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


represents an OrnsteinUhlenbeck process with initial condition x0.



open allclose all

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:

Introduced in 2012