Problem Types

AbstractStateSpaceProblem <: AbstractDEProblem

Abstract supertype for all discrete-time state-space problems in DifferenceEquations.jl.

Subtypes include LinearStateSpaceProblem, QuadraticStateSpaceProblem, PrunedQuadraticStateSpaceProblem, and StateSpaceProblem.

LinearStateSpaceProblem(A, B, u0, tspan[, p]; kwargs...)

Define a linear time-invariant state-space model:

\[u_{n+1} = A \, u_n + B \, w_{n+1}\]

with optional observation equation $z_n = C \, u_n + v_n$.

Positional Arguments

• A: Transition matrix (n×n).
• B: Noise input matrix (n×k), or nothing for deterministic dynamics.
• u0: Initial state vector, or a Distribution for random initial conditions.
• tspan: Time span as (t0, t_end) with integer distance (e.g., (0, T)).
• p: Parameters (default: NullParameters()).

Keyword Arguments

• C: Observation matrix (m×n). If nothing, no observations are computed.
• observables_noise: Observation noise covariance matrix (AbstractMatrix, e.g. Diagonal(d) or Symmetric(H * H')).
• observables: Observed data as Vector{Vector{T}} for likelihood computation.
• noise: Fixed noise sequence as Vector{Vector{T}}. If nothing, noise is drawn randomly.
• u0_prior_mean: Prior mean for Kalman filtering.
• u0_prior_var: Prior covariance matrix for Kalman filtering.
• syms: State variable names (e.g., (:x, :y)) for symbolic indexing.
• obs_syms: Observation variable names for symbolic indexing.

Notes

• Providing u0_prior_mean, u0_prior_var, observables, and observables_noise (with noise = nothing) triggers automatic selection of KalmanFilter.
• The observables timing convention: observations correspond to $z_1, z_2, \ldots$ (starting from the second state), so pass T observations for a tspan of (0, T).

See also: StateSpaceProblem, QuadraticStateSpaceProblem, DirectIteration, KalmanFilter.

QuadraticStateSpaceProblem(A_0, A_1, A_2, B, u0, tspan[, p]; kwargs...)

Define a second-order (quadratic) state-space model:

\[u_{n+1} = A_0 + A_1 \, u_n + u_n^\top A_2 \, u_n + B \, w_{n+1}\]

with optional observation equation $z_n = C_0 + C_1 \, u_n + u_n^\top C_2 \, u_n + v_n$.

Positional Arguments

• A_0: Constant drift vector (length n).
• A_1: Linear transition matrix (n×n).
• A_2: Quadratic transition tensor (n×n×n). Entry A_2[i,:,:] gives the matrix for the i-th element of the quadratic term.
• B: Noise input matrix (n×k), or nothing.
• u0: Initial state vector.
• tspan: Time span as (t0, t_end).

Keyword Arguments

• C_0, C_1, C_2: Observation equation coefficients (analogous to A_0, A_1, A_2).
• observables_noise, observables, noise, syms, obs_syms: Same as LinearStateSpaceProblem.

References

• Andreasen, Fernandez-Villaverde, and Rubio-Ramirez (2017), "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications."

See also: PrunedQuadraticStateSpaceProblem, LinearStateSpaceProblem.

PrunedQuadraticStateSpaceProblem(A_0, A_1, A_2, B, u0, tspan[, p]; kwargs...)

Define a pruned second-order state-space model. Unlike QuadraticStateSpaceProblem, the quadratic terms operate on a separate linear-part state $u_f$ rather than the full state:

\[u_f^{n+1} = A_1 \, u_f^n + B \, w_{n+1}\]

\[u_{n+1} = A_0 + A_1 \, u_n + (u_f^n)^\top A_2 \, u_f^n + B \, w_{n+1}\]

The observation equation similarly uses $u_f$: $z_n = C_0 + C_1 \, u_n + (u_f^n)^\top C_2 \, u_f^n + v_n$.

This pruning approach prevents explosive dynamics in higher-order perturbation solutions. Arguments are identical to QuadraticStateSpaceProblem.

References

• Andreasen, Fernandez-Villaverde, and Rubio-Ramirez (2017), "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications."

See also: QuadraticStateSpaceProblem.

StateSpaceProblem(transition, observation, u0, tspan[, p]; n_shocks, kwargs...)

Define a generic state-space model with user-provided callback functions:

\[u_{n+1} = f(u_n, w_{n+1}, p, t_n), \quad z_n = g(u_n, p, t_n)\]

Positional Arguments

• transition: Callback f!!(x_next, x, w, p, t) -> x_next. For mutable arrays, mutate x_next in place and return it; for immutable arrays (e.g., SVector), return a new value.
• observation: Callback g!!(y, x, p, t) -> y, or nothing for no observations.
• u0: Initial state vector, or a Distribution for random initial conditions.
• tspan: Time span as (t0, t_end) with integer distance.
• p: Parameters passed to the callbacks (default: NullParameters()).

Keyword Arguments

• n_shocks::Int: Number of noise dimensions (required).
• n_obs::Int: Number of observation dimensions (default: 0).
• observables_noise: Observation noise covariance matrix (AbstractMatrix, e.g. Diagonal(d) or Symmetric(H * H')).
• observables: Observed data as Vector{Vector{T}}.
• noise: Fixed noise sequence as Vector{Vector{T}}.
• syms: State variable names for symbolic indexing.
• obs_syms: Observation variable names for symbolic indexing.

See also: LinearStateSpaceProblem, DirectIteration.