On Explicit Stochastic Differential Algebraic Equations

The paper’s Corollary 3.2 states that for an UNSDAE with a noiseless constraint (Γ = 0), if 0 is a regular value of g and Img(σ(x)) is not contained in ker(Dg(x)) at some constraint point, then there is no Itô solution; the proof is by contrapositive of a geometric tangency condition (Lemma 3.1) derived from Itô differentiation and submersion-theoretic tangency of the noise to the constraint manifold . The model proves the same nonexistence by a direct Itô-formula argument on y(t)=g(x(t)): since y(t) must be 0 in the paper’s mean-square sense of solution , the associated local martingale term must vanish, forcing Dg(x_t)σ(x_t)=0 and yielding a contradiction at an initial point with Dgσ≠0. The conclusions match; the paper’s route is geometric (tangent-distribution), while the model’s is analytic (quadratic-variation of the martingale).