DiffEqNoiseProcess.jl: Noise Processes for Stochastic Modeling

DiffEqNoiseProcess.jl provides a comprehensive suite of noise processes for stochastic differential equations and random differential equations. The NoiseProcess types are distributionally-exact, meaning they are generated directly according to their analytical distributions rather than as solutions to SDEs.

Key Features

Mathematically rigorous: All processes maintain distributional exactness
Comprehensive collection: Wiener processes, Ornstein-Uhlenbeck, geometric Brownian motion, compound Poisson, and more
Flexible interface: Works seamlessly with the DifferentialEquations.jl ecosystem
Memory efficient: Includes virtual processes and wrappers for large-scale simulations
Bridging support: Conditional processes for specific boundary conditions

Installation

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

using Pkg
Pkg.add("DiffEqNoiseProcess")

Quick Start

Basic Wiener Process

Create and simulate a standard Brownian motion:

using DiffEqNoiseProcess, SciMLBase

# Create a Wiener process: WienerProcess(t0, W0, Z0)
W = WienerProcess(0.0, 0.0, 1.0)

# Simulate it over time interval [0, 1]
prob = NoiseProblem(W, (0.0, 1.0))
sol = solve(prob; dt = 0.01)

println("Final Brownian motion value: $(sol.u[end])")

Final Brownian motion value: 0.6132439379743437

Using Noise in SDE Problems

Noise processes integrate directly with SDE problems:

using SciMLBase

# Define SDE: dX = μX dt + σX dW (geometric Brownian motion)
f(u, p, t) = 0.05 * u  # 5% drift
g(u, p, t) = 0.2 * u   # 20% volatility

# Use custom geometric Brownian motion noise
gbm_noise = GeometricBrownianMotionProcess(0.05, 0.2, 0.0, 100.0, 100.0)

# Create SDE problem (would normally solve this)
# prob = SDEProblem(f, g, 100.0, (0.0, 1.0), noise = gbm_noise)

t: 1-element Vector{Float64}:
 0.0
u: 1-element Vector{Float64}:
 100.0

Ensemble Simulations

Generate multiple noise realizations:

# Create ensemble problem for Monte Carlo simulation
enprob = EnsembleProblem(prob)
ensemble_sol = solve(enprob; dt = 0.01, trajectories = 100)

println("Generated $(length(ensemble_sol.u)) noise trajectories")

Generated 100 noise trajectories

Available Noise Processes

DiffEqNoiseProcess.jl includes a rich collection of noise processes:

Classic Processes

WienerProcess: Standard Brownian motion
RealWienerProcess: Scalar Brownian motion
CorrelatedWienerProcess: Multi-dimensional correlated noise
OrnsteinUhlenbeckProcess: Mean-reverting process
GeometricBrownianMotionProcess: For financial modeling
CompoundPoissonProcess: Jump processes

Bridge Processes

BrownianBridge: Brownian motion with fixed endpoints
GeometricBrownianBridge: GBM with boundary conditions

Advanced Features

NoiseFunction: Custom noise from functions
NoiseGrid: Noise from pre-computed data
NoiseWrapper: Reuse previous simulations
VirtualBrownianTree: Memory-efficient alternative
NoiseApproximation: Colored noise via SDE solutions

Documentation Structure

This documentation is organized as follows:

Basic Usage Tutorial: Get started with fundamental concepts
Noise Process Types: Comprehensive guide to all available processes
Advanced Features: Custom processes, wrappers, and optimization
API Reference: Complete function and type documentation
Interface API: Step management and internal functions

Mathematical Foundation

All noise processes in this package maintain mathematical rigor through:

Distributional exactness: Processes follow their theoretical distributions exactly
Rejection Sampling with Memory (RSWM): Ensures correctness when adaptive stepping is used
Proper bridging: Conditional processes respect boundary conditions
Interpolation consistency: Values between grid points maintain distributional properties

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 `~/work/DiffEqNoiseProcess.jl/DiffEqNoiseProcess.jl/docs/Project.toml`
  [77a26b50] DiffEqNoiseProcess v5.33.0 `~/work/DiffEqNoiseProcess.jl/DiffEqNoiseProcess.jl`
  [31c24e10] Distributions v0.25.129
  [e30172f5] Documenter v1.17.0
  [0bca4576] SciMLBase v3.31.0
  [10745b16] Statistics v1.11.1
  [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: 4 × Intel(R) Xeon(R) Platinum 8370C CPU @ 2.80GHz
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, icelake-server)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)

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

Status `~/work/DiffEqNoiseProcess.jl/DiffEqNoiseProcess.jl/docs/Manifest.toml`
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.