REVIEW 1 major objections 4 minor 26 references
On approximations of stochastic optimal control problems with an application to climate equations
T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Section 4, Eq. (4.8)] 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.
minor comments (4)
- [Section 4, Theorem 4.2] 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 2.2, Theorem 2.4] 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 3.1, Assumption 3.2] 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.
- [Throughout] 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.
Circularity Check
No significant circularity: the main convergence theorem is derived from stated assumptions via an external BSDE stability estimate, and the cited self-papers are used for motivating examples rather than as load-bearing inputs.
full rationale
The central claim, Theorem 4.2, is not circular. The proof writes the difference BSDE (4.3)-(4.4), bounds its source term f^epsilon using (4.6), verifies a BMO bound for K^epsilon, then applies the external stability estimate (7) of [3] to obtain (4.8). The right-hand side of (4.8) contains only quantities whose convergence is proved from Assumption 4.1 by dominated convergence: E(|h(X^epsilon_T)-h(Xhat_T)|^{2p}) and E((int_0^T |f^epsilon_s|ds)^{2p}). The final control convergence (4.2) then follows from Assumption 3.2(2) and the already-proved L^p convergence of Z^epsilon - Zhat. No fitted parameter is fed back into the statement, and no conclusion is assumed as a hypothesis. The self-citations [1], [9], [16], and [17] are used for the motivating slow-fast example and for context; in particular, Remark 4.3 invokes [1, Theorem 2.2] only to instantiate Assumptions 2.3 and 4.1 for the finite-dimensional climate model, which is a separate published result rather than an ingredient needed to prove Theorem 4.2. The one genuine concern is correctness, not circularity: the text asserts 'assumption A3 in [3] is verified for any p>1' immediately before applying estimate (7) in [3], but it does not fully spell out the terminal integrability and exponent-matching hypotheses of that external estimate. This is an omitted-verification gap in the proof of (4.8), not a reduction of the theorem to its own conclusion. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (5)
- domain assumption Lipschitz coefficients and C0 semigroup for the state equation, Assumption 2.3, equations (2.2) and (2.3)
- domain assumption Coercive quadratic running cost, bounded terminal cost, and control map r with an anchor control u* such that r(x,u*)=0, Assumption 2.5, equations (2.7) through (2.14)
- ad hoc to paper The Hamiltonian infimum is achieved by a measurable Lipschitz minimizer u(x,z), Assumption 3.2, especially equation (3.7)
- domain assumption Pointwise convergence in probability of the uncontrolled solutions, Assumption 4.1
- standard math External stability estimate (7) of Briand and Confortola [3]
Cite this review
Pith. "Pith review of On approximations of stochastic optimal control problems with an application to climate equations." pith.science (2026). https://pith.science/paper/KR76B6OB
@misc{pith2026241116491,
author = {Pith},
title = {Pith review of: On approximations of stochastic optimal control problems with an application to climate equations},
year = {2026},
howpublished = {\url{https://pith.science/paper/KR76B6OB}},
note = {Machine review of arXiv:2411.16491}
}
read the original abstract
The paper is devoted to the optimal control of a system with two time-scales, in a regime when the limit equation is not of averaging type but, in the spirit of Wong-Zakai principle, it is a stochastic differential equation for the slow variable, with noise emerging from the fast one. It proves that it is possible to control the slow variable by acting only on the fast scales. The concrete problem, of interest for climate research, is embedded into an abstract framework in Hilbert spaces, with a stochastic process driven by an approximation of a given noise. The principle established here is that convergence of the uncontrolled problem is sufficient for convergence of both the optimal costs and the optimal controls. This target is reached using Girsanov transform and the representation of the optimal cost and the optimal controls using a Forward Backward System. A challenge in this program is represented by the generality considered here of unbounded control actions.
Reference graph
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