{"id":"ba6bb522-f983-43ca-a008-74af1d5a5be1","arxiv_id":"2411.16491","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a broad class of stochastic optimal control problems, convergence of the uncontrolled dynamics is enough to guarantee convergence of optimal costs and optimal controls.","lead":"This paper proves a general theorem: if the uncontrolled dynamics of a stochastic system converge to a limit, then the optimal controls and costs of the associated control problems also converge. The motivating application is climate-weather interaction, where acting on fast meteorological scales can control a slow climatic variable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2 rests on the external stability estimate (7) of [3], whose full hypotheses are only partially verified in the text; without a complete check, the proof of the key bound (4.8) is not fully justified.","rationale":"The central claim of the paper, Theorem 4.2, is a transfer principle: convergence of the uncontrolled forward equation implies convergence of optimal costs and optimal controls. The overall strategy is coherent, and the authors do a great deal of work to handle unbounded controls through localization and BMO-martingale estimates. My stress-test did not find a counterexample or a false step in the construction; the concern is more narrow and technical. The proof of Theorem 4.2 hinges on estimate (7) of [3], a cited external result. The manuscript verifies one named hypothesis, A3, and states the uniform BMO bound on the linearized coefficient K^ε, but it never enumerates or checks the remaining hypotheses of the external estimate. The reader's weakest-assumption report already flagged this, and I agree that it is the most load-bearing point: if estimate (7) carries an additional condition that is not met here, then (4.8) collapses and with it the convergence of Z^ε, which is the only route to the control convergence (4.2). This does not mean the theorem is false; it means the proof, as written, has a gap that a careful reader cannot close without going back to [3]. I therefore recommend a conditional acceptance: the theorem and its proof should be accepted once the hypotheses of [3, Eq. (7)] are explicitly verified for the difference BSDE. I did not treat the apparent misprint in Example 2.2's definition of Γ^ε (the exponent should presumably involve the fast semigroup e^{-(t-s)/ε} rather than e^{(t-s)A}) as load-bearing for the abstract theorem, although it should be corrected for the climate application to be rigorous.","tokens_in":19986,"tokens_out":27700,"duration_ms":237176,"concrete_test":"Obtain Briand-Confortola [3], locate estimate (7) on p. 831, and transcribe its complete statement together with every hypothesis. Then verify each hypothesis for the difference BSDE (4.3)-(4.4) with terminal condition ξ = h(X^ε_T)-h(Ŷ_T), source term f^ε, drift K^ε z, and martingale term ∫ z dW, using the bounds (4.6), (4.7), the uniform BMO estimate on ∫ K^ε dW, and the boundedness/lipschitz properties of h. In particular, check whether (7) requires an L^p bound on the generator at z=0 that is not supplied, and whether the exponent in 'A3 holds for any p>1' matches the exponent 2p in (4.8). If every hypothesis is satisfied, the proof of (4.8) is complete; if any fails, Theorem 4.2 is unproved as written and the conclusion (4.2) would need an additional argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 4.2 pivots on equation (4.8), obtained by applying the stability estimate (7) of Briand-Confortola [3] to the difference BSDE (4.3)-(4.4). The manuscript says that 'assumption A3 in [3] is verified for any p>1' and it shows the BMO bound on the stochastic-Lipschitz coefficient K^ε, but it never states or verifies the full set of hypotheses under which estimate (7) is proved. In particular, one needs to check explicitly: (i) the terminal condition ξ = h(X^ε_T) - h(Ŷ_T) belongs to the L^p space required by (7) for the p used in (4.8); (ii) the source term f^ε satisfies the exact integrability condition A3 with exponents matching the estimate, since the paper only asserts 'for any p>1' without aligning it with the exponent 2p appearing in (4.8); and (iii) the coefficient K^ε is admissible for [3] and its BMO norm indeed controls the constant appearing in (7), not merely the finiteness of that norm. If any of these hypotheses fails, the convergence (4.9) of Z^ε to Ẑ -- and therefore the optimal-control convergence (4.2) -- does not follow from the written proof. This step is load-bearing because it is the only mechanism that turns the assumed L^1-type convergence of the uncontrolled forward equation into the L^p convergence of the backward and control components.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an abstract framework, in Hilbert spaces, for the convergence of stochastic optimal control problems under approximations of the driving noise. For each ε, the uncontrolled state X^ε is driven by an approximation Γ^ε of the noise, while the limit state X̂ is driven by the actual noise; only convergence of these uncontrolled forward equations is assumed. The controlled problems are formulated in weak form through a Girsanov transformation with localization, because the controls are only square integrable and not bounded. The optimal cost and optimal control are represented through forward-backward systems, using BMO martingale theory to handle the quadratic, non-Lipschitz Hamiltonian. The main result, Theorem 4.2, states that convergence of the uncontrolled problems implies convergence of the optimal costs and, in L^2, of the optimal controls. The motivating application is a slow-fast climate model in which the control acts on the fast meteorological scale and the limit is a Wong-Zakai type reduced equation for the slow climate variable.","tokens_in":20336,"tokens_out":9501,"duration_ms":97928,"significance":"If the main theorem is fully justified, the paper makes a useful contribution: it reduces the convergence of nontrivial stochastic control problems to a property of uncontrolled equations, and it yields convergence of controls, not only of value functions. The proof is built from the stated assumptions and from BMO estimates; no fitted parameters or circular arguments were found. The weak formulation for unbounded controls and the use of forward-backward systems in Hilbert space are genuine technical ingredients. The paper also gives a credible climate-motivated application, although the verification of all structural assumptions for the concrete model is only partial. These strengths justify publication once the external-estimate hypotheses in the proof of Theorem 4.2 are made explicit and checked.","major_comments":[{"comment":"The proof of the key convergence (4.9) of Z^ε to Ẑ rests on estimate (7) of [3], but the manuscript does not state the hypotheses under which that estimate is proved. The text verifies only that 'assumption A3 in [3] is verified for any p>1' and asserts the BMO bound on K^ε. Please state the full set of hypotheses required by estimate (7) — including the integrability of the terminal condition h(X^ε_T)-h(X̂_T) at the exponent used in (4.8), the exact matching between the A3 exponents and the value p*=2p, and the admissibility of the stochastic Lipschitz coefficient K^ε with the constant in (7) controlled by its BMO norm — and verify each of them. Without this, the inequality (4.8), and therefore the passage from Assumption 4.1 to the convergence of the backward components, is not fully justified. The ingredients appear to be present, but the proof must be made explicit.","section":"Section 4, Eq. (4.8)"}],"minor_comments":[{"comment":"In the statement of Theorem 4.2, the notation U_ad is used for both the approximating admissible class U^ε_ad of Definition 2.7 and the limiting admissible class U_ad; please distinguish the two spaces, since the two infima are taken over different sets.","section":"Section 4, Theorem 4.2"},{"comment":"The existence result for (S^ε) is stated under Assumption 2.3 alone, but Assumption 2.1 does not contain a pathwise integrability condition on Γ^ε[GW] (for example Γ^ε[I] ∈ L^1([0,T];K) almost surely). Please add the minimal regularity needed for the mild-solution fixed point, or state explicitly that such regularity is part of the standing assumptions on Γ^ε.","section":"Section 2.2, Theorem 2.4"},{"comment":"Assumption 3.2(2) is stronger than the cost and control assumptions of Assumption 2.5 and is verified in the text only for the quadratic example of Example 3.3. Since the climate application in Section 1.2 allows a general Lipschitz map r(x,u), the paper should discuss for which classes of r and l the Hamiltonian minimizer u(x,z) is measurable and Lipschitz, or state clearly that the application requires this condition to be checked separately.","section":"Section 3.1, Assumption 3.2"},{"comment":"There are several presentation issues: 'Let us check that (3.13)' should read 'Let us check that (3.13) holds'; in Remark 2.10 the equation referred to as (2.4) is the controlled equation but the displayed line uses X^ε, not X^{ε,u}; and the notation E_T(r(X^ε,u^n)) is sometimes written with u and sometimes with u^n. These should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the main issue is the incomplete verification of the external stability estimate (7) of [3] in the proof of Theorem 4.2. I regard this as fixable with a precise statement and a verification paragraph, but it is load-bearing because it is the only mechanism producing convergence of the Z-components and hence of the optimal controls. I found no circularity and no fitted-parameter concerns; the self-citations are used as context and motivation, not as ingredients of the proof. The paper fits the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my honest take. The paper proves a genuine transfer principle: for a class of quadratic stochastic optimal control problems in Hilbert spaces, convergence in probability of the uncontrolled forward equation implies convergence of optimal costs and of optimal controls in L2. That is a clean, useful statement, and it goes beyond the earlier averaging-type results from the same group (Guatteri-Tessitore). The main technical novelty is the weak formulation with unbounded controls, handled by localization and BMO martingale estimates. That part is real work and it holds up.\n\nThe proof follows a reasonable route: represent the value and optimal feedback via a forward-backward SDE, then compare the two BSDEs. The difference BSDE argument in Theorem 4.2 is coherent. I checked for circularity or fitted parameters and found none. The cited self-references are used for motivation and context, not as load-bearing ingredients. Assumption 3.2, requiring a measurable Lipschitz Hamiltonian minimizer, is restrictive but explicitly stated and verified for the quadratic example. That is fair.\n\nThe soft spot is the use of the external stability estimate (7) of Briand-Confortola. The paper says \"assumption A3 in [3] is verified for any p>1\" and does not spell out the full hypotheses or the verification. You asked whether this is a load-bearing flaw. On reading, I think not: the needed ingredients are in the paper. The terminal increment h(X^epsilon_T)-h(hat X_T) is bounded because h is bounded; f^epsilon satisfies L^q integrability for all q via (4.7); and the coefficient K^epsilon has a uniform BMO bound, which controls the constant in (4.8). So the step is justified, but the presentation is too terse. A referee should ask the authors to state the hypotheses of [3] and check them item by item. That is a minor revision, not a fatal gap.\n\nThere are also a number of typos, \"Here exists\" and \"climat model\" for example, but nothing that obscures the mathematics.\n\nWho should read this: people working on stochastic optimal control, BSDEs, or slow-fast climate models. It is a serious paper and deserves a regular peer review. I would suggest the editor send it to a referee with instructions to check the Briand-Confortola application carefully. With that one clarification, the result stands.","headline":"A genuine transfer principle for Wong-Zakai-type approximations in stochastic control; the proof is sound, but the Briand-Confortola step needs a fuller verification in revision.","tokens_in":20840,"tokens_out":3452,"would_cite":false,"duration_ms":33138,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E20","60H10","49K45","60H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that convergence of the uncontrolled state equations alone—under a Wong–Zakai type approximation of the noise—is sufficient to guarantee convergence of both the optimal costs and the optimal controls, in the L2 sense.","keywords":["two-scale systems","stochastic optimal control","Wong-Zakai approximation","forward-backward SDE","BMO martingales","climate model reduction","optimal control convergence","unbounded controls"],"falsifier":"Find a system satisfying Assumptions 2.3, 2.5 and 4.1 where Assumption 3.2(2) fails (e.g., r(x,u) = r0(x)$u^{2}$ or |u|^p with p ≠ 2) and check numerically or analytically whether E∫|uε − u|^2 dt still converges to 0; if it does, the Lipschitz condition is not necessary, and if it does not, the theorem's assumption is sharp. Alternatively, verify all hypotheses of the stability estimate (7) in [3] for the BSDE with coefficient ψ; if one of them is not satisfied, the proof has a gap.","tokens_in":19785,"feed_emoji":"🌍","tokens_out":5242,"duration_ms":43654,"temperature":0.7,"pith_summary":"The paper proves a general principle for approximating stochastic optimal control problems: if the uncontrolled state equations converge under a Wong–Zakai type approximation of the noise, then the optimal costs and the optimal controls of the corresponding control problems also converge. This matters because it lets a modeler check only the uncontrolled dynamics, which is much easier than re-solving control problems at every scale. The result is stated in abstract Hilbert spaces, covers unbounded controls via a weak formulation, and is applied to a slow–fast climate model where control acts on the fast (meteorological) scale and the limit reduced equation is a stochastic differential equation for the slow (climatic) variable. The theorem is proved through a forward–backward SDE representation: the optimal cost is the initial value of the backward component and the optimal control is a feedback of the backward component.","feed_headline":"Uncontrolled convergence alone forces optimal controls to converge","feed_subtitle":"Paper proves it for stochastic control with unbounded controls, via forward–backward SDEs.","key_machinery":"The forward–backward SDE representation (systems 3.12 and 3.23). For each ε, the optimal cost is Yε_0 and the optimal control is u(Xε, Zε), where u is the measurable minimizer of the Hamiltonian ψ(x,z) = inf_u { l(x,u) − ⟨z, r(x,u)⟩ } (Assumption 3.2). The proof compares the two backward equations via a stability estimate for BSDEs with stochastic Lipschitz coefficient (estimate (7) of [3]), using BMO-martingale properties to control the non-Lipschitz quadratic growth of ψ in z. This turns the uncontrolled forward convergence into convergence of (Yε, Zε), and then, via the Lipschitz property of u, into convergence of the controls.","core_discovery":"The central theorem (Theorem 4.2) asserts that, under Assumptions 2.3, 2.5, 3.2 and 4.1, lim ε→0 inf Jε = inf J and the optimal controls converge in L2(Ω×[0,T]) to the limit optimal control. The only coupling hypothesis is Assumption 4.1: the solution of the uncontrolled forward equation driven by the approximated noise converges in probability, for each time, to the solution driven by the true Wiener noise. No additional convergence hypothesis is imposed on the control problems. The proof routes everything through a forward–backward system: the optimal cost at level ε equals Yε_0, the initial value of the backward component, and the optimal control is the feedback u(Xε_s, Zε_s). Comparing the backward components at the two levels reduces the control convergence to convergence of the forward process and of the integrands in the BSDE.","pith_inferences":["The principle is likely generic: any approximation scheme for the noise that yields convergence of the uncontrolled solution—such as piecewise linear interpolation, convolution, or colored noise—should automatically transfer to the control problem, as long as the Hamiltonian minimizer is regular enough.","The Lipschitz condition on u(x,z) could perhaps be relaxed to continuity plus a growth bound, because the L2 convergence of (Xε,Zε) and the quadratic structure of the cost might still force convergence of the feedback; this is testable by constructing a counterexample with a merely Hölder minimizer.","For climate applications, the result suggests that model intercomparison and control design can be performed at the level of the reduced stochastic equation without re-solving the full two-scale problem, provided the reduced equation is known to be the limit of the uncontrolled slow variable."],"forward_implications":["In the motivating slow–fast climate model (1.1)-(1.2), the finite-dimensional case yields convergence of optimal costs and controls when costs satisfy Assumptions 2.5 and 3.2 (Remark 4.3).","In the Wong–Zakai type example (Section 5.1), mollified noise approximations give L2 convergence of optimal controls to the Stratonovich-corrected limit equation.","The same abstract framework extends to quadratic fast–fast interactions (Section 5.2), where the reduced equation contains the average of the fast self-interaction under the invariant Gaussian measure.","Optimal controls at every scale are square integrable and admit the feedback form u(Xε,Zε), so the convergence is in the strong L2 sense of the actual processes, not just of the costs."],"supporting_citations":[{"why":"Supplies the BSDE stability estimate (7) and BMO-martingale bounds used to compare the backward components of the two problems.","marker":"[3]"},{"why":"Proves the stochastic model reduction convergence (Theorem 2.2) that verifies Assumption 4.1 for the finite-dimensional motivating example.","marker":"[1]"},{"why":"Provides BMO-martingale properties and exponential-martingale integrability used to control the quadratic Hamiltonian.","marker":"[19]"},{"why":"Gives existence and uniqueness for quadratic-growth BSDEs, used in Theorem 3.4.","marker":"[21]"},{"why":"Establishes the mathematical framework for stochastic climate models that motivates the slow–fast setting and the model reduction.","marker":"[24]"},{"why":"Used in Section 5.1 for the Wong–Zakai approximation convergence of the mollified noise.","marker":"[5]"}],"fun_headline_variants":["Uncontrolled convergence alone guarantees optimal control convergence","For optimal control convergence, only uncontrolled convergence is needed","Convergence of uncontrolled problem suffices for optimal controls to converge","One hypothesis: uncontrolled convergence. Then optimal cost and control converge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Hamiltonian minimizer u(x,z) must be Lipschitz in both variables (Assumption 3.2(2)); the paper verifies this only for the quadratic-cost example, and the BSDE stability estimate invoked from [3] is checked only partially because the text verifies Assumption A3 but does not restate the remaining hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Uncontrolled convergence alone guarantees optimal control convergence","For optimal control convergence, only uncontrolled convergence is needed","Convergence of uncontrolled problem suffices for optimal controls to converge","One hypothesis: uncontrolled convergence. Then optimal cost and control converge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000643,"raw_usage":{"total_tokens":2926,"prompt_tokens":883,"completion_tokens":2043,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":1977}},"tokens_in":499,"tokens_out":2043,"duration_ms":15895,"temperature":1.0,"reasoning_tokens":1977,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:03:58.673472+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a system satisfying Assumptions 2.3, 2.5 and 4.1 where Assumption 3.2(2) fails (e.g., r(x,u) = r0(x)$u^{2}$ or |u|^p with p ≠ 2) and check numerically or analytically whether E∫|uε − u|^2 dt still converges to 0; if it does, the Lipschitz condition is not necessary, and if it does not, the theorem's assumption is sharp. Alternatively, verify all hypotheses of the stability estimate (7) in [3] for the BSDE with coefficient ψ; if one of them is not satisfied, the proof has a gap.","supporting_citations":[{"cited_title":"Briand, F","cited_arxiv_id":null,"evidence_quote":"Supplies the BSDE stability estimate (7) and BMO-martingale bounds used to compare the backward components of the two problems."},{"cited_title":"Assing, F","cited_arxiv_id":null,"evidence_quote":"Proves the stochastic model reduction convergence (Theorem 2.2) that verifies Assumption 4.1 for the finite-dimensional motivating example."},{"cited_title":"Kazamaki, Continuous exponential martingales and BMO , Lecture Notes in Mathematics, Springer- Verlag, 1994","cited_arxiv_id":null,"evidence_quote":"Provides BMO-martingale properties and exponential-martingale integrability used to control the quadratic Hamiltonian."},{"cited_title":"Kobylanski, Backward stochastic differential equat ions and partial differential equations with quadratic growth, The Annals of Probability , 28, No","cited_arxiv_id":null,"evidence_quote":"Gives existence and uniqueness for quadratic-growth BSDEs, used in Theorem 3.4."},{"cited_title":"Majda, I","cited_arxiv_id":null,"evidence_quote":"Establishes the mathematical framework for stochastic climate models that motivates the slow–fast setting and the model reduction."},{"cited_title":"Caravenna, M","cited_arxiv_id":null,"evidence_quote":"Used in Section 5.1 for the Wong–Zakai approximation convergence of the mollified noise."}],"review_version":1}