SDE Problems

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

Parameters are optional, and if not given then a NullParameters() singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback in the problem, then that callback will be added in every solve call.

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

Fields

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:

#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.SDEProblemLibrary
# load problems
SDEProblemLibrary.importsdeproblems()
prob = SDEProblemLibrary.prob_sde_linear
sol = solve(prob)
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