Ornstein-Uhlenbeck Process

📐 Definition

For $\theta > 0$ , $\mu \in \mathbb{R}$ , and $\sigma > 0$ , the Ornstein–Uhlenbeck process solves

dX_t = -\theta (X_t-\mu)\,dt + \sigma\,dW_t

where $W_t$ is standard Brownian motion.

Domain and Codomain

Time $t \ge 0$ . Real-valued Gaussian process; complex-valued variants use independent driving noises for real and imaginary parts.

⚙️ Key Properties

Stationary mean $\mathbb{E}[X_t] = \mu$ and variance $\operatorname{Var}[X_t] = \sigma^2/(2\theta)$ in equilibrium. Autocovariance

\mathbb{E}\!\big[(X_t-\mu)(X_s-\mu)\big] = \frac{\sigma^2}{2\theta} e^{-\theta |t-s|}

As $\theta \to 0^+$ , $X_t$ approaches Brownian motion scaled by $\sigma$ . As $\theta \to \infty$ , mean reversion dominates and $X_t$ collapses to $\mu$ .

🎯 Special Cases and Limits

$\theta\to 0^+$ approaches (shifted) Brownian motion behavior.
Large $\theta$ yields fast relaxation to the mean.

Generated by filtering Gaussian white noise through a stable linear SDE; fractional Brownian motion generalizes temporal correlation beyond the exponential kernel.

Usage in Oakfield

Oakfield implements OU-style colored noise as an operator rather than as a standalone SDE textbook object:

stochastic_noise operator maintains per-sample noise state with parameters like sigma (strength) and tau (correlation time), and applies it as an additive forcing term each step.
Noise law (Itô vs Stratonovich) is tracked in the operator’s configuration for consistency with stochastic calculus semantics in the engine.

Historical Foundations

📜 Mean Reversion and Relaxation

The OU process is a classical Gaussian Markov process modeling mean-reverting dynamics, with exponential correlation and tractable stationary statistics.

🌍 Modern Perspective

It is a standard generator of colored noise and a common stochastic thermostat component in numerical modeling.

📚 References

Øksendal, Stochastic Differential Equations
Gardiner, Stochastic Methods