Mathematical Specification of a SDE Problem
To define an SDE Problem, you simply need to give the forcing function $f$, the noise function
g, and the initial condition $u₀$ which define an SDE:
g should be specified as
g(t,u) respectively, and
u₀ should be an AbstractArray whose geometry matches the desired geometry of
u. Note that we are not limited to numbers or vectors for
u₀; one is allowed to provide
u₀ as arbitrary matrices / higher dimension tensors as well. A vector of
gs can also be defined to determine an SDE of higher Ito dimension.
Wraps the data which defines an SDE problem
with initial condition $u0$.
SDEProblem(f,g,u0,tspan,noise=WHITE_NOISE,noise_rate_prototype=nothing) : Defines the SDE with the specified functions. The default noise is
f: The drift function in the SDE.
g: The noise function in the SDE.
u0: The initial condition.
tspan: The timespan for the problem.
noise: The noise process applied to the noise upon generation. Defaults to Gaussian white noise. For information on defining different noise processes, see the noise process documentation page
noise_rate_prototype: A prototype type instance for the noise rates, that is the output
g. It can be any type which overloads
A_mul_B!with itself being the middle argument. Commonly, this is a matrix or sparse matrix. If this is not given, it defaults to
nothing, which means the problem should be interpreted as having diagonal noise.
callback: A callback to be applied to every solver which uses the problem. Defaults to nothing.
mass_matrix: The mass-matrix. Defaults to
Examples problems can be found in DiffEqProblemLibrary.jl.
To use a sample problem, such as
prob_sde_linear, you can do something like:
# Pkg.add("DiffEqProblemLibrary") using DiffEqProblemLibrary prob = prob_sde_linear sol = solve(prob)
where β=1.01, α=0.87, and initial condtion u0=1/2, with solution
8 linear SDEs (as a 4x2 matrix):
where β=1.01, α=0.87, and initial condtion u0=1/2 with solution
and initial condition
u0=1.0 with solution
Lorenz Attractor with additive noise
with $σ=10$, $ρ=28$, $β=8/3$, $α=3.0$ and inital condition $u0=[1;1;1]$.
and initial condtion u0=1/2, with solution
Additive noise problem
and initial condition u0=1.0 with α=0.1 and β=0.05, with solution
A multiple dimension extension of