Introduction (Available as Jupyter notebook at https://github.com/mschauer/MicrostructureNoise.jl/blob/master/example/MicrostructureNoise.ipynb)
MicrostructureNoise is a Julia package for Bayesian volatility estimation in presence of market microstructure noise implementing the methodology described in our new preprint:
Shota Gugushvili, Frank van der Meulen, Moritz Schauer, and Peter Spreij: Nonparametric Bayesian volatility learning under microstructure noise. arxiv:1805.05606, 2018. This blogpost gives a short introduction.
Description MicrostructureNoise estimates the volatility function $s$ of the stochastic differential equation

This IJulia script converted blog-post gives a tour for my package Bridge with focus on multivariate stochastic differential equations. I use Makie.jl for the visualisations.
Installation To run this IJulia script install Bridge and checkout the master branch to run.
# Pkg.add("Bridge") # Pkg.checkout("Bridge") The installation of Makie is a bit tricky and is detailed in the README.md file of Makie.
Setting the scene The next few lines load the needed packages and some scripts.

Introduction Mathieu Besançon made a nice blog post on Chris’ DifferentialEquations ecosystem, https://mbesancon.github.io/post/2017-12-14-diffeq-julia/ .
One of the examples is simple model tracking individuals in a population with three states $S$, $I$, $R$.
Let $u = (u_1, u_2, u_2)$ be the vector of individuals in these states.
At any moment an individual of state $A$ can convince a individual of state $B$ to join the fraction $A$ $$ S,I \mapsto 2I $$ This reaction happens proportional to the number of possible pairs $A, B$ which can match up, so proportional to $u_1 u_2$ with proportionality constant $\alpha$.