Weak order one convergence of structure-preserving stochastic theta methods for stochastic differential algebraic equations with time-dependent singular matrices

This paper studies the weak convergence order of structure-preserving stochastic theta methods for a class of index-$1$ stochastic differential algebraic equations with time-dependent singular matrices. The singular matrix is allowed to vary in time but preserves a fixed differential-algebraic splitting, thereby extending the constant singular-matrix setting while retaining the projector structure required for constraint preservation. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we establish an abstract weak convergence theorem for constraint-preserving one-step approximations and apply it to the stochastic theta method with $\theta \in (0,1]$. Under global Lipschitz, linear growth, and suitable smoothness assumptions, the considered method is proved to be well posed, to preserve the algebraic constraints at all time levels, and to converge with weak order one. Numerical experiments are finally presented to confirm the structure-preserving property and the theoretical convergence order.