Bridge.jl

Stochastic calculus and univariate and multivariate stochastic processes/Markov processes in continuous time. See ./example/tutorial.jl for an introduction. I am personally interested in simulating diffusion bridges and doing Bayesian inference on discretely observed diffusion processes, but this package is written to be of general use and contributions are welcome.

• 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
• Ornstein-Uhlenbeck processes
• Geometric Brownian motion
• Bessel processes
• Gamma processes
• Basic stochastic calculus functionality (Ito integral, quadratic variation)
• Euler-Scheme and implicit methods (Rungekutta)

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.

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?> euler
search: euler euler! eulergamma default_worker_pool schedule @schedule

  euler(u, W, P) -> X

  Solve stochastic differential equation ``dX_t = b(t, X_t)dt + σ(t, X_t)dW_t, X_0 = u``
  using the Euler scheme.

Pre-defined processes defined are Wiener, WienerBridge, Gamma, LinPro (linear diffusion/generalized Ornstein-Uhlenbeck) and others.

It is also quite transparent how to add a new process:

# Define a diffusion process
struct OrnsteinUhlenbeck  <: ContinuousTimeProcess{Float64}
    β::Float64 # drift parameter (also known as inverse relaxation time)
    σ::Float64 # diffusion parameter
    function OrnsteinUhlenbeck(β::Float64, σ::Float64)
        isnan(β) || β > 0. || error("Parameter λ must be positive.")
        isnan(σ) || σ > 0. || error("Parameter σ must be positive.")
        new(β, σ)
    end
end

# define drift and diffusion coefficient of OrnsteinUhlenbeck
import Bridge: b, σ, a, transitionprob
Bridge.b(t,x, P::OrnsteinUhlenbeck) = -P.β*x
Bridge.σ(t, x, P::OrnsteinUhlenbeck) = P.σ
Bridge.a(t, x, P::OrnsteinUhlenbeck) = P.σ^2

# simulate OrnsteinUhlenbeck using Euler scheme
W = sample(0:0.01:10, Wiener()) 
X = euler(0.1, W, OrnsteinUhlenbeck(20.0, 1.0))

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 [email protected] and able to give some support.

Literature

F. v. d. Meulen, M. Schauer: Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Electronic Journal of Statistics 11 (1), 2017, doi:10.1214/17-EJS1290.

M. Schauer, F. v. d. Meulen, H. v. Zanten: Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli 23 (4A), 2017, doi:10.3150/16-BEJ833.