cognitive/knowledge_base/mathematics/stochastic_processes.md

title	type	status	created	complexity	processing_priority	tags	semantic_relations
Stochastic Processes	concept	stable	2024-03-15	advanced	1	mathematics probability dynamics random_processes uncertainty	type links foundation_for dynamical_systems network_science statistical_physics type links implements probability_theory measure_theory differential_equations type links relates information_theory optimization_theory control_theory complex_systems

Stochastic Processes

Overview

Stochastic Processes provide a mathematical framework for analyzing systems with random dynamics. They form the foundation for understanding uncertainty and variability in complex systems, from molecular dynamics to ecological populations and neural activity.

Mathematical Foundation

Probability Spaces

Filtered Space

(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geq 0}, \mathbb{P})

where:

• \Omega is sample space
• \mathcal{F} is σ-algebra
• \{\mathcal{F}_t\} is filtration
• \mathbb{P} is probability measure

Martingales

\mathbb{E}[X_{t+s}|\mathcal{F}_t] = X_t

where:

• X_t is martingale process
• \mathcal{F}_t is filtration at time t

Stochastic Differential Equations

Itô Process

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

where:

• \mu is drift term
• \sigma is diffusion term
• W_t is Wiener process

Implementation

Stochastic Simulator

class StochasticProcess:
    def __init__(self,
                 drift: Callable,
                 diffusion: Callable,
                 dimension: int,
                 seed: Optional[int] = None):
        """Initialize stochastic process.
        
        Args:
            drift: Drift function μ(x,t)
            diffusion: Diffusion function σ(x,t)
            dimension: State dimension
            seed: Random seed
        """
        self.drift = drift
        self.diffusion = diffusion
        self.dim = dimension
        
        # Initialize random state
        self.rng = np.random.RandomState(seed)
        
        # Initialize process state
        self.state = np.zeros(dimension)
        self.time = 0.0
    
    def step(self,
            dt: float = 0.01) -> np.ndarray:
        """Evolve process one step using Euler-Maruyama.
        
        Args:
            dt: Time step
            
        Returns:
            state: Updated state
        """
        # Generate Wiener increment
        dW = self.rng.normal(0, np.sqrt(dt), self.dim)
        
        # Compute drift and diffusion
        drift = self.drift(self.state, self.time)
        diff = self.diffusion(self.state, self.time)
        
        # Update state
        self.state += drift * dt + diff * dW
        self.time += dt
        
        return self.state
    
    def simulate(self,
                duration: float,
                dt: float = 0.01) -> Tuple[np.ndarray, np.ndarray]:
        """Simulate process trajectory.
        
        Args:
            duration: Simulation duration
            dt: Time step
            
        Returns:
            times: Time points
            states: State trajectory
        """
        n_steps = int(duration / dt)
        times = np.linspace(0, duration, n_steps)
        states = np.zeros((n_steps, self.dim))
        
        for i in range(n_steps):
            states[i] = self.step(dt)
        
        return times, states

Markov Chain

class MarkovChain:
    def __init__(self,
                 transition_matrix: np.ndarray,
                 state_space: List[Any]):
        """Initialize Markov chain.
        
        Args:
            transition_matrix: State transition probabilities
            state_space: List of possible states
        """
        self.P = transition_matrix
        self.states = state_space
        self.n_states = len(state_space)
        
        # Validate transition matrix
        assert np.allclose(np.sum(self.P, axis=1), 1)
        
        # Initialize state
        self.current_state = 0
    
    def step(self) -> Any:
        """Take one step in chain.
        
        Returns:
            state: New state
        """
        # Sample next state
        self.current_state = np.random.choice(
            self.n_states,
            p=self.P[self.current_state]
        )
        
        return self.states[self.current_state]
    
    def compute_stationary(self) -> np.ndarray:
        """Compute stationary distribution.
        
        Returns:
            pi: Stationary distribution
        """
        # Solve eigenvalue problem
        eigenvals, eigenvecs = np.linalg.eig(self.P.T)
        
        # Find eigenvector for eigenvalue 1
        idx = np.argmin(np.abs(eigenvals - 1))
        pi = np.real(eigenvecs[:, idx])
        
        # Normalize
        pi = pi / np.sum(pi)
        
        return pi

Stochastic Differential Equations

class SDESolver:
    def __init__(self,
                 sde: StochasticProcess,
                 method: str = 'euler'):
        """Initialize SDE solver.
        
        Args:
            sde: Stochastic process
            method: Integration method
        """
        self.sde = sde
        self.method = method
    
    def solve(self,
             x0: np.ndarray,
             duration: float,
             dt: float = 0.01) -> Tuple[np.ndarray, np.ndarray]:
        """Solve SDE numerically.
        
        Args:
            x0: Initial condition
            duration: Integration duration
            dt: Time step
            
        Returns:
            times: Time points
            solution: Solution trajectory
        """
        # Initialize
        n_steps = int(duration / dt)
        times = np.linspace(0, duration, n_steps)
        solution = np.zeros((n_steps, self.sde.dim))
        solution[0] = x0
        
        # Integration loop
        for i in range(1, n_steps):
            if self.method == 'euler':
                solution[i] = self._euler_step(
                    solution[i-1], times[i-1], dt
                )
            elif self.method == 'milstein':
                solution[i] = self._milstein_step(
                    solution[i-1], times[i-1], dt
                )
        
        return times, solution
    
    def _euler_step(self,
                   x: np.ndarray,
                   t: float,
                   dt: float) -> np.ndarray:
        """Euler-Maruyama step."""
        dW = np.random.normal(0, np.sqrt(dt), self.sde.dim)
        
        return (x + 
                self.sde.drift(x, t) * dt +
                self.sde.diffusion(x, t) * dW)

Applications

Physical Systems

Molecular Dynamics

• Brownian motion
• Diffusion processes
• Chemical reactions
• Thermal fluctuations

Quantum Systems

• Quantum noise
• Open systems
• Decoherence
• Measurement

Biological Systems

Population Dynamics

• Birth-death processes
• Competition models
• Epidemic spread
• Genetic drift

Neural Systems

• Spike trains
• Synaptic noise
• Population coding
• Decision making

Financial Systems

Market Models

• Price processes
• Option pricing
• Risk assessment
• Portfolio theory

Economic Systems

• Agent-based models
• Game theory
• Strategic behavior
• Market dynamics

Advanced Topics

Random Fields

• Spatial processes
• Gaussian fields
• Point processes
• Lattice models

Filtering Theory

• Kalman filtering
• Particle filters
• State estimation
• Data assimilation

Large Deviations

• Rate functions
• Asymptotic behavior
• Rare events
• Phase transitions

Best Practices

Modeling

1. Choose appropriate noise
2. Validate assumptions
3. Consider timescales
4. Handle boundaries

Implementation

1. Numerical stability
2. Error control
3. Efficient sampling
4. Convergence checks

Analysis

1. Statistical tests
2. Uncertainty quantification
3. Robustness checks
4. Validation methods

Common Issues

Technical Challenges

1. Numerical instability
2. Rare event sampling
3. High dimensionality
4. Long-time behavior

Solutions

1. Adaptive stepping
2. Importance sampling
3. Dimension reduction
4. Multi-scale methods

Related Documentation

• probability_theory
• dynamical_systems
• statistical_physics
• information_theory
• control_theory
• complex_systems