ASYMPTOTIC ANALYSIS OF ESTIMATORS OF ERGODIC STOCHASTIC DIFFERENTIAL EQUATIONS

The paper’s Theorem 3.5 proves the CLT for the (penalized) approximate MLE of the drift, with plug-in diffusion estimate, under Conditions 2.1 and 2.6, the mesh scaling ∆̃(ε)/ε→0, and a small penalty gradient, yielding ε−1/2(μ̂ε−μ0) ⇒ N(0, Σθ0−1) (statement and assumptions summarized in Theorem 3.5) . Its proof explicitly uses the discretized score representation (2.5)–(2.7) and the scaled formulation (3.14), then decomposes the score at μ̂ε, identifies a martingale term with covariance converging to Σθ0 (via Proposition 4.10) and shows all discretization/plug-in remainders are negligible at the √ε–scale (using Corollary 4.5 and Lemma 4.7), followed by a Taylor expansion and an information-matrix limit ∫Gε(u)du→Σθ0 to solve the first-order condition (5.3–5.6) . The candidate solution follows the same route: (i) score = martingale + discretization remainder, (ii) √ε-negligibility of the remainder under ∆̃(ε)/ε→0 and the Hölder/polynomial bounds of Condition 2.6, (iii) convergence of the predictable bracket to Σθ0 by ergodic averaging and plug-in consistency ς̂ε→ς0, (iv) CLT for the score, (v) Hessian convergence and (vi) Taylor/Slutsky to conclude. Small differences are present only in presentation: the paper instantiates the technical bounds via Corollary 4.5 and Lemma 4.7 and uses Proposition 4.10 for the martingale CLT, whereas the model appeals to the conditional Gaussianity of Itô integrals. Both rely on the same assumptions (Condition 2.1, Condition 2.6; consistency of ς̂ε and μ̂ε; penalty scaling ε−1/2∇γ→0; and ∆̃(ε)/ε→0) . Hence both are correct with substantially the same proof.