# 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),g!,
ones(3),(0.0,1.0))