OrnsteinUhlenbeck

Systems / SDEs / Noise-driven systems

The canonical mean-reverting process — linear restoring drift plus additive noise, the Gaussian stationary benchmark.

stochastic · SDE1 dimensionmean-reverting

sample path

Definition

\[ \begin{aligned} dx &= \left(\theta \left(\mu - x\right)\right)\,dt + \left(\sigma\right)\,dW_{x} \end{aligned} \]

Parameters

Symbol	Default	Role
theta	1	reversion rate
mu	0	long-run mean
sigma	0.3	noise amplitude

State variables: x

Properties

Lyapunov spectrum

TODO — SDE Lyapunov not computed at build time

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - \theta$
constant

Equilibria

TODO — equilibria of a stochastic system not enumerated

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.OrnsteinUhlenbeck()
traj = sys.integrate(final_time=100.0, dt=0.01)

Reference

Uhlenbeck & Ornstein (1930), Phys. Rev. 36, 823-841

BibTeX

@misc{ornsteinuhlenbeck,
  title = {OrnsteinUhlenbeck system},
  note = {Uhlenbeck & Ornstein (1930), Phys. Rev. 36, 823-841}
}