Divergence-Kernel Method for Scores of Random Systems

The paper’s discrete-time divergence–kernel identity (Theorem 4.6) is rigorously derived and clearly stated, but the continuous-time SDE version (Theorem 1.1) is explicitly presented as a formal limit, relying on heuristic expansions and ‘ignoring cross terms’ during the passage to the limit, with the notion of integrating a backward-adapted αt defined only via a limiting prescription (Equation (10)), rather than as a standard Itô/Skorokhod integral. Hence, the SDE-level result is not fully rigorous in the paper . The candidate solution aims to supply a rigorous continuous-time proof via Euler–Maruyama plus Gaussian integration by parts, but it silently treats the dB-integral with backward-adapted α as an Itô integral (anticipating integrand) and differentiates αk with respect to Xk+1 during Stein’s identity without imposing any smoothness of α, both of which are nontrivial and unaddressed. Therefore, the model’s proof also leaves critical gaps. In short: the paper is rigorous in discrete time but only formal in continuous time; the model attempts a rigorous SDE proof but misses necessary assumptions and definitions for anticipating terms and the differentiability of α.