Continuous stochastic processes

A continuous stochastic process is a stochastic state variable whose class bundles both the discretized grid and its transition mechanism. Unlike ordinary grids, a process computes its own grid points and transition matrix from a distribution and its parameters — so you place it in states and never in state_transitions.

Process classes follow the naming convention <Distribution><Kind>Process and are imported directly from lcm:

from lcm import NormalIIDProcess, TauchenAR1Process

*IIDProcess — independent draws each period.
*AR1Process — an AR(1) process with a chosen discretization scheme.

IID Processes¶

Processes whose draws are independent across periods.

NormalIIDProcess¶

Discretized normal distribution $N(\mu, \sigma^2)$ .

NormalIIDProcess(n_points=7, gauss_hermite=False, mu=0.0, sigma=1.0, n_std=2.0)

Parameters:

n_points: Number of grid points.
gauss_hermite: If True, use Gauss-Hermite quadrature nodes and weights. If False, use equally spaced points spanning $\mu \pm n_\text{std} \cdot \sigma$ .
mu: Mean of the distribution.
sigma: Standard deviation.
n_std: Number of standard deviations for the grid boundary. Mutually exclusive with gauss_hermite=True.

LogNormalIIDProcess¶

Discretized log-normal distribution where $\ln X \sim N(\mu, \sigma^2)$ .

LogNormalIIDProcess(n_points=7, gauss_hermite=False, mu=0.0, sigma=0.5, n_std=2.0)

Same parameters as NormalIIDProcess. Grid points are exp() of the underlying normal grid.

UniformIIDProcess¶

Discretized uniform distribution $U(\text{start}, \text{stop})$ . Both endpoints are included in the grid.

UniformIIDProcess(n_points=5, start=0.0, stop=1.0)

Equally spaced points with uniform probabilities (all 1/n_points).

NormalMixtureIIDProcess¶

Two-component normal mixture: $\varepsilon \sim p_1 \, N(\mu_1, \sigma_1^2) + (1 - p_1) \, N(\mu_2, \sigma_2^2)$ .

NormalMixtureIIDProcess(
    n_points=9,
    n_std=2.0,
    p1=0.9,
    mu1=0.0,
    sigma1=0.1,
    mu2=0.0,
    sigma2=1.0,
)

Grid spans the mixture mean $\pm n_\text{std}$ mixture standard deviations.

AR(1) Processes¶

Processes with serial correlation. The process is $y_t = \mu + \rho \, y_{t-1} + \varepsilon_t$ . The innovation distribution depends on the class:

TauchenAR1Process and RouwenhorstAR1Process: $\varepsilon_t \sim N(0, \sigma^2)$
TauchenNormalMixtureAR1Process: $\varepsilon_t \sim p_1 \, N(\mu_1, \sigma_1^2) + (1 - p_1) \, N(\mu_2, \sigma_2^2)$

TauchenAR1Process¶

Discretization via Tauchen (1986). Uses CDF-based transition probabilities.

TauchenAR1Process(
    n_points=7,
    gauss_hermite=False,
    rho=0.9,
    sigma=0.1,
    mu=0.0,
    n_std=2.0,
)

gauss_hermite: If True, use Gauss-Hermite quadrature nodes.
n_std: Number of unconditional standard deviations for the grid boundary. Mutually exclusive with gauss_hermite=True.

RouwenhorstAR1Process¶

Discretization via Rouwenhorst (1995) / Kopecky & Suen (2010). Better for highly persistent processes ( $\rho$ close to 1).

RouwenhorstAR1Process(n_points=7, rho=0.95, sigma=0.1, mu=0.0)

TauchenNormalMixtureAR1Process¶

AR(1) with mixture-of-normals innovations, discretized via Tauchen. Following Fella et al. (2019).

TauchenNormalMixtureAR1Process(
    n_points=9,
    rho=0.9,
    mu=0.0,
    n_std=2.0,
    p1=0.9,
    mu1=0.0,
    sigma1=0.1,
    mu2=0.0,
    sigma2=1.0,
)

Using a Continuous Stochastic Process in a Regime¶

A process goes in states. It must not appear in state_transitions — it manages its own transition:

from lcm import LinSpacedGrid, NormalIIDProcess, Regime

working = Regime(
    transition=next_regime,
    states={
        "wealth": LinSpacedGrid(start=0, stop=100, n_points=50),
        "income_shock": NormalIIDProcess(
            n_points=5,
            gauss_hermite=False,
            mu=0.0,
            sigma=1.0,
            n_std=2.0,
        ),
    },
    state_transitions={
        "wealth": next_wealth,
        # income_shock does NOT appear here — it manages its own transitions
    },
    actions={...},
    functions={
        "utility": utility,
        "earnings": lambda wage, income_shock: wage * jnp.exp(income_shock),
    },
)

Key Rules¶

A process goes in states — it defines the values the shock can take.
A process must not appear in state_transitions — placing it there is a validation error.
Process parameters can be specified at construction or deferred to runtime (set to None).
Runtime params follow the same hierarchy as other params (see Parameters).

Runtime Parameters¶

Set distribution parameters to None at construction to supply them at runtime:

NormalIIDProcess(n_points=5, gauss_hermite=False, mu=None, sigma=None, n_std=None)

Then supply the values in the params dict, keyed by regime name:

params = {
    "regime_name": {
        "mu": 0.0,
        "sigma": 1.0,
        "n_std": 2.0,
    },
}

n_points and gauss_hermite are structural, not distribution parameters — they must always be given at construction.