In this paper, we study the well-posedness and regularity of non-autonomous stochastic differential algebraic equations (SDAEs) with nonlinear, locally Lipschitz and monotone (2) coefficients of the form (1). The main difficulty is the fact that the operator A(.) is non-autonomous, i.~e. depends on t and the matrix $A(t)$ is singular for all $t\in \left[0,T\right]$. Our interest is in SDAE of index-1. This means that in order to solve the problem, we can transform the initial SDAEs into an ordinary stochastic differential equation with algebraic constraints. Under appropriate hypothesizes, the main result establishes the existence and uniqueness of the solution in $\mathcal{M}^p(\left[0, T\right], \mathbb{R}^n)$, $p\geq 2$, $p\in \mathbb{N}$. Several strong estimations and regularity results are also provided. Note that, in this paper, we use various techniques such as It\^o's lemma, Burkholder-Davis-Gundy inequality, and Young inequality.

翻译：本文研究了具有形式(1)的非线性、局部Lipschitz且单调(2)系数的非自治随机微分代数方程(SDAEs)的适定性和正则性。主要困难在于算子A(·)是非自治的，即依赖于t，且矩阵A(t)对所有t∈[0,T]均奇异。我们关注的是指标为1的SDAE，这意味着要解决问题，可将初始SDAE转化为带代数约束的普通随机微分方程。在适当假设下，主要结果建立了解在M^p([0,T],R^n)、p≥2、p∈N空间中的存在唯一性，并给出了若干强估计和正则性结果。值得注意的是，本文使用了多种技术，如Itô引理、Burkholder-Davis-Gundy不等式和Young不等式。