SDE Problems

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:

\[du = f(u,p,t)dt + Σgᵢ(u,p,t)dWⁱ\]

f and g should be specified as f(u,p,t) and g(u,p,t) 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.

Problem Type

Wraps the data which defines an SDE problem

\[u = f(u,p,t)dt + Σgᵢ(u,p,t)dWⁱ\]

with initial condition $u0$.

Constructors

SDEProblem(f::SDEFunction,g,u0,tspan,noise=WHITE_NOISE,noise_rate_prototype=nothing)
SDEProblem{isinplace}(f,g,u0,tspan,noise=WHITE_NOISE,noise_rate_prototype=nothing) : Defines the SDE with the specified functions. The default noise is WHITE_NOISE. isinplace optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

For specifying Jacobians and mass matrices, see the DiffEqFunctions page.

Fields

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.

Example Problems

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)

DiffEqProblemLibrary.prob_sde_linear
DiffEqProblemLibrary.prob_sde_2Dlinear
DiffEqProblemLibrary.prob_sde_wave
DiffEqProblemLibrary.prob_sde_lorenz
DiffEqProblemLibrary.prob_sde_cubic
DiffEqProblemLibrary.prob_sde_additive
DiffEqProblemLibrary.prob_sde_additivesystem