systems.builtin.stochastic.continuous.OrnsteinUhlenbeck

systems.builtin.stochastic.continuous.OrnsteinUhlenbeck(*args, **kwargs)

Ornstein-Uhlenbeck process with mean reversion and additive noise.

The fundamental mean-reverting stochastic process, combining deterministic relaxation toward an equilibrium with random fluctuations. This is the stochastic analog of a damped harmonic oscillator and the continuous-time limit of AR(1) processes.

Stochastic Differential Equation

Centered form (mean = 0): dX = -α·X·dt + σ·dW

With control: dX = (-α·X + u)·dt + σ·dW

where: - X(t) ∈ ℝ: State (velocity, rate, deviation) - α > 0: Mean reversion rate (1/time) - σ > 0: Volatility (noise intensity, state/√time) - u ∈ ℝ: External forcing/control - W(t): Standard Wiener process

Key Features

Mean Reversion: Drift term -α·X creates restoring force toward zero: - Above zero: Negative drift pushes down - Below zero: Positive drift pushes up - Strength proportional to displacement

Stationarity: Unique stationary distribution: X(∞) ~ N(u/α, σ²/(2α))

Unlike Brownian motion or GBM, variance bounded.

Additive Noise: Diffusion σ constant (state-independent). Simplifies analysis and simulation.

Ergodicity: Time averages equal ensemble averages. Can estimate statistics from single long trajectory.

Mathematical Properties

Exact Solution (u = 0, X(0) = X₀): X(t) = X₀·e^(-α·t) + ∫₀ᵗ σ·e^(-α·(t-s))·dW(s)

Moments: Mean: E[X(t)] = X₀·e^(-α·t) + (u/α)·(1 - e^(-α·t))

Variance: Var[X(t)] = (σ²/2α)·(1 - e^(-2α·t))

Asymptotic (t → ∞): Mean: E[X(∞)] = u/α Variance: Var[X(∞)] = σ²/(2α) Distribution: N(u/α, σ²/(2α))

Autocorrelation: For stationary process: Cov[X(t), X(t+τ)] = (σ²/2α)·e^(-α·τ)

Exponential decay with rate α.

Physical Interpretation

Mean Reversion Rate α: - Controls speed of return to equilibrium - Units: [1/time] - Time constant: τ = 1/α - Half-life: t₁/₂ = ln(2)/α ≈ 0.693/α

Examples: - α = 0.1: Slow reversion, τ = 10s - α = 1.0: Moderate, τ = 1s - α = 10.0: Fast, τ = 0.1s

Volatility σ: - Instantaneous noise intensity - Units: [state]/√[time] - Stationary std: σ/√(2α)

Ratio σ/α: Effective noise level: - Large: Weak reversion, large fluctuations - Small: Strong reversion, tight around mean

Stationary Standard Deviation: σ_stat = σ/√(2α)

Balance between noise injection (σ) and dissipation (α).

State Space

State: x ∈ ℝ (unbounded) - Can take any real value - Equilibrium at x = u/α - Fluctuates around equilibrium

Control: u ∈ ℝ (optional) - Shifts equilibrium to u/α - Examples: External force, policy intervention

Parameters

Name	Type	Description	Default
alpha	float	Mean reversion rate (must be positive for stability) - Larger α: Faster reversion, smaller steady-state variance - Time constant τ = 1/α - Typical range: 0.1 to 10.0	`1.0`
sigma	float	Volatility (must be positive) - Controls noise intensity - Stationary std: σ/√(2α) - Typical: 0.1 to 2.0	`1.0`

Stochastic Properties

Noise Type: ADDITIVE
Diffusion: g(x) = σ (constant, state-independent)
SDE Type: Itô (standard)
Noise Dimension: nw = 1
Stationary: Yes (unique equilibrium distribution)
Ergodic: Yes (time averages = ensemble averages)

Applications

1. Physics: - Langevin equation (particle velocity in fluid) - Thermal equilibrium (velocity distribution) - Optical tweezers (trapped particle) - Molecular dynamics

2. Finance: - Vasicek interest rate model - Commodity price spreads - Pairs trading (spread between correlated assets) - Credit spreads

3. Neuroscience: - Neural membrane potential between spikes - Leaky integrate-and-fire neurons - Synaptic conductances

4. Biology: - Population fluctuations around carrying capacity - Gene expression dynamics - Ecological systems with noise

5. Control & Signal Processing: - Colored noise generation - Autoregressive processes (continuous AR(1)) - Kalman filtering benchmark

Numerical Simulation

Euler-Maruyama: X[k+1] = (1 - α·Δt)·X[k] + u·Δt + σ·√Δt·Z[k]

Requires α·Δt < 1 for stability.

Exact Discretization (Preferred): X[k+1] = e^(-α·Δt)·X[k] + (u/α)·(1-e^(-α·Δt)) + σ_eff·Z[k]

where σ_eff = √((σ²/2α)·(1-e^(-2α·Δt)))

Advantages: - Exact (no discretization error) - Unconditionally stable - Preserves stationary distribution

Statistical Analysis

Parameter Estimation: From discrete observations, estimate: α̂ = -ln(autocorr(Δt)) / Δt σ̂ = sample_std · √(2α̂)

Model Validation: - Check exponential autocorrelation - Verify Gaussian residuals - Test stationarity (KPSS) - Test mean reversion (ADF unit root)

Comparison with Other Processes

vs. Brownian Motion: - BM: No reversion (non-stationary) - OU: Mean-reverting (stationary)

vs. Geometric Brownian Motion: - GBM: Multiplicative noise, for prices - OU: Additive noise, for rates/deviations

vs. CIR Process: - CIR: Multiplicative noise √X (stays positive) - OU: Additive noise (can be negative)

Limitations

Can be negative (problem for rates/prices)
Constant volatility (unrealistic for some applications)
Linear mean reversion (may be non-linear in reality)
Gaussian (real data may have fat tails)

Solutions: - CIR for positive quantities - State-dependent diffusion - Non-linear drift functions - Jump extensions

Methods

Name	Description
define_system	Define Ornstein-Uhlenbeck process dynamics.
get_half_life	Get half-life t₁/₂ = ln(2)/α ≈ 0.693/α.
get_stationary_std	Compute theoretical stationary standard deviation.
get_time_constant	Get mean reversion time constant τ = 1/α.

define_system

systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.define_system(
    alpha=1.0,
    sigma=1.0,
)

Define Ornstein-Uhlenbeck process dynamics.

Sets up the stochastic differential equation: dX = (-α·X + u)·dt + σ·dW

with mean reversion and additive noise.

Parameters

Name	Type	Description	Default
alpha	float	Mean reversion rate (should be positive) - α > 0: Stable (mean-reverting) - α = 0: Brownian motion (no reversion) - α < 0: Unstable (explosive)	`1.0`
sigma	float	Volatility (must be positive) - Controls noise intensity - Stationary std: σ/√(2α)	`1.0`

Raises

Name	Type	Description
	ValueError	If sigma ≤ 0
	UserWarning	If alpha ≤ 0 (unstable/non-reverting)

Notes

Stability Condition: Require α > 0 for mean reversion and stationarity. - α > 0: Process stable, reverts to mean - α = 0: Becomes Brownian motion - α < 0: Unstable, diverges exponentially

Time Scales: - Time constant: τ = 1/α - Half-life: t₁/₂ = ln(2)/α ≈ 0.693/α - Settling time: t_settle ≈ 5/α (1% of initial deviation)

After time τ: Deviation reduced by factor e ≈ 0.368 After time 5τ: Deviation reduced by factor e⁻⁵ ≈ 0.007 (~99% reverted)

Stationary Statistics: For u = 0: - Mean: E[X(∞)] = 0 - Variance: Var[X(∞)] = σ²/(2α) - Std: σ/√(2α)

For u ≠ 0: - Mean: E[X(∞)] = u/α - Variance: σ²/(2α) (same)

Parameter Selection: - Fast reversion: α = 5-10 (τ = 0.1-0.2s) - Moderate: α = 1-2 (τ = 0.5-1s) - Slow: α = 0.1-0.5 (τ = 2-10s)

Noise Level: Effective noise (stationary std): σ/√(2α) - To achieve desired std s: set σ = s·√(2α)

Fluctuation-Dissipation: In physical systems at temperature T: σ² = 2α·kᵦ·T/m Balance between thermal noise and friction.

get_half_life

systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_half_life()

Get half-life t₁/₂ = ln(2)/α ≈ 0.693/α.

Time for deviation from mean to reduce by 50%. More intuitive than time constant for some applications.

Returns

Name	Type	Description
	float	Half-life [time units]

get_stationary_std

systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_stationary_std()

Compute theoretical stationary standard deviation.

For OU process, stationary distribution is: X(∞) ~ N(u/α, σ²/(2α))

Returns

Name	Type	Description
	float	Stationary standard deviation: σ/√(2α)

Notes

Interpretation: This is the long-term standard deviation around the mean, reached after transient dies out (t >> 1/α).

Relationship to Parameters: - Increases with σ (more noise) - Decreases with α (stronger reversion) - Ratio σ/α sets scale

Comparison with σ: - σ: Instantaneous noise intensity - σ/√(2α): Accumulated noise effect - Always: σ_stat < σ (for α > 0.5)

Design: To achieve target std s: σ = s · √(2α)

get_time_constant

systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_time_constant()

Get mean reversion time constant τ = 1/α.

The time constant is the characteristic time scale for the process to revert to its mean. After time τ, the deviation from mean is reduced by factor e ≈ 0.368.

Returns

Name	Type	Description
	float	Time constant [time units]

Notes

Physical Meaning: - Time to reduce deviation by ~63% - Reciprocal of mean reversion rate - Sets time scale of dynamics

Related Quantities: - Half-life: t₁/₂ = ln(2)·τ ≈ 0.693·τ - 99% settling: t_99 ≈ 5·τ - 1% settling: t_01 ≈ 0.01·τ

Rule of Thumb: Process reaches stationarity after 3-5 time constants. - t < τ: Still in transient - t ≈ 5τ: Essentially stationary

Examples: - τ = 0.1s: Fast dynamics (α = 10) - τ = 1.0s: Moderate (α = 1) - τ = 10s: Slow (α = 0.1)

# systems.builtin.stochastic.continuous.OrnsteinUhlenbeck { #cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck } ```python systems.builtin.stochastic.continuous.OrnsteinUhlenbeck(*args, **kwargs) ``` Ornstein-Uhlenbeck process with mean reversion and additive noise. The fundamental mean-reverting stochastic process, combining deterministic relaxation toward an equilibrium with random fluctuations. This is the stochastic analog of a damped harmonic oscillator and the continuous-time limit of AR(1) processes. ## Stochastic Differential Equation {.doc-section .doc-section-stochastic-differential-equation} Centered form (mean = 0): dX = -α·X·dt + σ·dW With control: dX = (-α·X + u)·dt + σ·dW where: - X(t) ∈ ℝ: State (velocity, rate, deviation) - α > 0: Mean reversion rate (1/time) - σ > 0: Volatility (noise intensity, state/√time) - u ∈ ℝ: External forcing/control - W(t): Standard Wiener process ## Key Features {.doc-section .doc-section-key-features} **Mean Reversion:** Drift term -α·X creates restoring force toward zero: - Above zero: Negative drift pushes down - Below zero: Positive drift pushes up - Strength proportional to displacement **Stationarity:** Unique stationary distribution: X(∞) ~ N(u/α, σ²/(2α)) Unlike Brownian motion or GBM, variance bounded. **Additive Noise:** Diffusion σ constant (state-independent). Simplifies analysis and simulation. **Ergodicity:** Time averages equal ensemble averages. Can estimate statistics from single long trajectory. ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Exact Solution (u = 0, X(0) = X₀):** X(t) = X₀·e^(-α·t) + ∫₀ᵗ σ·e^(-α·(t-s))·dW(s) **Moments:** Mean: E[X(t)] = X₀·e^(-α·t) + (u/α)·(1 - e^(-α·t)) Variance: Var[X(t)] = (σ²/2α)·(1 - e^(-2α·t)) **Asymptotic (t → ∞):** Mean: E[X(∞)] = u/α Variance: Var[X(∞)] = σ²/(2α) Distribution: N(u/α, σ²/(2α)) **Autocorrelation:** For stationary process: Cov[X(t), X(t+τ)] = (σ²/2α)·e^(-α·τ) Exponential decay with rate α. ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Mean Reversion Rate α:** - Controls speed of return to equilibrium - Units: [1/time] - Time constant: τ = 1/α - Half-life: t₁/₂ = ln(2)/α ≈ 0.693/α **Examples:** - α = 0.1: Slow reversion, τ = 10s - α = 1.0: Moderate, τ = 1s - α = 10.0: Fast, τ = 0.1s **Volatility σ:** - Instantaneous noise intensity - Units: [state]/√[time] - Stationary std: σ/√(2α) **Ratio σ/α:** Effective noise level: - Large: Weak reversion, large fluctuations - Small: Strong reversion, tight around mean **Stationary Standard Deviation:** σ_stat = σ/√(2α) Balance between noise injection (σ) and dissipation (α). ## State Space {.doc-section .doc-section-state-space} State: x ∈ ℝ (unbounded) - Can take any real value - Equilibrium at x = u/α - Fluctuates around equilibrium Control: u ∈ ℝ (optional) - Shifts equilibrium to u/α - Examples: External force, policy intervention ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | alpha | float | Mean reversion rate (must be positive for stability) - Larger α: Faster reversion, smaller steady-state variance - Time constant τ = 1/α - Typical range: 0.1 to 10.0 | `1.0` | | sigma | float | Volatility (must be positive) - Controls noise intensity - Stationary std: σ/√(2α) - Typical: 0.1 to 2.0 | `1.0` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - Noise Type: ADDITIVE - Diffusion: g(x) = σ (constant, state-independent) - SDE Type: Itô (standard) - Noise Dimension: nw = 1 - Stationary: Yes (unique equilibrium distribution) - Ergodic: Yes (time averages = ensemble averages) ## Applications {.doc-section .doc-section-applications} **1. Physics:** - Langevin equation (particle velocity in fluid) - Thermal equilibrium (velocity distribution) - Optical tweezers (trapped particle) - Molecular dynamics **2. Finance:** - Vasicek interest rate model - Commodity price spreads - Pairs trading (spread between correlated assets) - Credit spreads **3. Neuroscience:** - Neural membrane potential between spikes - Leaky integrate-and-fire neurons - Synaptic conductances **4. Biology:** - Population fluctuations around carrying capacity - Gene expression dynamics - Ecological systems with noise **5. Control & Signal Processing:** - Colored noise generation - Autoregressive processes (continuous AR(1)) - Kalman filtering benchmark ## Numerical Simulation {.doc-section .doc-section-numerical-simulation} **Euler-Maruyama:** X[k+1] = (1 - α·Δt)·X[k] + u·Δt + σ·√Δt·Z[k] Requires α·Δt < 1 for stability. **Exact Discretization (Preferred):** X[k+1] = e^(-α·Δt)·X[k] + (u/α)·(1-e^(-α·Δt)) + σ_eff·Z[k] where σ_eff = √((σ²/2α)·(1-e^(-2α·Δt))) **Advantages:** - Exact (no discretization error) - Unconditionally stable - Preserves stationary distribution ## Statistical Analysis {.doc-section .doc-section-statistical-analysis} **Parameter Estimation:** From discrete observations, estimate: α̂ = -ln(autocorr(Δt)) / Δt σ̂ = sample_std · √(2α̂) **Model Validation:** - Check exponential autocorrelation - Verify Gaussian residuals - Test stationarity (KPSS) - Test mean reversion (ADF unit root) ## Comparison with Other Processes {.doc-section .doc-section-comparison-with-other-processes} **vs. Brownian Motion:** - BM: No reversion (non-stationary) - OU: Mean-reverting (stationary) **vs. Geometric Brownian Motion:** - GBM: Multiplicative noise, for prices - OU: Additive noise, for rates/deviations **vs. CIR Process:** - CIR: Multiplicative noise √X (stays positive) - OU: Additive noise (can be negative) ## Limitations {.doc-section .doc-section-limitations} - Can be negative (problem for rates/prices) - Constant volatility (unrealistic for some applications) - Linear mean reversion (may be non-linear in reality) - Gaussian (real data may have fat tails) **Solutions:** - CIR for positive quantities - State-dependent diffusion - Non-linear drift functions - Jump extensions ## See Also {.doc-section .doc-section-see-also} BrownianMotion : No mean reversion (α=0 limit) GeometricBrownianMotion : Multiplicative noise CoxIngersollRoss : Mean-reverting with √X diffusion ## Methods | Name | Description | | --- | --- | | [define_system](#cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.define_system) | Define Ornstein-Uhlenbeck process dynamics. | | [get_half_life](#cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_half_life) | Get half-life t₁/₂ = ln(2)/α ≈ 0.693/α. | | [get_stationary_std](#cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_stationary_std) | Compute theoretical stationary standard deviation. | | [get_time_constant](#cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_time_constant) | Get mean reversion time constant τ = 1/α. | ### define_system { #cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.define_system } ```python systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.define_system( alpha=1.0, sigma=1.0, ) ``` Define Ornstein-Uhlenbeck process dynamics. Sets up the stochastic differential equation: dX = (-α·X + u)·dt + σ·dW with mean reversion and additive noise. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | alpha | float | Mean reversion rate (should be positive) - α > 0: Stable (mean-reverting) - α = 0: Brownian motion (no reversion) - α < 0: Unstable (explosive) | `1.0` | | sigma | float | Volatility (must be positive) - Controls noise intensity - Stationary std: σ/√(2α) | `1.0` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|-------------|---------------------------------------| | | ValueError | If sigma ≤ 0 | | | UserWarning | If alpha ≤ 0 (unstable/non-reverting) | #### Notes {.doc-section .doc-section-notes} **Stability Condition:** Require α > 0 for mean reversion and stationarity. - α > 0: Process stable, reverts to mean - α = 0: Becomes Brownian motion - α < 0: Unstable, diverges exponentially **Time Scales:** - Time constant: τ = 1/α - Half-life: t₁/₂ = ln(2)/α ≈ 0.693/α - Settling time: t_settle ≈ 5/α (1% of initial deviation) After time τ: Deviation reduced by factor e ≈ 0.368 After time 5τ: Deviation reduced by factor e⁻⁵ ≈ 0.007 (~99% reverted) **Stationary Statistics:** For u = 0: - Mean: E[X(∞)] = 0 - Variance: Var[X(∞)] = σ²/(2α) - Std: σ/√(2α) For u ≠ 0: - Mean: E[X(∞)] = u/α - Variance: σ²/(2α) (same) **Parameter Selection:** - Fast reversion: α = 5-10 (τ = 0.1-0.2s) - Moderate: α = 1-2 (τ = 0.5-1s) - Slow: α = 0.1-0.5 (τ = 2-10s) **Noise Level:** Effective noise (stationary std): σ/√(2α) - To achieve desired std s: set σ = s·√(2α) **Fluctuation-Dissipation:** In physical systems at temperature T: σ² = 2α·kᵦ·T/m Balance between thermal noise and friction. ### get_half_life { #cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_half_life } ```python systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_half_life() ``` Get half-life t₁/₂ = ln(2)/α ≈ 0.693/α. Time for deviation from mean to reduce by 50%. More intuitive than time constant for some applications. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|------------------------| | | float | Half-life [time units] | ### get_stationary_std { #cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_stationary_std } ```python systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_stationary_std() ``` Compute theoretical stationary standard deviation. For OU process, stationary distribution is: X(∞) ~ N(u/α, σ²/(2α)) #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|----------------------------------------| | | float | Stationary standard deviation: σ/√(2α) | #### Notes {.doc-section .doc-section-notes} **Interpretation:** This is the long-term standard deviation around the mean, reached after transient dies out (t >> 1/α). **Relationship to Parameters:** - Increases with σ (more noise) - Decreases with α (stronger reversion) - Ratio σ/α sets scale **Comparison with σ:** - σ: Instantaneous noise intensity - σ/√(2α): Accumulated noise effect - Always: σ_stat < σ (for α > 0.5) **Design:** To achieve target std s: σ = s · √(2α) ### get_time_constant { #cdesym.systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_time_constant } ```python systems.builtin.stochastic.continuous.OrnsteinUhlenbeck.get_time_constant() ``` Get mean reversion time constant τ = 1/α. The time constant is the characteristic time scale for the process to revert to its mean. After time τ, the deviation from mean is reduced by factor e ≈ 0.368. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|----------------------------| | | float | Time constant [time units] | #### Notes {.doc-section .doc-section-notes} **Physical Meaning:** - Time to reduce deviation by ~63% - Reciprocal of mean reversion rate - Sets time scale of dynamics **Related Quantities:** - Half-life: t₁/₂ = ln(2)·τ ≈ 0.693·τ - 99% settling: t_99 ≈ 5·τ - 1% settling: t_01 ≈ 0.01·τ **Rule of Thumb:** Process reaches stationarity after 3-5 time constants. - t < τ: Still in transient - t ≈ 5τ: Essentially stationary **Examples:** - τ = 0.1s: Fast dynamics (α = 10) - τ = 1.0s: Moderate (α = 1) - τ = 10s: Slow (α = 0.1)