Existence and uniqueness for the solutions of non-autonomous stochastic differential algebraic equations with locally Lipschitz coefficients

The paper’s argument transforms an index-1 SDAE into an inherent SDE via time-dependent projectors, solves the algebraic constraint by the implicit function theorem using the bounded inverse of J, and then lifts back; this is consistent and carefully justified in four parts, with the needed assumption P(t)=A^-(t)A(t)∈C^1 and a precise use of Itô’s formula and classical SDE theory. The candidate solution follows the same blueprint but makes a critical algebraic error asserting invertibility of D_xG_t=P+R f'_X from invertibility of J=A+R f'_X by multiplying with A; this step is invalid in general. It also incorrectly claims P and R are complementary projectors, and it silently assumes P∈C^1 without stating it. Aside from these issues, the rest of the outline aligns with the paper. Minor note: the paper’s Theorem 3 includes a supremum-moment bound and invokes a result that uses Assumption (A2.1); that assumption is not listed in the statement of Theorem 3, suggesting a small presentational oversight.