An Ornstein-Uhlenbeck (OU) process represents a continuous time Markov chain parameterized by an initial state $$x_0$$, selection strength $$\alpha>0$$, long-term mean $$\theta$$, and time-unit variance $$\sigma^2$$. Given $$x_0$$, at time $$t$$, the state of the process is characterized by a normal distribution with mean $$x_0 exp(-\alpha t) + \theta (1 - exp(-\alpha t))$$ and variance $$\sigma^2 (1-exp(-2 \alpha t)) / (2 \alpha)$$. In the limit $$\alpha -> 0$$, the OU process converges to a Brownian motion process with initial state $$x_0$$ and time-unit variance $$\sigma^2$$ (at time $$t$$, this process is characterized by a normal distribution with mean $$x_0$$ and variance $$t \sigma^2$$.

dOU(z, z0, t, alpha, theta, sigma, log = TRUE)

rOU(n, z0, t, alpha, theta, sigma)

meanOU(z0, t, alpha, theta)

varOU(t, alpha, sigma)

sdOU(t, alpha, sigma)

## Arguments

z Numeric value or vector of size n. Numeric value or vector of size n, initial value(s) to condition on. Numeric value or vector of size n, denoting the time-step. Numeric values or n-vectors, parameters of the OU process; alpha and sigma must be non-negative. A zero alpha is interpreted as the Brownian motion process in the limit alpha -> 0. Logical indicating whether the returned density should is on the logarithmic scale. Integer, the number of values to sample.

## Value

dOU returns the conditional probability density(ies) of the elements in z, given the initial state(s) z0, time-step(s) t and OU-parameters by alpha, theta and sigma.

rOU returns a numeric vector of length n, a random sample from the conditional distribution(s) of one or n OU process(es) given initial value(s) and time-step(s).

meanOU returns the expected value of the OU-process at time t.

varOU returns the expected variance of the OU-process at time t.

sdOU returns the standard deviation of the OU-process at time t.

## Details

Similar to dnorm and rnorm, the functions described in this help-page support single values as well as vectors for the parameters z, z0, t, alpha, theta and sigma.

## Functions

• dOU: probability density

• rOU: random generator

• meanOU: mean value

• varOU: variance

• sdOU: standard deviation

## Examples

z0 <- 8
t <- 10
n <- 100000
sample <- rOU(n, z0, t, 2, 3, 1)
dens <- dOU(sample, z0, t, 2, 3, 1)
var(sample)  # around 1/4#> [1] 0.2492913varOU(t, 2, 1)#> [1] 0.25