Ornstein–Uhlenbeck process

File:Leonard Ornstein mural, Oosterkade, Utrecht, 2021 - 1 (cropped) - Ornstein's 1930 random walk formula.jpg

File:Leonard Ornstein mural, Oosterkade, Utrecht, 2021 - 1.jpg) 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 $x\_t$ is defined by the following stochastic differential equation:

:$dx\_t\; =\; -\backslash theta\; \backslash ,\; x\_t\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash ,\; dW\_t$

where $\backslash theta\; >\; 0$ and $\backslash sigma\; >\; 0$ are parameters and $W\_t$ denotes the Wiener process.{{sfn | Karatzas | Shreve | 1991 | p = 358}}{{sfn | Gard | 1988 | p = 115}}{{sfn | Gardiner | 1985 | p=}}

An additional term is sometimes added:

:$dx\_t\; =\; \backslash theta\; (\backslash mu\; -\; x\_t)\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash ,\; dW\_t$

where $\backslash mu$ is a constant called the (long-term) mean.

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

:$$

\frac{dx_t}{dt} = -\theta \, x_t + \sigma \, \eta(t)

where $\backslash eta(t)$, also known as white noise, stands in for the supposed derivative $d\; W\_t\; /\; dt$ of the Wiener process.{{sfn | Risken | 1989 | p=}} However, $d\; W\_t\; /\; dt$ does not exist because the Wiener process is nowhere differentiable,{{sfn | Lawler | 2006 | p=}} 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.

The Ornstein–Uhlenbeck process can also be described in terms of a probability density function, $P(x,t)$, which specifies the probability of finding the process in the state $x$ at time $t$.{{sfn | Risken | 1989 | p=}} This function satisfies the Fokker–Planck equation

: $\backslash frac\{\backslash partial\; P\}\{\backslash partial\; t\}\; =\; \backslash theta\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; (x\; P)\; +\; D\; \backslash frac\{\backslash partial^2\; P\}\{\backslash partial\; x^2\}$

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

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

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

Conditioned on a particular value of $x\_0$, the mean is

: $\backslash operatorname\; \backslash mathbb\{E\}(x\_t\; \backslash mid\; x\_0)=x\_0\; e^\{-\backslash theta\; t\}+\backslash mu(1-e^\{-\backslash theta\; t\})$

and the covariance is

:$$

\operatorname{cov}(x_s,x_t) = \frac{\sigma^2}{2\theta} \left( e^{-\theta|t-s|} - e^{-\theta(t+s)} \right).

For the stationary (unconditioned) process, the mean of $x\_t$ is $\backslash mu$, and the covariance of $x\_s$ and $x\_t$ is $\backslash frac\{\backslash sigma^2\}\{2\backslash theta\}\; e^\{-\backslash theta|t-s|\}$.

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."

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

:$$

x_t = \frac{\sigma}{\sqrt{2\theta}} e^{-\theta t} W_{e^{2 \theta t}}

where $W\_t$ is the standard Wiener process. This is roughly Theorem 1.2 in {{harvnb | Doob | 1942 | p=}}. Equivalently, with the change of variable $s\; =\; e^\{2\; \backslash theta\; t\}$ this becomes

:$$

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 $W\_t$ into corresponding statements for $x\_t$. For instance, the law of the iterated logarithm for $W\_t$ becomes{{sfn | Doob | 1942 | p=}}

:$$

\limsup_{t \to \infty} \frac{x_t}{\sqrt{(\sigma^2 / \theta) \ln t}} = 1, \quad \text{with probability 1.}

The stochastic differential equation for $x\_t$ can be formally solved by variation of parameters.{{sfn | Gardiner | 1985 | p = 106}} Writing

: $f(x\_t,\; t)\; =\; x\_t\; e^\{\backslash theta\; t\}\; \backslash ,$

we get

: $$

\begin{align}

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{align}

Integrating from $0$ to $t$ we get

: $x\_t\; e^\{\backslash theta\; t\}\; =\; x\_0\; +\; \backslash int\_0^t\; e^\{\backslash theta\; s\}\backslash theta\backslash ,\backslash mu\; \backslash ,\; ds\; +\; \backslash int\_0^t\; \backslash sigma\backslash ,e^\{\backslash theta\; s\}\backslash ,\; dW\_s\; \backslash ,$

whereupon we see

: $x\_t\; =\; x\_0\backslash ,e^\{-\backslash theta\; t\}\; +\; \backslash mu\backslash ,(1-e^\{-\backslash theta\; t\})\; +\; \backslash sigma\; \backslash int\_0^t\; e^\{-\backslash theta\; (t-s)\}\backslash ,\; dW\_s.\; \backslash ,$

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

: $\backslash operatorname\; E(x\_t)=x\_0\; e^\{-\backslash theta\; t\}+\backslash mu(1-e^\{-\backslash theta\; t\})\; \backslash !\backslash$

assuming $x\_0$ is constant. Moreover, the Itō isometry can be used to calculate the covariance function by

: $$

\begin{align}

\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{align}

Since the Itô integral of deterministic integrand is normally distributed, it follows that{{citation needed | date = February 2023}}

:$$

x_t = x_0 e^{-\theta t}+\mu(1-e^{-\theta t}) + \tfrac{\sigma}{\sqrt{2\theta}} W_{1-e^{-2 \theta t}}

The infinitesimal generator of the process is{{Cite web |last=Holmes-Cerfon |first=Miranda |date=2022 |title=Lecture 12: Detailed balance and Eigenfunction methods |url=https://cims.nyu.edu/~holmes/teaching/asa22/handout-Lecture12_2022.pdf}}$$Lf\; =\; -\backslash theta\; (x-\backslash mu)\; f\text{'}\; +\; \backslash frac\; 12\; \backslash sigma^2\; f\text{'}\text{'}$$If we let $y\; =\; (x-\; \backslash mu)\backslash sqrt\{\backslash frac\{2\backslash theta\}\{\backslash sigma^2\}\}$, then the eigenvalue equation simplifies to:

$$\backslash frac\{d^2\}\{dy^2\}\backslash phi\; -\; y\backslash frac\{d\}\{dy\}\backslash phi\; -\; \backslash frac\{\backslash lambda\}\{\backslash theta\}\; \backslash phi\; =\; 0$$which is the defining equation for Hermite polynomials. Its solutions are $\backslash phi(y)\; =\; He\_n(y)$, with $\backslash lambda\; =\; -n\backslash theta$, which implies that the mean first passage time for a particle to hit a point on the boundary is on the order of $\backslash theta^\{-1\}$.

By using discretely sampled data at time intervals of width $t$, the maximum likelihood estimators for the parameters of the Ornstein–Uhlenbeck process are asymptotically normal to their true values.{{sfn | Aït-Sahalia | 2002 | pp=223–262}} More precisely,{{Failed verification|date=March 2019}}$$\backslash sqrt\{n\}\; \backslash left(\; \backslash begin\{pmatrix\}\; \backslash widehat\backslash theta\_n\; \backslash \backslash \; \backslash widehat\backslash mu\_n\; \backslash \backslash \; \backslash widehat\backslash sigma\_n^2\; \backslash end\{pmatrix\}\; -$$

\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^{2 t \theta}-1}{t^2} & 0 & \frac{\sigma^2( e^{2 t \theta}-1-2 t \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^{2 t \theta}-1-2 t \theta )}{ t^2 \theta } & 0 &

\frac{\sigma^4 \left[ \left( e^{2 t \theta} - 1 \right)^2 + 2 t^2 \theta^2 \left( e^{2 t \theta} + 1 \right) + 4 t \theta \left( e^{2 t \theta} - 1 \right)\right] }

{ t^2 \left(e^{2 t \theta}-1\right) \theta^2}

\end{pmatrix}

\right)

[[Image:Ornstein-Uhlenbeck-traces-a-mu.svg|thumb|450px|Four sample paths of different OU-processes with θ = 1, σ = $\backslash sqrt2$:

blue: initial value a = 10, μ = 0

orange: initial value a = 0, μ = 0

green: initial value a = −10, μ = 0

red: initial value a = 0, μ = −10]]To simulate an OU process numerically with standard deviation $\backslash Sigma$ and correlation time $\backslash tau\; =\; 1/\backslash Theta$, one method is to apply the finite-difference formula

x(t+dt) = x(t) - \Theta \, dt \, x(t) + \Sigma \sqrt{2 \, dt \, \Theta} \nu_i

where $\backslash nu\_i$ is a normally distributed random number with zero mean and unit variance, sampled independently at every time-step $dt$.{{sfn | Kloeden | Platen | Schurz | 1994 | p=}}

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 $n$ black and white balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let $X\_k$ be the number of black balls in the urn after $k$ steps. Then $\backslash frac\{X\_\{[nt]\}\; -\; n/2\}\{\backslash sqrt\{n\}\}$ converges in law to an Ornstein–Uhlenbeck process as $n$ tends to infinity. This was obtained by Mark Kac.{{sfn | Iglehart | 1968}}

Heuristically one may obtain this as follows.

Let $X^\{(n)\}\_t:=\; \backslash frac\{X\_\{[nt]\}\; -\; n/2\}\{\backslash sqrt\{n\}\}$, and we will obtain the stochastic differential equation at the $n\backslash to\; \backslash infty$ limit. First deduce

$$\backslash Delta\; t\; =\; 1/n,\backslash quad\; \backslash Delta\; X^\{(n)\}\_t\; =\; X^\{(n)\}\_\{t+\backslash Delta\; t\}\; -X^\{(n)\}\_t.$$

With this, we can calculate the mean and variance of $\backslash Delta\; X^\{(n)\}\_t$, which turns out to be $-2\; X^\{(n)\}\_t\; \backslash Delta\; t$ and $\backslash Delta\; t$. Thus at the $n\backslash to\; \backslash infty$ limit, we have $dX\_t\; =\; -2X\_t\backslash ,dt\; +\; dW\_t$, with solution (assuming $X\_0$ distribution is standard normal) $X\_t\; =\; e^\{-2t\}W\_\{e^\{4t\}\}$.

The Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. A canonical example is a Hookean spring (harmonic oscillator) with spring constant $k$ whose dynamics is overdamped

with friction coefficient $\backslash gamma$. In the presence of thermal fluctuations with temperature $T$, the length $x(t)$ of the spring fluctuates around the spring rest length $x\_0$; its stochastic dynamics is described by an Ornstein–Uhlenbeck process with

:$$

\begin{align}

\theta &=k/\gamma, \\

\mu & =x_0, \\

\sigma &=\sqrt{2k_B T/\gamma},

\end{align}

where $\backslash sigma$ is derived from the Stokes–Einstein equation $D=\backslash sigma^2/2=k\_B\; T/\backslash gamma$ for the effective diffusion constant.{{sfn | Nørrelykke | Flyvbjerg | 2011 | p=}}{{sfn | Goerlich | Li | Albert | Manfredi | 2021 | p=}} This model has been used to characterize the motion of a Brownian particle in an optical trap.{{sfn | Goerlich | Li | Albert | Manfredi | 2021 | p=}}{{sfn | Li | Sentissi | Azzini | Schnoering | 2019 | p=}}

At equilibrium, the spring stores an average energy $\backslash langle\; E\backslash rangle\; =\; k\; \backslash langle\; (x-x\_0)^2\; \backslash rangle\; /2=k\_B\; T/2$ in accordance with the equipartition theorem.{{sfn | Nelson | 1967 | p=}}

The Ornstein–Uhlenbeck process is used in the Vasicek model of the interest rate.{{sfn | Björk | 2009 | pp = 375, 381}} 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 $\backslash mu$ represents the equilibrium or mean value supported by fundamentals; $\backslash sigma$ the degree of volatility around it caused by shocks, and $\backslash theta$ 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.{{sfn | Leung | Li | 2016 | p=}}[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]

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.{{cite web |url=https://www.risk.net/regulation/1528679/detecting-market-abuse |title=Detecting Market Abuse |date=2 November 2004 | publisher=Risk Magazine}}{{cite web |url=https://www.consob.it/documents/1912911/2006254/qdf54en.pdf/d31b160c-9ba5-e08d-d39d-a78bddfc698e |title=The detection of Market Abuse on financial markets: a quantitative approach | publisher=Consob – The Italian Securities and Exchange Commission}}

The Ornstein–Uhlenbeck process has been proposed as an improvement over a Brownian motion model for modeling the change in organismal phenotypes over time.{{sfn | Martins | 1994 | pp=193–209}} 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.{{sfn | Hunt | 2007 | p=}} 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.{{sfn | Cornuault | 2022}}

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:{{sfn | Jespersen | Metzler | Fogedby | 1999}}{{sfn | Fink | Klüppelberg | 2011 | p=}}

:$dx\_t\; =\; -\backslash theta\; \backslash ,\; x\_t\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash ,\; dL\_t$

Here, the differential of the Wiener process $W\_t$ has been replaced with the differential of a Lévy process $L\_t$.

In addition, in finance, stochastic processes are used where the volatility increases for larger values of $X$. In particular, the CKLS process (Chan–Karolyi–Longstaff–Sanders){{sfn | Chan | Karolyi | Longstaff | Sanders | 1992}} with the volatility term replaced by $\backslash sigma\backslash ,x^\backslash gamma\backslash ,\; dW\_t$ can be solved in closed form for $\backslash gamma=1$, as well as for $\backslash gamma=0$, which corresponds to the conventional OU process. Another special case is $\backslash gamma=1/2$, which corresponds to the Cox–Ingersoll–Ross model (CIR-model).

A multi-dimensional version of the Ornstein–Uhlenbeck process, denoted by the N-dimensional vector $\backslash mathbf\{x\}\_t$, can be defined from

:$$

d \mathbf{x}_t = -\boldsymbol{\beta} \, \mathbf{x}_t \, dt + \boldsymbol{\sigma} \, d\mathbf{W}_t.

where $\backslash mathbf\{W\}\_t$ is an N-dimensional Wiener process, and $\backslash boldsymbol\{\backslash beta\}$ and $\backslash boldsymbol\{\backslash sigma\}$ are constant N×N matrices.{{sfn | Gardiner | 1985 | p = 109}} The solution is

:$$

\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

:$$

\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 $P(\backslash mathbf\{x\},t)$, which satisfies the Fokker–Planck equation{{sfn | Gardiner | 1985 | p = 97}}

:$$

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

:$$

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 $\backslash boldsymbol\{\backslash omega\}$ is determined from the Lyapunov equation $\backslash boldsymbol\{\backslash beta\}\; \backslash boldsymbol\{\backslash omega\}\; +\; \backslash boldsymbol\{\backslash omega\}\; \backslash boldsymbol\{\backslash beta\}^T\; =\; 2\; \backslash boldsymbol\{D\}$.{{sfn | Risken | 1989 | p=}}

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

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• {{cite journal | last1=Li | first1=Minghao | last2=Sentissi | first2=Oussama | last3=Azzini | first3=Stefano | last4=Schnoering | first4=Gabriel | last5=Canaguier-Durand | first5=Antoine | last6=Genet | first6=Cyriaque | title=Subfemtonewton force fields measured with ergodic Brownian ensembles | journal=Physical Review A | volume=100 | issue=6 | date=2019-12-10 | page=063816 | issn=2469-9926 | doi=10.1103/physreva.100.063816 | arxiv = 1908.00610

| bibcode=2019PhRvA.100f3816L | s2cid=199405409 }}

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{{refend}}

• [https://ssrn.com/abstract=1109160 A Stochastic Processes Toolkit for Risk Management], Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki
• [https://web.archive.org/web/20150619164944/http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/ Simulating and Calibrating the Ornstein–Uhlenbeck process], M. A. van den Berg
• [http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf Maximum likelihood estimation of mean reverting processes], Jose Carlos Garcia Franco
• {{cite web | url=http://turingfinance.com/interactive-stochastic-processes/ | title=Interactive Web Application: Stochastic Processes used in Quantitative Finance | access-date=2015-07-03 | archive-url=https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/ | archive-date=2015-09-20 | url-status=dead }}

{{Stochastic processes}}

{{DEFAULTSORT:Ornstein-Uhlenbeck process}}

Category:Stochastic differential equations

Category:Markov processes

Category:Variants of random walks