DiffusionDefinition

Utility functions for defining diffusion processes in the fewest lines of code. The package is designed to work seamlessly with BridgeSDEInference.jl.

Minimal example

For instance, to define a Lorenz system it is enough to write

@diffusion_process Lorenz begin
    :dimensions
    process --> 3
    wiener --> 3

    :parameters
    _ --> (3, Float64)
    σ --> Float64

    :additional
    constdiff --> true
end

b(t, x, P::Lorenz) = @SVector [
    P.p1*(x[2]-x[1]),
    P.p2*x[1] - x[2] - x[1]*x[3],
    x[1]*x[2] - P.p3*x[3]
]

σ(t, x, P::Lorenz) = SDiagonal(P.σ, P.σ, P.σ)

It is also possible to include template parameters, for instance:

@diffusion_process Lorenz{T} begin
    :dimensions
    process --> 3
    wiener --> 3

    :parameters
    _ --> (3, T)
    σ --> Float64

    :additional
    constdiff --> true
end

We also provide some examples of pre-defined diffusion processes that can be imported without having to write any code with:

@load_diffusion :lorenz

To see a list of all pre-defined examples call

@load_diffusion

We additionally provide some functionality for sampling trajectories. For instance, to sample a three-dimensional standard Brownian motion use:

const DD = DiffusionDefinition
tt = collect(0.0:0.01:1.0)
wiener_path = rand(tt, Wiener(), zero(DD.ℝ{3}))

The wiener path can then be used in an Euler-Maruyama scheme to compute a trajectory under a given diffusion law:

XX = trajectory(tt, DD.ℝ{3})
P = Lorenz(28.0, 10.0, 8.0/3.0, 2.0)
DD.solve!(XX, wiener_path, P, zero(DD.ℝ{3}))

See the documentation for a comprehensive overview.