium, the spring stores an average energy ${\displaystyle \langle E\rangle =k\langle (x-x_{0})^{2}\rangle /2=k_{B}T/2}$ in accordance with the equipartition theorem.

In financial mathematics[]

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 ${\displaystyle \mu }$ represents the equilibrium or mean value supported by fundamentals; ${\displaystyle \sigma }$ the degree of volatility around it caused by shocks, and ${\displaystyle \theta }$ 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.^[10][11][12]

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.^[13] 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.^[14]

Generalizations[]

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 ${\displaystyle X}$. In particular, the CKLS (Chan–Karolyi–Longstaff–Sanders) process^[15] with the volatility term replaced by ${\displaystyle \sigma \,x^{\gamma }\,dW_{t}}$ can be solved in closed form for ${\displaystyle \gamma =1}$, as well as for ${\displaystyle \gamma =0}$, which corresponds to the conventional OU process. Another special case is ${\displaystyle \gamma =1/2}$, 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 ${\displaystyle \mathbf {x} _{t}}$, can be defined from

${\displaystyle d\mathbf {x} _{t}=-{\boldsymbol {\beta }}\,\mathbf {x} _{t}\,dt+{\boldsymbol {\sigma }}\,d\mathbf {W} _{t}.}$

where ${\displaystyle \mathbf {W} _{t}}$ is an N-dimensional Wiener process, and ${\displaystyle {\boldsymbol {\beta }}}$ and ${\displaystyle {\boldsymbol {\sigma }}}$ are constant N×N matrices.^[16] The solution is

${\displaystyle \mathbf {x} _{t}=e^{-{\boldsymbol {\beta }}t}\mathbf {x} _{0}+\int _{0}^{t}e^{-{\boldsymbol {\beta }}(t-t')}{\boldsymbol {\sigma }}\,d\mathbf {W} _{t'}}$

and the mean is

${\displaystyle \operatorname {E} (\mathbf {x} _{t})=e^{-{\boldsymbol {\beta }}t}\operatorname {E} (\mathbf {x} _{0}).}$

These expressions make use of the matrix exponential.

The process can also be described in terms of the probability density function ${\displaystyle P(\mathbf {x} ,t)}$, which satisfies the Fokker–Planck equation^[17]

${\displaystyle {\frac {\partial P}{\partial t}}=\sum _{i,j}\beta _{ij}{\frac {\partial }{\partial x_{i}}}(x_{j}P)+\sum _{i,j}D_{ij}{\frac {\partial ^{2}P}{\partial x_{i}\,\partial x_{j}}}.}$

where the matrix ${\displaystyle {\boldsymbol {D}}}$ with components ${\displaystyle D_{ij}}$ is defined by ${\displaystyle {\boldsymbol {D}}={\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{T}/2}$. 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 ${\displaystyle P(\mathbf {x} ,t\mid \mathbf {x} ',t')}$ is a Gaussian which can be written down explicitly. If the real parts of the eigenvalues of ${\displaystyle {\boldsymbol {\beta }}}$ are larger than zero, a stationary solution ${\displaystyle P_{\text{st}}(\mathbf {x} )}$ moreover exists, given by

${\displaystyle P_{\text{st}}(\mathbf {x} )=(2\pi )^{-N/2}(\det {\boldsymbol {\omega }})^{-1/2}\exp \left(-{\frac {1}{2}}\mathbf {x} ^{T}{\boldsymbol {\omega }}^{-1}\mathbf {x} \right)}$

where the matrix ${\displaystyle {\boldsymbol {\omega }}}$ is determined from the Lyapunov equation ${\displaystyle {\boldsymbol {\beta }}{\boldsymbol {\omega }}+{\boldsymbol {\omega }}{\boldsymbol {\beta }}^{T}=2{\boldsymbol {D}}}$.^[6]

See also[]

• Stochastic calculus
• Wiener process
• Gaussian process
• Mathematical finance
• The Vasicek model of interest rates
• Short-rate model
• Diffusion
• Fluctuation-dissipation theorem
• Klein–Kramers equation

Notes[]

References[]

External links[]

• A Stochastic Processes Toolkit for Risk Management, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki
• Simulating and Calibrating the Ornstein–Uhlenbeck process, M. A. van den Berg
• Maximum likelihood estimation of mean reverting processes, Jose Carlos Garcia Franco
• "Interactive Web Application: Stochastic Processes used in Quantitative Finance". Archived from the original on 2015-09-20. Retrieved 2015-07-03.