Simulate an Ornstein–Uhlenbeck process with a random initial condition:

With fixed initial condition:

Mean and variance functions:

With fixed initial condition:

Covariance function:

With fixed initial condition:

Quadratic transformation of an Ornstein–Uhlenbeck process:

Simulate the transformed process:

Mean and variance functions are constant:

Verify the results by simulating a slice of the process:

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 Ornstein–Uhlenbeck process:

Ornstein–Uhlenbeck process has a well-defined StationaryDistribution:

With a fixed starting point:

Ornstein–Uhlenbeck 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 Ornstein–Uhlenbeck process with a fixed initial condition is a special ItoProcess:

As well as StratonovichProcess:

Ornstein–Uhlenbeck process is a solution of the stochastic differential equation :

Compare with the corresponding smooth solution:

Ornstein–Uhlenbeck 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:

Ornstein–Uhlenbeck 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:

Simulate an Ornstein–Uhlenbeck process in two dimensions:

Simulate an Ornstein–Uhlenbeck process in three dimensions:

Simulate 500 paths from an Ornstein–Uhlenbeck process:

Take a slice at 1 and visualize its distribution:

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