Compute stationary distribution for Ornstein-Uhlenbeck model

Compute the stationary (long-run equilibrium) distribution of the Ornstein-Uhlenbeck process, including its mean, standard deviation, and (for multivariate models) the full covariance and correlation structure.

Usage

# S3 method for class 'affectOU'
stationary(object, ...)

Arguments

object: An affectOU model object.
...: Additional arguments (unused).

Value

A list of class stationary_affectOU containing:

is_stable: Logical, TRUE if a stationary distribution exists
mean: Stationary (long-run) mean, or NULL if non-stable
sd: Stationary standard deviations, or NULL if non-stable
cov: Stationary covariance matrix (NULL for 1D or non-stable)
cor: Stationary correlation matrix (NULL for 1D or non-stable)
ndim: Dimensionality of the process

Details

A stationary distribution exists only when the system is stable (see stability()). For non-stable systems, the function returns is_stable = FALSE with NULL distribution properties.

Stationary distribution

When $\theta > 0$ (1D) or all eigenvalues of $\Theta$ have positive real parts (multivariate), the process converges to a stationary distribution: $$X_\infty \sim N\!\left(\mu,\; \frac{\gamma^2}{2\theta}\right)$$ The stationary variance $\gamma^2/(2\theta)$ depends on both $\gamma$ and $\theta$. Different parameter combinations can produce the same long-run spread but very different dynamics (see examples).

Stationary covariance (multivariate)

For multivariate models, the stationary covariance matrix $\Sigma_\infty$ solves the Lyapunov equation: $$\Theta \Sigma_\infty + \Sigma_\infty \Theta^\top = \Gamma \Gamma^T$$

Off-diagonal elements in $\Theta$ (coupling between dimensions) can induce correlation at equilibrium even when the noise is independent.

Formula reference

Key theoretical quantities for the 1D case:

Quantity	Formula	Interpretation
Stationary mean	$\mu$	Long-run center
Stationary variance	$\gamma^2 / (2\theta)$	Long-run spread
Half-life	$\log(2) / \theta$	Persistence of perturbations
ACF at lag $\tau$	$e^{-\theta\tau}$	Predictability over time
Conditional mean	$\mu + (x - \mu) e^{-\theta \Delta t}$	Expected next value given current

Stationary properties depend on both $\gamma$ and $\theta$; temporal dynamics depend mainly on $\theta$. Two processes can share stationary distributions but differ in dynamics, or vice versa.

Examples

# 1D model
model <- affectOU(theta = 0.5, mu = 0, gamma = 1)
stationary(model)
#> 
#> ── Stationary distribution of 1D Ornstein-Uhlenbeck Model ──
#> 
#> Mean: 0
#> SD: 1
#> 95% interval: [-2, 2]

# All components
unclass(stationary(model))
#> $is_stable
#> [1] TRUE
#> 
#> $mean
#> [1] 0
#> 
#> $sd
#> [1] 1
#> 
#> $cov
#> NULL
#> 
#> $cor
#> NULL
#> 
#> $ndim
#> [1] 1
#> 

# Different dynamics, same stationary distribution
model_slow <- affectOU(theta = 0.5, mu = 0, gamma = 1)
model_fast <- affectOU(theta = 2.0, mu = 0, gamma = 2)
stationary(model_slow)$sd
#> [1] 1
stationary(model_fast)$sd
#> [1] 1

# Non-stable model
model_rw <- affectOU(theta = 0)
stationary(model_rw)
#> 
#> ── Stationary distribution of 1D Ornstein-Uhlenbeck Model ──
#> 
#> Does not exist (system is not stable).

# 2D: coupling induces stationary correlation
theta_2d <- matrix(c(0.5, 0.0, 0.3, 0.5), nrow = 2, byrow = TRUE)
model_2d <- affectOU(ndim = 2, theta = theta_2d, mu = 0, gamma = 1)
stationary(model_2d)$cor # non-zero off-diagonal
#>            [,1]       [,2]
#> [1,]  1.0000000 -0.2761724
#> [2,] -0.2761724  1.0000000