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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: -1.0876461595783575

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:

Mathematical Foundation

All noise processes in this package maintain mathematical rigor through:

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

Contributing

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
Status `~/work/DiffEqNoiseProcess.jl/DiffEqNoiseProcess.jl/docs/Project.toml`
  [77a26b50] DiffEqNoiseProcess v5.26.0 `~/work/DiffEqNoiseProcess.jl/DiffEqNoiseProcess.jl`
  [31c24e10] Distributions v0.25.123
  [e30172f5] Documenter v1.16.1
  [0bca4576] SciMLBase v2.132.0
  [10745b16] Statistics v1.11.1
  [9a3f8284] Random v1.11.0
and using this machine and Julia version.
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 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.
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You can also download the manifest file and the project file.