SDAE Problems

Mathematical Specification of a Stochastic Differential-Algebraic Equation (SDAE) Problem

To define an SDAE, you simply define an SDE Problem with the forcing function f, the noise function g, a mass matrix M and the initial condition u₀ which define the SDAE in mass matrix form:

\[M du = f(u,p,t)dt + Σgᵢ(u,p,t)dWⁱ\]

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.

Nonsingular mass matrices correspond to constraint equations and thus a stochastic DAE.

Example

const mm_A = [-2.0 1 4
              4 -2 1
              0 0 0]
function f!(du, u, p, t)
    du[1] = u[1]
    du[2] = u[2]
    du[3] = u[1] + u[2] + u[3] - 1
end

function g!(du, u, p, t)
    @. du = 0.1
end

prob = SDEProblem(SDEFunction(f!, g!; mass_matrix = mm_A),
    ones(3), (0.0, 1.0))