# OrnsteinUhlenbeckProcess

OrnsteinUhlenbeckProcess[μ,σ,θ]

represents a stationary OrnsteinUhlenbeck process with long-term mean μ, volatility , and mean reversion speed θ.

OrnsteinUhlenbeckProcess[μ,σ,θ,x0]

represents an OrnsteinUhlenbeck process with initial condition x0.

# Examples

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## Basic Examples(3)

Simulate an OrnsteinUhlenbeck process with a random initial condition:

With fixed initial condition:

Mean and variance functions:

With fixed initial condition:

Covariance function:

With fixed initial condition:

## Scope(12)

### Basic Uses(7)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different values of the mean reversion speed:

Simulate the process for various starting points:

Process parameter estimation:

The following estimation methods are available:

Correlation function:

With fixed starting point:

Absolute correlation function:

With fixed starting point:

### Process Slice Properties(5)

Univariate SliceDistribution:

Probability density function does not depend on time:

With a fixed starting point:

Probability density function does depend on time:

Multivariate slice distribution:

With a fixed starting point:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

## Generalizations & Extensions(1)

Quadratic transformation of an OrnsteinUhlenbeck process:

Simulate the transformed process:

Mean and variance functions are constant:

Verify the results by simulating a slice of the process:

## Properties & Relations(9)

OrnsteinUhlenbeckProcess starting at a random value is weakly stationary:

The process starting at a fixed value is not weakly stationary:

Power spectrum of a stationary OrnsteinUhlenbeck process:

OrnsteinUhlenbeck process has a well-defined StationaryDistribution:

With a fixed starting point:

OrnsteinUhlenbeck process does not have independent increments:

Compare to the product of expectations:

With a fixed starting point:

Compare to the product of expectations:

Conditional cumulative distribution function:

An OrnsteinUhlenbeck process with a fixed initial condition is a special ItoProcess:

As well as StratonovichProcess:

OrnsteinUhlenbeck process is a solution of the stochastic differential equation :

Compare with the corresponding smooth solution:

OrnsteinUhlenbeck with three arguments is mean ergodic:

The process is weakly stationary:

Calculate absolute correlation function:

Find value of the strip integral:

The process is mean ergodic only for μ=0:

OrnsteinUhlenbeck process at integer times behaves as a first-order ARProcess:

Create moment equations to find parameters for an ARProcess:

Create an ARProcess:

Check for moment agreements:

## Neat Examples(3)

Simulate an OrnsteinUhlenbeck process in two dimensions:

Simulate an OrnsteinUhlenbeck process in three dimensions:

Simulate 500 paths from an OrnsteinUhlenbeck process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1:

Wolfram Research (2012), OrnsteinUhlenbeckProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html.

#### Text

Wolfram Research (2012), OrnsteinUhlenbeckProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html.

#### CMS

Wolfram Language. 2012. "OrnsteinUhlenbeckProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html.

#### APA

Wolfram Language. (2012). OrnsteinUhlenbeckProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html

#### BibTeX

@misc{reference.wolfram_2024_ornsteinuhlenbeckprocess, author="Wolfram Research", title="{OrnsteinUhlenbeckProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html}", note=[Accessed: 07-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_ornsteinuhlenbeckprocess, organization={Wolfram Research}, title={OrnsteinUhlenbeckProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/OrnsteinUhlenbeckProcess.html}, note=[Accessed: 07-September-2024 ]}