DifferenceEquations.jl

DifferenceEquations.jl solves initial value problems for deterministic and stochastic difference equations, with differentiable solvers and filters. Automatic differentiation is powered by Enzyme.jl (reverse and forward mode). The package is part of the SciML ecosystem.

Features

Linear, quadratic, and generic state-space models – LinearStateSpaceProblem, QuadraticStateSpaceProblem, PrunedQuadraticStateSpaceProblem, and StateSpaceProblem with user-defined callbacks.
Kalman filter for computing the marginal log-likelihood of linear Gaussian models via KalmanFilter.
Differentiable via Enzyme.jl and ForwardDiff.jl – Enzyme reverse/forward mode for all problem sizes; ForwardDiff as a lightweight alternative for small models (N ≤ 5).
StaticArrays support for small models where heap allocations dominate runtime.
Workspace API – StateSpaceWorkspace with init / solve! for allocation-free repeated solves (useful inside AD and tight loops).
SciML ecosystem integration – EnsembleProblem for Monte Carlo, plot recipes, DataFrame conversion, symbolic indexing, and remake.

Installation

To install DifferenceEquations.jl, use the Julia package manager:

using Pkg
Pkg.add("DifferenceEquations")

Quick Example

using DifferenceEquations, LinearAlgebra
A = [0.95 0.1; 0.0 0.2]
B = [0.0; 0.01;;]
u0 = zeros(2)
T = 10
prob = LinearStateSpaceProblem(A, B, u0, (0, T))
sol = solve(prob)
sol.u[end]  # final state

2-element Vector{Float64}:
 -0.0012340755937890878
 -0.0012274738683213534

Mathematical Background

The general class of discrete-time state-space models supported by this package takes an initial condition $u_0$ and an evolution equation

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

for transition function $f$, noise coefficient $g$, and IID noise shocks $w_{n+1}$. The parameter vector $p$ is potentially differentiable.

An optional observation equation relates the latent state to measured data:

\[z_n = h(u_n, p, t_n) + v_n\]

where $v_n$ is observation noise and $z_n$ may have a different dimension from $u_n$.

Specializations

Linear: $f(u) = A\,u$, $g(u) = B$, $h(u) = C\,u$. Solved by DirectIteration or KalmanFilter. See LinearStateSpaceProblem.
Quadratic: Adds second-order terms $u^\top A_2\, u$ to both transition and observation. Useful for pruned perturbation solutions of DSGE models. See QuadraticStateSpaceProblem and PrunedQuadraticStateSpaceProblem.
Generic: User-supplied transition and observation callbacks. See StateSpaceProblem.

When the system is linear, the shocks are Gaussian, and a Gaussian prior is provided, the KalmanFilter computes the exact marginal log-likelihood. For all other cases, DirectIteration iterates the state forward and (optionally) accumulates a joint log-likelihood.

Note

Boundary value problems and difference-algebraic equations are not in scope.

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/github-runners/demeter4-22/_work/DifferenceEquations.jl/DifferenceEquations.jl/docs/Project.toml`
  [a93c6f00] DataFrames v1.8.2
  [2b5f629d] DiffEqBase v7.5.5
  [e0ca9c66] DifferenceEquations v1.1.0 `~/github-runners/demeter4-22/_work/DifferenceEquations.jl/DifferenceEquations.jl`
  [31c24e10] Distributions v0.25.126
  [e30172f5] Documenter v1.17.0
  [d12716ef] DocumenterInterLinks v1.1.0
  [7da242da] Enzyme v0.13.157
  [f6369f11] ForwardDiff v1.4.1
  [7f7a1694] Optimization v5.6.1
  [36348300] OptimizationOptimJL v0.4.14
  [91a5bcdd] Plots v1.41.6
  [90137ffa] StaticArrays v1.9.18
  [37e2e46d] LinearAlgebra v1.12.0
  [9a3f8284] Random v1.11.0

and using this machine and Julia version.

Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 64 × AMD EPYC 9354 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 64 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/chrisrackauckas/github-runners/demeter4-22/.julia

A more complete overview of all dependencies and their versions is also provided.

Status `~/github-runners/demeter4-22/_work/DifferenceEquations.jl/DifferenceEquations.jl/docs/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

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