Consider a system of quasi-linear stochastic differential-algebraic equations (SDAEs) of the form

with a singular matrix

$\mathit{E}\in {\mathbb{R}}^{n\times n}$ of rank

$d<n$. The function

${\mathit{f}}_{0}\in {\mathcal{C}}^{k}({\mathbb{D}}_{\mathit{x}},{\mathbb{R}}^{n})$ (for some

$k\ge 1$) is known as drift, and

${\mathit{f}}_{1},\dots ,{\mathit{f}}_{m}\in {\mathcal{C}}^{k+1}({\mathbb{D}}_{\mathit{x}},{\mathbb{R}}^{n})$ are the diffusions. Here,

$\mathbb{I}:=[{t}_{0},{t}_{f}]\subseteq {\mathbb{R}}^{+}$ is a closed time interval and

${\mathbb{D}}_{\mathit{x}}\subseteq {\mathbb{R}}^{n}$ is an open set. Furthermore,

${w}_{t}^{j}$ (for

$j=1,\dots ,m$) form an

m-dimensional Wiener process defined on the complete probability space

$(\Omega ,\mathcal{F},\mathbb{P})$ with a filtration

${({\mathcal{F}}_{t})}_{t\ge {t}_{0}}$, where

$\mathcal{F}=\mathcal{B}(\Omega )$ is the

$\sigma $-algebra of Borel sets in

$\Omega $. Each

j-th Wiener process

${w}_{t}$ is understood as a process such that

${w}_{t}(\omega )=\omega (t)$, where

$\omega \in \Omega $, i.e., the elements of

$\Omega $ are identified with the paths. We recall that the Wiener process has stationary independent increments, such that

$({w}_{t}-{w}_{s})\sim \mathcal{N}(0,t-s)$, i.e., is a Gaussian random variable for all

$0\le s<t$, such that

We assume consistent initial values

${\mathit{x}}_{{t}_{0}}={\mathit{x}}_{0}$ independent of the Wiener processes

${w}_{t}^{j}$ and with finite second moments [

6]. A solution

${\mathit{x}}_{t}=\mathit{x}(t,\omega )$ of (

2) is an

n-dimensional vector-valued Markovian stochastic process depending on

$t\in \mathbb{I}$ and

$\omega \in \Omega $ (the parameter

$\omega $ is commonly omitted in the notation of

$\mathit{x}$). Such a solution can be defined as strong solution if it fulfills the following conditions, see e.g., [

9,

13].

Because of the presence of the algebraic equations that are associated with the kernel of

$\mathit{E}$, the solution components associated with these equations would be directly affected by white noise and not integrated. In order to avoid this, a reasonable restriction is to ensure that the noise sources do not appear in the algebraic constraints. According to [

8,

9], this assumption can be accomplished in SDAE systems whose deterministic part

is a DAE with tractability index-1 [

9,

22], in which the constraints are regularly and globally uniquely solvable for parts of the solution vector. We slightly modify this assumption and consider SDAE systems whose deterministic part (

3) is a regular strangeness-free DAE [

3] i.e., it has differentiation index-1. A system with these characteristics can be transformed into a semi-explicit form by means of an appropriate kinematic equivalence transformation [

1,

19], i.e., there exist pointwise orthogonal matrix functions

$\mathcal{P}$ and

$\mathcal{Q}$ such that, pre-multiplying (

1) by

$\mathcal{P}$, and changing the variables

${\mathit{x}}_{t}$ according to the transformation

${\mathit{x}}_{t}=\mathcal{Q}{\widehat{\mathit{x}}}_{t}$ one obtains a system in semi-explicit form

where

${\widehat{\mathit{x}}}_{t}^{D}$ and

${\widehat{\mathit{x}}}_{t}^{A}$ is a separation of the transformed state into differential and algebraic variables, respectively, which is performed in such a way that the Jacobian of the function

${\widehat{\mathit{f}}}_{0}^{A}$ with respect to the algebraic variables is nonsingular, see [

3] for details of the construction. The condition that the noise sources do not appear in the constraints, implies that

${\sum}_{j=1}^{m}{\widehat{\mathit{f}}}_{j}^{A}\equiv 0$, so that the algebraic equation in (

4b) can be solved as

${\widehat{\mathit{x}}}_{t}^{A}={\mathit{F}}^{A}({\widehat{\mathit{x}}}_{t}^{D})$ and inserted in the dynamic Equation (

4a) yielding an ordinary SDE

This equation is called underlying SDE of the strangeness-free SDAE. It acts in the lower-dimensional subspace

${\mathbb{R}}^{d}$, with

$d=n-a$ (where

a denotes the number of algebraic equations). The SDE system (

5) preserves the inherent dynamics of a strangeness-free SDAE system [

22]. Note that, in this way, the algebraic equations have been removed from the system, but, whenever a numerical method is used for the numerical integration, then one has to make sure that the algebraic equations are properly solved at each time step, so that the back-transformation to the original state variables can be performed.