Bridge.jl
Statistics and stochastic calculus for Markov processes in continuous time, include univariate and multivariate stochastic processes such as stochastic differential equations or diffusions (SDE's) or Levy processes.
I am personally interested in doing Bayesian inference on discretely observed diffusion processes, but this package is written to be of general use and contributions are welcome. Specifically for our code for parameter inference for diffusion processes from discrete data or passage times, check out the dependent package BridgeSDEInference.jl. The statistical method relies a lot on simulating conditional diffusions (so called "difffusion bridges"). See ./example/tutorial.jl for a more general introduction into working with this package.
 Define and simulate diffusion processes in one or more dimension
 Continuous and discrete likelihood using Girsanovs theorem and transition densities
 Monte Carlo sample diffusion bridges, diffusion processes conditioned to hit a point v at a prescribed time T
 Brownian motion in one and more dimensions
 OrnsteinUhlenbeck processes and OrnsteinUhlenbeck bridges
 Bessel processes
 Gamma processes
 Inhomogenous poisson process
 Basic stochastic calculus functionality (Ito integral, quadratic variation)
 EulerScheme and implicit methods (RungeKutta)
 Levydriven SDEs
 Continuousdiscrete filtering for partially observed diffusion processes
The layout/api was originally written to be compatible with Simon Danisch's package FixedSizeArrays.jl. It was refactored to be compatible with StaticArrays.jl by Dan Getz.
Some SDE and ODE solvers in Bridge are accessible with the JuliaDiffEq
common interface via BridgeDiffEq.jl.
The example programs in the example/ directory have additional dependencies: ConjugatePriors and a plotting library.
Introduction
The key objects introduced are the abstract type ContinuousTimeProcess{T}
parametrised by the state space of the path, for example T == Float64
and various structs
suptyping it, for example Wiener{Float64}
for a real Brownian motion. These play roughly a similar role as types subtyping Distribution
in the Distributions.jl package.
Secondly, the struct
struct SamplePath{T}
tt::Vector{Float64}
yy::Vector{T}
SamplePath{T}(tt, yy) where {T} = new(tt, yy)
end
serves as container for sample path returned by direct and approximate samplers (sample
, euler
, ...).
tt
is the vector of the grid points of the simulation and yy
the corresponding vector of states.
Help is available at the REPL:
help?> GammaProcess
search: GammaProcess LocalGammaProcess VarianceGammaProcess
GammaProcess 

A GammaProcess with jump rate ν(x) = γ x⁻¹ exp(λ x), Here Examples

Predefined processes defined are
Wiener
, WienerBridge
, Gamma
, LinPro
(linear diffusion/generalized OrnsteinUhlenbeck) and others.
It is also quite transparent how to add a new process:
using Bridge
using Plots
# Define a diffusion process
struct OrnsteinUhlenbeck <: ContinuousTimeProcess{Float64}
β::Float64 # drift parameter (also known as inverse relaxation time)
σ::Float64 # diffusion parameter
end
# define drift and diffusion coefficient of OrnsteinUhlenbeck
Bridge.b(t, x, P::OrnsteinUhlenbeck) = P.β*x
Bridge.σ(t, x, P::OrnsteinUhlenbeck) = P.σ
# simulate OrnsteinUhlenbeck using Euler scheme
W = sample(0:0.01:10, Wiener())
X = solve(EulerMaruyama(), 0.1, W, OrnsteinUhlenbeck(2.0, 1.0))
plot(X, label="X")
# Levy (DifferenceGamma process) driven OrnsteinUhlenbeck
Z = sample(0:0.01:10, GammaProcess(100.0,10.0))
Z.yy .= sample(0:0.01:10, GammaProcess(100.0,10.0)).yy
Y = solve(EulerMaruyama(), 0.1, Z, OrnsteinUhlenbeck(2.0, 1.0))
plot(Y, label="Y")
Feedback and Contributing
See the documentation for more functionality and issue #12 (Feedback and Contribution) for coordination of the development. Bridge is free software under the MIT licence. If you use Bridge.jl in a closed environment I’d be happy to hear about your use case in a mail to moritzschauer@web.de and able to give some support.
Literature
F. v. d. Meulen, M. Schauer: Bayesian estimation of discretely observed multidimensional diffusion processes using guided proposals. Electronic Journal of Statistics 11 (1), 2017, doi:10.1214/17EJS1290.
M. Schauer, F. v. d. Meulen, H. v. Zanten: Guided proposals for simulating multidimensional diffusion bridges. Bernoulli 23 (4A), 2017, doi:10.3150/16BEJ833.