Overview

These are the companion notes for the paper "Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises", by Peter E. Kloeden and Ricardo M. S. Rosa.

We briefly review the main results of the paper and reveal the numerical codes used for the examples presented in the paper.

The codes are written in the Julia programming language. The examples are based on the local package RODEConvergence.jl, residing on the folder src/ of the github repository. It contains the implementation of the Euler method for scalar equations and systems of equations and all the helper functions needed to defined the noises, setup the problem, check the convergence of the method, and plot the desired figures. The methods defined in this local package can be seen in the section API.

The local package RODEConvergence.jl used here is not a registered package in Julia, as it is only used here as a companion code for the paper, with the bare minimum needed for it. For a much more complete package for solving Random ODEs and other types of differential equations, check the SciML: Open Source Software for Scientific Machine Learning ecosystem.

For the code used here, it is illustrative to see the first example Homogenous linear RODE with a Wiener process noise coefficient, in which all the steps are explained in more details.

We use a few standard libraries (Random, LinearAlgebra, Statistics, Test) and a few packages (JuliaStats/Distributions.jl, JuliaMath/FFTW.jl, JuliaPlots/Plots.jl).

This documentation makes use of Documenter.jl and Literate.jl, with the help of LiveServer.jl and Revise.jl, during development.

Some extra material uses JuliaCI/BenchmarkTools.jl.