Optimal response for stochastic differential equations by local kernel perturbations

The paper’s Proposition 28 asserts that, for a closed, bounded, strictly convex feasible set P ⊂ L2(D×D) with 0 in its relative interior, and a continuous linear J that is not uniformly vanishing on P, the maximization max_{κ̇∈P} J(κ̇) has a unique solution; the existence and uniqueness are delegated to Appendix A (Propositions 30 and 31) and applied to Problem 27 (closed unit L2-ball) via continuity of the linear-response map R and continuity of J (from Lemma 26 and the definition of J) . The candidate solution establishes the same two pillars: existence by weak compactness (Hilbert reflexivity + P convex and norm-closed ⇒ weakly closed; hence P is weakly compact inside a closed ball) and uniqueness by the interior-point argument for linear functionals on strictly convex sets with 0 ∈ ri(P), ruling out the degenerate case J ≡ 0 on P. This matches the paper’s logic and hypotheses. The paper goes a step further by invoking Riesz’s representation to identify the optimizer explicitly on the unit ball, κ̇opt = g/∥g∥2 when J(κ̇) = ⟨g, κ̇⟩ . No missing hypotheses or logical gaps are apparent in either argument.