systems.builtin.stochastic.continuous.BrownianMotion

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

Standard Brownian motion (Wiener process) - pure diffusion process.

This is the fundamental stochastic process underlying all stochastic differential equations. It represents pure random walk with no deterministic drift, making it the stochastic analog of a zero dynamics system.

Stochastic Differential Equation

Continuous-time SDE: dX = σ·dW

where: - X(t): State (position) at time t - σ > 0: Diffusion coefficient (volatility/noise intensity) - W(t): Standard Wiener process - dW: Brownian motion increment (infinitesimal random variable) - No drift term: f(X) = 0 - Additive noise: σ is constant (state-independent)

Key Distinction from Other SDEs: Unlike most SDEs, Brownian motion has: - Zero drift (f = 0): No deterministic tendency - Constant diffusion (g = σ): Noise intensity independent of state - This makes it the “simplest” non-trivial stochastic process

Mathematical Properties

Distribution: Starting at X(0) = x₀, the solution is: X(t) ~ N(x₀, σ²·t)

Moments: - Mean: E[X(t)] = x₀ (constant for all time) - Variance: Var[X(t)] = σ²·t (grows linearly) - Standard Deviation: Std[X(t)] = σ·√t - Skewness: 0 (symmetric) - Kurtosis: 3 (Gaussian)

Path Properties: - Continuous everywhere - Differentiable nowhere (fractal) - Hölder continuous: |X(t) - X(s)| ≤ C·|t-s|^α for α < 1/2 - Unbounded variation: paths are infinitely “wiggly” - Quadratic variation: ∫(dX)² = σ²·t (fundamental to Itô calculus)

Markov Property: The future evolution depends only on current state, not history: P(X(t) | X(s), s ≤ u) = P(X(t) | X(u)) for t > u

Martingale Property: Expected future value equals current value: E[X(t) | X(s)] = X(s) for t > s

This makes Brownian motion “fair” - no expected gain or loss.

Self-Similarity: For any c > 0: {σ·W(c²·t) : t ≥ 0} has the same distribution as {c·σ·W(t) : t ≥ 0}

Physical Interpretation

Diffusion Coefficient σ: Controls the “speed” of diffusion: - Larger σ: Faster diffusion, more volatile paths - Smaller σ: Slower diffusion, smoother paths - Diffusion length scale: ℓ ~ σ·√t

Relationship to Temperature: In physical systems: σ = √(2·D) where D = kᵦT/γ - D: Diffusion coefficient - kᵦ: Boltzmann constant - T: Temperature - γ: Friction coefficient

Higher temperature → larger σ → faster diffusion

Parameters

Name	Type	Description	Default
sigma	float	Diffusion coefficient (volatility) [units depend on application] - Must be positive: σ > 0 - Controls noise intensity - Units: [state]/√[time] - Standard Brownian motion: σ = 1 Physical Meaning: - σ² = 2D where D is diffusion coefficient - Variance growth rate: dVar/dt = σ² - RMS displacement in unit time: √(σ²·1) = σ	`1.0`

State Space

State: x ∈ ℝ (unbounded) - Can take any real value: -∞ < x < +∞ - No equilibria (system is non-stationary) - No attractors or repellers - All points equally likely in long-time limit (if unbounded)

Control: None (autonomous system) - nu = 0: No control input - Purely noise-driven - Cannot be “steered” by external input

Boundary Conditions: For physical applications, boundaries may be imposed: - Absorbing: Process stops upon hitting boundary - Reflecting: Process bounces back from boundary - Periodic: Wraps around (equivalent to circle topology)

Stochastic Properties

Noise Type: ADDITIVE - Diffusion matrix g = σ (constant) - Does not depend on state x - Simplest form of stochastic dynamics

SDE Type: Itô (standard interpretation) - Can also use Stratonovich (equivalent for additive noise) - No Itô correction needed (f independent of x)

Noise Dimension: - nw = 1: Single Wiener process drives the system - Scalar noise in scalar system

Simulation Methods

Exact Discrete-Time Solution: For time step Δt: X[k+1] = X[k] + σ·√(Δt)·Z[k] where Z[k] ~ N(0,1) is standard normal.

This is EXACT - no discretization error for Brownian motion!

Statistical Analysis

Hypothesis Testing: To verify simulated paths are truly Brownian:

Mean Test: E[X(T) - X(0)] = 0
- Sample mean should be ≈0
- Standard error: σ/√(n_samples)
Variance Test: Var[X(T) - X(0)] = σ²·T
- Sample variance should grow linearly with T
Normality Test: X(T) - X(0) ~ N(0, σ²·T)
- Use Shapiro-Wilk or Kolmogorov-Smirnov test
- Should not reject normality
Independence Test: Increments uncorrelated
- Autocorrelation of ΔX should be zero
- Ljung-Box test for white noise
Quadratic Variation Test:
- Σ(ΔX)² should approach σ²·T as Δt → 0
- Fundamental property distinguishing from smooth functions

Applications

1. Physics: - Particle diffusion in fluids - Thermal noise in electronic circuits - Quantum fluctuations (simplified model) - Langevin equation foundation

2. Finance: - Building block for stock price models - Interest rate dynamics (Vasicek, Hull-White) - Option pricing (Black-Scholes foundation) - Risk-free asset path (money market account noise)

3. Biology: - Molecular diffusion in cells - Random walk of bacteria (before chemotaxis) - Genetic drift in population genetics - Neural membrane potential fluctuations

4. Signal Processing: - White noise integration - Random signal generation - Filter design and testing - Noise modeling

5. Mathematics: - Foundation of stochastic calculus - Benchmark for SDE solvers - Example of martingales - Study of stochastic processes

6. Machine Learning: - Stochastic gradient descent noise - Diffusion models (score-based generative models) - Langevin dynamics sampling - Variational inference with reparametrization

Comparison with Other Processes

vs. Geometric Brownian Motion: - GBM: dX = μ·X·dt + σ·X·dW (multiplicative noise) - BM: dX = σ·dW (additive noise) - GBM stays positive, BM can be negative - GBM for stock prices, BM for price returns

vs. Ornstein-Uhlenbeck: - OU: dX = -α·X·dt + σ·dW (mean-reverting) - BM: dX = σ·dW (no mean reversion) - OU has stationary distribution, BM does not - OU for interest rates, temperatures

vs. Random Walk: - Random walk: discrete time, discrete space - Brownian motion: continuous time, continuous space - BM is limit of random walk as Δt, Δx → 0 with Δx² ~ Δt

vs. Lévy Process: - Lévy: Can have jumps (Poisson, stable processes) - Brownian: Continuous paths only - Brownian is special case of Lévy (Gaussian Lévy)

Theoretical Importance

Why Brownian Motion is Fundamental:

Donsker’s Theorem (Functional CLT): Random walk → Brownian motion as step size → 0 Makes BM the universal limit of random walks
Lévy Characterization: Any continuous martingale with quadratic variation t must be Brownian motion (up to time change)
Feynman-Kac Formula: Connects SDEs to PDEs via expectation Solution to heat equation = expected value over Brownian paths
Girsanov Theorem: Change of measure for SDEs Converts drift to diffusion via change of probability measure
Reflection Principle: P(max X(t) > a) = 2·P(X(T) > a) Used in barrier option pricing

Limitations and Extensions

Standard Brownian Motion Limitations: - No memory (Markov property may be unrealistic) - Gaussian increments (real data often heavy-tailed) - Continuous paths (no jumps) - Constant volatility (time-varying in reality) - Linear variance growth (subdiffusion/superdiffusion in complex media)

Extensions to Address Limitations: - Fractional Brownian motion: Long-range dependence (Hurst parameter) - Lévy processes: Jumps and heavy tails - Stochastic volatility: Time-varying σ(t) - Rough volatility: σ follows rough process - Anomalous diffusion: Variance ~ t^α, α ≠ 1

Methods

Name	Description
define_system	Define standard Brownian motion dynamics.

define_system

systems.builtin.stochastic.continuous.BrownianMotion.define_system(sigma=1.0)

Define standard Brownian motion dynamics.

This sets up the stochastic differential equation: dX = σ·dW

with zero drift and constant diffusion.

Parameters

Name	Type	Description	Default
sigma	float	Diffusion coefficient (must be positive) - Standard Brownian motion: σ = 1 - Units: [state]/√[time] - Controls noise intensity - Variance growth rate: σ²	`1.0`

Raises

Name	Type	Description
	ValueError	If sigma ≤ 0 (diffusion coefficient must be positive)

Notes

System Structure: This is a minimal stochastic system with: - Zero drift: f(x) = 0 - Constant diffusion: g(x) = σ - Single noise source: nw = 1 - No control input: nu = 0 - First-order dynamics: order = 1

Special Properties: - Pure diffusion process (no drift component) - Autonomous (no time dependence) - Additive noise (σ independent of state) - Unbounded state space (x ∈ ℝ) - No equilibrium points (non-stationary)

Standard vs. Scaled Brownian Motion: When σ = 1, this is the standard Wiener process W(t). For σ ≠ 1, we have scaled Brownian motion: X(t) = σ·W(t)

All Brownian motions can be written as: X(t) = x₀ + σ·W(t) where W(t) is standard Brownian motion.

Discretization: The discrete-time equivalent is: X[k+1] = X[k] + σ·√(dt)·w[k] where w[k] ~ N(0,1) and dt is the time step.

This is exact (no discretization error) for Brownian motion.

Comparison with Other SDEs: Unlike most SDEs: - No drift means E[X(t)] = x₀ for all t (constant mean) - Constant diffusion means integration is particularly simple - Euler-Maruyama method is exact (no approximation error) - No stability concerns regardless of time step size

# systems.builtin.stochastic.continuous.BrownianMotion { #cdesym.systems.builtin.stochastic.continuous.BrownianMotion } ```python systems.builtin.stochastic.continuous.BrownianMotion(*args, **kwargs) ``` Standard Brownian motion (Wiener process) - pure diffusion process. This is the fundamental stochastic process underlying all stochastic differential equations. It represents pure random walk with no deterministic drift, making it the stochastic analog of a zero dynamics system. ## Stochastic Differential Equation {.doc-section .doc-section-stochastic-differential-equation} Continuous-time SDE: dX = σ·dW where: - X(t): State (position) at time t - σ > 0: Diffusion coefficient (volatility/noise intensity) - W(t): Standard Wiener process - dW: Brownian motion increment (infinitesimal random variable) - No drift term: f(X) = 0 - Additive noise: σ is constant (state-independent) **Key Distinction from Other SDEs:** Unlike most SDEs, Brownian motion has: - Zero drift (f = 0): No deterministic tendency - Constant diffusion (g = σ): Noise intensity independent of state - This makes it the "simplest" non-trivial stochastic process ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Distribution:** Starting at X(0) = x₀, the solution is: X(t) ~ N(x₀, σ²·t) **Moments:** - Mean: E[X(t)] = x₀ (constant for all time) - Variance: Var[X(t)] = σ²·t (grows linearly) - Standard Deviation: Std[X(t)] = σ·√t - Skewness: 0 (symmetric) - Kurtosis: 3 (Gaussian) **Path Properties:** - Continuous everywhere - Differentiable nowhere (fractal) - Hölder continuous: |X(t) - X(s)| ≤ C·|t-s|^α for α < 1/2 - Unbounded variation: paths are infinitely "wiggly" - Quadratic variation: ∫(dX)² = σ²·t (fundamental to Itô calculus) **Markov Property:** The future evolution depends only on current state, not history: P(X(t) | X(s), s ≤ u) = P(X(t) | X(u)) for t > u **Martingale Property:** Expected future value equals current value: E[X(t) | X(s)] = X(s) for t > s This makes Brownian motion "fair" - no expected gain or loss. **Self-Similarity:** For any c > 0: {σ·W(c²·t) : t ≥ 0} has the same distribution as {c·σ·W(t) : t ≥ 0} ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Diffusion Coefficient σ:** Controls the "speed" of diffusion: - Larger σ: Faster diffusion, more volatile paths - Smaller σ: Slower diffusion, smoother paths - Diffusion length scale: ℓ ~ σ·√t **Relationship to Temperature:** In physical systems: σ = √(2·D) where D = kᵦT/γ - D: Diffusion coefficient - kᵦ: Boltzmann constant - T: Temperature - γ: Friction coefficient Higher temperature → larger σ → faster diffusion ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | sigma | float | Diffusion coefficient (volatility) [units depend on application] - Must be positive: σ > 0 - Controls noise intensity - Units: [state]/√[time] - Standard Brownian motion: σ = 1 **Physical Meaning:** - σ² = 2D where D is diffusion coefficient - Variance growth rate: dVar/dt = σ² - RMS displacement in unit time: √(σ²·1) = σ | `1.0` | ## State Space {.doc-section .doc-section-state-space} State: x ∈ ℝ (unbounded) - Can take any real value: -∞ < x < +∞ - No equilibria (system is non-stationary) - No attractors or repellers - All points equally likely in long-time limit (if unbounded) Control: None (autonomous system) - nu = 0: No control input - Purely noise-driven - Cannot be "steered" by external input **Boundary Conditions:** For physical applications, boundaries may be imposed: - Absorbing: Process stops upon hitting boundary - Reflecting: Process bounces back from boundary - Periodic: Wraps around (equivalent to circle topology) ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} **Noise Type:** ADDITIVE - Diffusion matrix g = σ (constant) - Does not depend on state x - Simplest form of stochastic dynamics **SDE Type:** Itô (standard interpretation) - Can also use Stratonovich (equivalent for additive noise) - No Itô correction needed (f independent of x) **Noise Dimension:** - nw = 1: Single Wiener process drives the system - Scalar noise in scalar system ## Simulation Methods {.doc-section .doc-section-simulation-methods} **Exact Discrete-Time Solution:** For time step Δt: X[k+1] = X[k] + σ·√(Δt)·Z[k] where Z[k] ~ N(0,1) is standard normal. This is EXACT - no discretization error for Brownian motion! ## Statistical Analysis {.doc-section .doc-section-statistical-analysis} **Hypothesis Testing:** To verify simulated paths are truly Brownian: 1. **Mean Test**: E[X(T) - X(0)] = 0 - Sample mean should be ≈0 - Standard error: σ/√(n_samples) 2. **Variance Test**: Var[X(T) - X(0)] = σ²·T - Sample variance should grow linearly with T 3. **Normality Test**: X(T) - X(0) ~ N(0, σ²·T) - Use Shapiro-Wilk or Kolmogorov-Smirnov test - Should not reject normality 4. **Independence Test**: Increments uncorrelated - Autocorrelation of ΔX should be zero - Ljung-Box test for white noise 5. **Quadratic Variation Test**: - Σ(ΔX)² should approach σ²·T as Δt → 0 - Fundamental property distinguishing from smooth functions ## Applications {.doc-section .doc-section-applications} **1. Physics:** - Particle diffusion in fluids - Thermal noise in electronic circuits - Quantum fluctuations (simplified model) - Langevin equation foundation **2. Finance:** - Building block for stock price models - Interest rate dynamics (Vasicek, Hull-White) - Option pricing (Black-Scholes foundation) - Risk-free asset path (money market account noise) **3. Biology:** - Molecular diffusion in cells - Random walk of bacteria (before chemotaxis) - Genetic drift in population genetics - Neural membrane potential fluctuations **4. Signal Processing:** - White noise integration - Random signal generation - Filter design and testing - Noise modeling **5. Mathematics:** - Foundation of stochastic calculus - Benchmark for SDE solvers - Example of martingales - Study of stochastic processes **6. Machine Learning:** - Stochastic gradient descent noise - Diffusion models (score-based generative models) - Langevin dynamics sampling - Variational inference with reparametrization ## Comparison with Other Processes {.doc-section .doc-section-comparison-with-other-processes} **vs. Geometric Brownian Motion:** - GBM: dX = μ·X·dt + σ·X·dW (multiplicative noise) - BM: dX = σ·dW (additive noise) - GBM stays positive, BM can be negative - GBM for stock prices, BM for price returns **vs. Ornstein-Uhlenbeck:** - OU: dX = -α·X·dt + σ·dW (mean-reverting) - BM: dX = σ·dW (no mean reversion) - OU has stationary distribution, BM does not - OU for interest rates, temperatures **vs. Random Walk:** - Random walk: discrete time, discrete space - Brownian motion: continuous time, continuous space - BM is limit of random walk as Δt, Δx → 0 with Δx² ~ Δt **vs. Lévy Process:** - Lévy: Can have jumps (Poisson, stable processes) - Brownian: Continuous paths only - Brownian is special case of Lévy (Gaussian Lévy) ## Theoretical Importance {.doc-section .doc-section-theoretical-importance} **Why Brownian Motion is Fundamental:** 1. **Donsker's Theorem (Functional CLT):** Random walk → Brownian motion as step size → 0 Makes BM the universal limit of random walks 2. **Lévy Characterization:** Any continuous martingale with quadratic variation t must be Brownian motion (up to time change) 3. **Feynman-Kac Formula:** Connects SDEs to PDEs via expectation Solution to heat equation = expected value over Brownian paths 4. **Girsanov Theorem:** Change of measure for SDEs Converts drift to diffusion via change of probability measure 5. **Reflection Principle:** P(max X(t) > a) = 2·P(X(T) > a) Used in barrier option pricing ## Limitations and Extensions {.doc-section .doc-section-limitations-and-extensions} **Standard Brownian Motion Limitations:** - No memory (Markov property may be unrealistic) - Gaussian increments (real data often heavy-tailed) - Continuous paths (no jumps) - Constant volatility (time-varying in reality) - Linear variance growth (subdiffusion/superdiffusion in complex media) **Extensions to Address Limitations:** - Fractional Brownian motion: Long-range dependence (Hurst parameter) - Lévy processes: Jumps and heavy tails - Stochastic volatility: Time-varying σ(t) - Rough volatility: σ follows rough process - Anomalous diffusion: Variance ~ t^α, α ≠ 1 ## See Also {.doc-section .doc-section-see-also} BrownianMotion2D : Two-dimensional independent Brownian motions BrownianBridge : Brownian motion conditioned on endpoints GeometricBrownianMotion : Multiplicative noise (stock prices) OrnsteinUhlenbeck : Mean-reverting Brownian motion ## Methods | Name | Description | | --- | --- | | [define_system](#cdesym.systems.builtin.stochastic.continuous.BrownianMotion.define_system) | Define standard Brownian motion dynamics. | ### define_system { #cdesym.systems.builtin.stochastic.continuous.BrownianMotion.define_system } ```python systems.builtin.stochastic.continuous.BrownianMotion.define_system(sigma=1.0) ``` Define standard Brownian motion dynamics. This sets up the stochastic differential equation: dX = σ·dW with zero drift and constant diffusion. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-----------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | sigma | float | Diffusion coefficient (must be positive) - Standard Brownian motion: σ = 1 - Units: [state]/√[time] - Controls noise intensity - Variance growth rate: σ² | `1.0` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|-------------------------------------------------------| | | ValueError | If sigma ≤ 0 (diffusion coefficient must be positive) | #### Notes {.doc-section .doc-section-notes} **System Structure:** This is a minimal stochastic system with: - Zero drift: f(x) = 0 - Constant diffusion: g(x) = σ - Single noise source: nw = 1 - No control input: nu = 0 - First-order dynamics: order = 1 **Special Properties:** - Pure diffusion process (no drift component) - Autonomous (no time dependence) - Additive noise (σ independent of state) - Unbounded state space (x ∈ ℝ) - No equilibrium points (non-stationary) **Standard vs. Scaled Brownian Motion:** When σ = 1, this is the standard Wiener process W(t). For σ ≠ 1, we have scaled Brownian motion: X(t) = σ·W(t) All Brownian motions can be written as: X(t) = x₀ + σ·W(t) where W(t) is standard Brownian motion. **Discretization:** The discrete-time equivalent is: X[k+1] = X[k] + σ·√(dt)·w[k] where w[k] ~ N(0,1) and dt is the time step. This is exact (no discretization error) for Brownian motion. **Comparison with Other SDEs:** Unlike most SDEs: - No drift means E[X(t)] = x₀ for all t (constant mean) - Constant diffusion means integration is particularly simple - Euler-Maruyama method is exact (no approximation error) - No stability concerns regardless of time step size