{"id":"8d08de69-6459-477b-ad2c-cfd80f4e7980","arxiv_id":"2508.18744","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Claims well-posedness for quadratic G-BSDEs with double mean reflections, but the proof silently drops the f term and does not prove the stated f-inclusive theorem.","lead":"This paper claims existence and uniqueness for quadratic backward stochastic differential equations driven by G-Brownian motion with double mean reflections, for bounded and unbounded terminal conditions. The main theorem as stated includes a drift term, but the proofs only treat the drift-free case, so the central claim is not fully supported.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'WLOG f=0' reduction is unjustified: all proofs of Theorem 3.5 solve (3.2), never restore f ds; an explicit f=1 test case shows the fixed point cannot be a solution of (3.1).","rationale":"The reader's weakest_assumption identifies exactly the f=0 WLOG issue, and I agree it is structural. All displays after Lemma 3.3—(3.6)–(3.9), Step 2's interval equation—omit ∫ f ds, and no transformation restores it. The explicit f=1 case shows the fixed-point map gives Γ(0)=0 while (3.1) has solution 1−t, so the proof's output is for (3.2), not (3.1). The circular definition ξ(m)=Y(m)_{t0+h} in Theorem 3.7 Step 1 is an additional independent defect, but the f-gap already invalidates Theorem 3.5. Since the paper's headline result is stated for (3.1), the current REJECT verdict stands; no change is needed.","tokens_in":30222,"tokens_out":10786,"duration_ms":114683,"concrete_test":"Run the admissible f≡1 example through the proof: compute Γ(0) from (3.2) (it is Y=0, A=0) and compare with the explicit solution Y_t=1−t of (3.1). If Γ fixes 0 and the proof's global solution is 0, the f-drift has been dropped; alternatively, add the ∫ f ds term into (3.6)/(3.8) and show the contraction estimate still closes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1's 'Without loss of generality, we assume f≡0 in (3.1)' is load-bearing and unsupported. Lemma 3.3 is stated for (3.2), the fixed-point map Γ in Step 1 solves (3.2), and the backward-induction equations in Step 2 again omit the f ds term. No Girsanov/transform is supplied that maps solutions of (3.2) to solutions of (3.1). To see this is not cosmetic, take f≡1, g≡0, ξ=0, L(t,x)=x−2, R(t,x)=x+2, T=1, which satisfies Assumptions 3.1–3.2. Equation (3.1) has solution Y_t=1−t (Z=K=A=0), while Γ(0) from (3.2) is Y=0; hence the contraction fixed point produced by the proof solves the f=0 equation, not (3.1). The proof therefore does not establish Theorem 3.5 as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quadratic backward stochastic differential equations driven by G-Brownian motion with double mean reflections, i.e. constraints of the form E[L(t,Y_t)] ≤ 0 ≤ E[R(t,Y_t)] with Skorokhod-type conditions on a deterministic bounded-variation process A. The main results are Theorem 3.5, asserting well-posedness in S^∞_G × BMO_G × ∩_{α≥2} S^α_G × AD for bounded terminal conditions under Assumptions 3.1–3.2, and Theorem 3.7, asserting existence and uniqueness of a 'deterministic flat solution' in EG(R) × HG(R) × LG(R) × AD for unbounded terminal conditions under Assumptions 3.1 and 3.6. The method combines G-BMO martingale techniques, a fixed-point argument, the backward Skorokhod problem, and the θ-method. The bounded-terminal proof relies on decomposing the solution as the sum of a solution to a standard quadratic G-BSDE and a deterministic Skorokhod reflection process; the unbounded-terminal proof uses truncation, uniform exponential estimates, and the θ-method.","tokens_in":30471,"tokens_out":3914,"duration_ms":45746,"significance":"If established, the result would be a useful extension of mean-reflected G-BSDE theory: double mean reflections with quadratic generators and unbounded terminal data have not previously been treated in the G-framework. The paper's intended construction — representing the reflecting process A via the backward Skorokhod problem and isolating the quadratic part through G-BMO techniques — is natural and, in the f=0 case, appears viable. The paper also makes use of existing results rather than fitting parameters, and the high-level architecture is coherent. However, the central well-posedness claim as stated is not proved: the reduction to f=0 is asserted rather than justified, and the proof of the bounded-terminal theorem actually solves a different equation. This is a load-bearing gap, not a presentation issue.","major_comments":[{"comment":"The sentence 'Without loss of generality, we assume f ≡ 0 in (3.1)' is load-bearing and unsupported. Lemma 3.3 is stated and proved for (3.2), the fixed-point map Γ in Step 1 of Theorem 3.5 solves (3.2), and the backward-induction equations in Step 2 again contain only the g d⟨B⟩ term. No Girsanov-type transformation, change of measure, or other argument is supplied that recovers the ∫ f ds term from solutions of (3.2). This is not cosmetic: take f≡1, g≡0, ξ=0, L(t,x)=x−2, R(t,x)=x+2, T=1, which satisfies Assumptions 3.1–3.2. Equation (3.1) has the solution Y_t=1−t (Z=K=A=0), while Γ(0) computed from (3.2) is Y=0. Hence the contraction fixed point produced by the proof solves the f=0 equation, not (3.1), and Theorem 3.5 as stated is unproved.","section":"Section 3.1, Eq. (3.1)–(3.2)"},{"comment":"The definition 'Denote ξ^(m) = Y^(m)_{t0+h}' is self-referential: ξ^(m) is the terminal condition of the very process (Y^(m),Z^(m),K^(m),A^(m)) being constructed. In the truncation construction one would expect ξ^(m) to be a truncated version of the original terminal variable ξ (as suggested by the preceding line 'l^(m)=(l∧m)∨(−m) for l=ξ,g0(t)'), but the displayed equality makes the terminal condition depend on the unknown solution. If read literally, this is circular; if it is a typo, the subsequent estimates and convergence proof do not establish the claimed result for the intended truncated data. This needs to be corrected and the argument rewritten.","section":"Theorem 3.7, Step 1"},{"comment":"The global backward-induction step is stated too tersely to be checkable. The line 'Y^{n+1}_T = Y^n_T = ξ' is confusing: Y^{n+1} is not otherwise defined, and the patching of Y,Z,K,A at the grid points T_k requires compatibility of the terminal value of the k-th local solution with the terminal data of the (k+1)-th local solution. The text asserts existence of local solutions on [T_{k−1},T_k] with terminal Y^{k+1}_{T_k}, but the displayed system is not fully specified and the matching of the Skorokhod conditions across intervals is not proved. This is secondary to the f=0 issue, but it further obscures the global claim.","section":"Theorem 3.5, Step 2"},{"comment":"Assumption 3.2(ii) includes a bound on ∫|f(t,ω,0,0)|^2 dt, but after the WLOG f=0 reduction this term is never used. More importantly, the uniqueness proof of Lemma 3.3 invokes 'the same argument as Proposition 3.4 in [14]' for the deterministic process A without giving the argument; since the uniqueness of A is an essential part of Theorem 3.5, this reliance on an external proposition should be made precise, especially because the present setting has quadratic g and double mean reflections.","section":"Assumption 3.2 and Lemma 3.3"}],"minor_comments":[{"comment":"There are numerous typos and inconsistent notation: 'calss', 'provids', 'Bsrownian', 'G-Bsrownian', 'admits most one solution' (Theorem 3.9), and inconsistent use of bE versus \\hat E. The term 'deterministic flat solution' is used without definition.","section":"Throughout"},{"comment":"Equation (3.10) is introduced as the simplification of (3.1) with f omitted, but no statement explains whether the theorem is intended for the original equation (3.1) or only for (3.10). This ambiguity matters because Theorem 3.7 is stated for (3.10).","section":"Section 3.2, Eq. (3.10)"},{"comment":"The notation K_t = K^k_t + ∑_{j=1}^{k−1} K^j_{T_j} is not precisely defined: the superscript k on K^k_t should be understood as the k-th local component, but the indexing is not consistent with the previous display.","section":"Step 2 of Theorem 3.5"},{"comment":"Reference [38] is given as 'Possamaï, D., Zhou, C.' and used for G-Girsanov results; the paper does not clearly identify which results are quoted from [16] versus [38] in the preliminary section, making verification harder.","section":"References"}],"recommendation":"reject","confidential_remarks":"The explicit f=1 counterexample in the report is decisive: the proof of Theorem 3.5 solves (3.2) and never returns to (3.1), so the main bounded-terminal theorem is not established. The self-referential ξ^(m)=Y^(m)_{t0+h} in Theorem 3.7 is a further sign that the unbounded-terminal proof is not yet in a reliable form. The paper's reliance on [14] for the key uniqueness argument also warrants scrutiny. These are not merely local defects; they concern the central claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper's headline result is not proved as stated. The \"without loss of generality, we assume f≡0\" in Section 3.1 is doing a lot of work, and it doesn't hold. The stress-test example is exactly right: with f≡1, g≡0, ξ=0 and constraints x−2, x+2, equation (3.1) has solution Y_t=1−t, while the fixed point built from (3.2) gives Y≡0. So the contraction in Step 1 solves a different equation. I don't see any Girsanov or other transform in the paper that would restore the f ds term; the proof of Theorem 3.5 never goes back to (3.1). That is a structural gap, not a typo.\n\nWhat's genuinely new is the combination of double mean reflection with quadratic growth in the G-framework. The paper identifies the right gap in the literature (He-Li was Lipschitz, Gu-Lin-Xu was single reflection), and the machinery—G-BMO, backward Skorokhod, θ-method—is appropriate. If the f=0 case is what is proved, that is still a meaningful result; Lemma 3.3 and the unbounded section are about (3.10), which has no f. The writing suggests the authors know the f=0 case is the real contribution.\n\nThere is also a self-referential line in the proof of Theorem 3.7: ξ^(m)=Y^(m)_{t0+h} is circular as written; presumably a typo for a truncated terminal, but as it stands it's undefined. There are also many typos and several \"similar to\" arguments that skip substantial details. These are minor relative to the f=0 issue, but they add to the difficulty.\n\nWho is this for? Specialists in G-BSDEs and mean reflection. A reader in that area will see the potential but should not rely on Theorem 3.5 as stated. The paper deserves a serious referee because the question is legitimate and the flaw is identifiable and possibly repairable; a good referee could push the authors to either justify the reduction or state the f=0 version. I would not cite it until fixed.","headline":"The claimed result for quadratic double-mean-reflected G-BSDEs is not proven as stated: the WLOG f=0 reduction in Section 3.1 is unsupported and changes the solution, so Theorem 3.5 is out of reach without a real argument.","tokens_in":30971,"tokens_out":3599,"would_cite":false,"duration_ms":38612,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Quadratic G-BSDEs with double mean reflections have unique solutions for bounded terminal values, and for unbounded terminals under convexity and exponential moments.","keywords":["G-Brownian motion","quadratic BSDE","double mean reflection","backward Skorokhod problem","G-BMO martingale","theta-method","volatility ambiguity"],"falsifier":"Take a nonzero drift, for example f(t, y, z) = L1 y under Assumption 3.2, and check whether the map Γ obtained from the f = 0 equation remains a contraction: restoring f should introduce an extra term of order L1 T in the estimate (3.8), and for fixed T > 0 the contraction constant may fail to stay below 1. Equivalently, compute the difference between solutions of (3.1) and (3.2) on a small interval; if the difference is not exactly the integral of the f term, the claimed reduction is invalid.","tokens_in":30068,"feed_emoji":"📉","tokens_out":5751,"duration_ms":66971,"temperature":0.7,"pith_summary":"This paper proves well-posedness for quadratic backward stochastic differential equations driven by G-Brownian motion when the solution is constrained in expectation by two nonlinear loss functions, one from below and one from above. Under bounded terminal data, the solution quadruple exists and is unique in spaces that allow the generator to grow quadratically in the control variable. Under unbounded terminal data, existence and uniqueness hold when the generator is convex or concave in the control variable and the terminal value has exponential moments. The proofs combine G-BMO martingale theory, Girsanov transformation, the backward Skorokhod problem, fixed-point contraction on small intervals, and the θ-method. The result matters because such equations appear in pricing and risk problems where volatility is ambiguous and constraints are imposed only on averages rather than path by path.","feed_headline":"Double-constrained quadratic G-BSDEs get unique solutions","feed_subtitle":"A Skorokhod correction plus the θ-method proves well-posedness under volatility ambiguity.","key_machinery":"The central representation is Y^U_t = y^U_t + A^U_T − A^U_t, where y is the solution of a standard quadratic G-BSDE and A is the deterministic solution of the backward Skorokhod problem with boundaries l(t,x) = E[L(t, y_t − E[y_t] + x)] and r(t,x) = E[R(t, y_t − E[y_t] + x)]. G-BMO martingales and the G-Girsanov transform linearize the difference of two solutions, making the fixed-point map Γ a strict contraction on sufficiently small intervals. The θ-method handles unbounded terminals by truncating terminal values and generators, proving uniform exponential-moment estimates and Cauchy convergence.","core_discovery":"The paper establishes that the doubly mean-reflected quadratic G-BSDE (3.1) admits a unique solution (Y, Z, K, A) in S∞_G × BMO_G × ∩_{α≥2} S^α_G × AD when the terminal value is bounded (Theorem 3.5), and a unique deterministic flat solution in E_G(R) × H_G(R) × L_G(R) × AD when the terminal value is unbounded but satisfies exponential-moment assumptions and the generator is convex or concave (Theorem 3.7). The construction represents the reflected solution as the sum of a standard quadratic G-BSDE and a deterministic correction A obtained from the backward Skorokhod problem with barriers built from G-expectations of the loss functions. Uniqueness is obtained by comparison arguments and Skor","pith_inferences":["The paper asserts that setting f ≡ 0 in (3.1) is without loss of generality, but no transformation or construction is given to recover the f ds term; as written, the proof of Theorem 3.5 covers the f = 0 equation (3.2), and a reader should treat the general claim as conditional on that reduction.","With f restored, the small-interval contraction estimate would likely acquire an extra Lipschitz term proportional to L1 T, so the admissible interval length may shrink or the contraction may fail for some admissible f.","The representation separates the stochastic and deterministic components of the solution, suggesting that continuity of A with respect to perturbations of L, R, and ξ could be tested numerically under the stated assumptions.","A natural extension is to multi-dimensional G-Brownian motion or to drop convexity/concavity via comparison principles; the same exponential-moment controls would likely be needed."],"forward_implications":["The explicit representation Y = y + (A_T − A_t) gives a numerical strategy: solve a standard quadratic G-BSDE and a deterministic Skorokhod problem separately, then add the two pieces.","Quadratic generators are now compatible with double mean reflections, extending earlier Lipschitz and single-constraint results to problems such as robust pricing with average-loss constraints.","For unbounded terminals, the θ-method produces a convergent sequence of truncated bounded-terminal solutions, so approximations inherit the same Skorokhod conditions in the limit.","Uniqueness of the reflection process A is forced by the flat-off Skorokhod conditions, so the correction process is fully determined by the average constraints."],"supporting_citations":[{"why":"Supplies Theorem 5.1, used to solve the standard quadratic G-BSDE and to identify Y − (A_T − A), Z, K.","marker":"[4]"},{"why":"Provides G-BMO martingale theory, the G-Girsanov transformation, and quadratic G-BSDE well-posedness underlying the contraction argument.","marker":"[16]"},{"why":"Defines and solves the backward Skorokhod problem used to construct the deterministic correction A.","marker":"[24]"},{"why":"Gives the double-mean-reflection G-BSDE framework under Lipschitz conditions and the comparison result used for uniqueness of A.","marker":"[14]"},{"why":"Supplies the θ-method and exponential-moment estimates for quadratic G-BSDEs with unbounded terminal conditions.","marker":"[18]"},{"why":"Provides Lemma 3.8, the exponential estimate used throughout the unbounded-terminal existence proof.","marker":"[10]"},{"why":"Introduced the mean-reflection representation on which the A-representation is based.","marker":"[1]"}],"fun_headline_variants":["Double-constrained G-BSDEs: unique solutions proven","Quadratic G-BSDEs with double constraints: well-posed","Unique solutions for doubly reflected quadratic G-BSDEs","Skorokhod method proves unique G-BSDE solutions under double constraints"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof assumes, with the sentence 'Without loss of generality, we assume f ≡ 0 in (3.1)', that the drift term f can be dropped without changing the problem; no transformation is shown, so the proof as written applies to the f = 0 equation.","fun_headline_variants_meta":{"raw":{"variants":["Double-constrained G-BSDEs: unique solutions proven","Quadratic G-BSDEs with double constraints: well-posed","Unique solutions for doubly reflected quadratic G-BSDEs","Skorokhod method proves unique G-BSDE solutions under double constraints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1446,"prompt_tokens":635,"completion_tokens":811,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":379,"completion_tokens_details":{"reasoning_tokens":745}},"tokens_in":379,"tokens_out":811,"duration_ms":9237,"temperature":1.0,"reasoning_tokens":745,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T16:14:51.042413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a nonzero drift, for example f(t, y, z) = L1 y under Assumption 3.2, and check whether the map Γ obtained from the f = 0 equation remains a contraction: restoring f should introduce an extra term of order L1 T in the estimate (3.8), and for fixed T > 0 the contraction constant may fail to stay below 1. Equivalently, compute the difference between solutions of (3.1) and (3.2) on a small interval; if the difference is not exactly the integral of the f term, the claimed reduction is invalid.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 5.1, used to solve the standard quadratic G-BSDE and to identify Y − (A_T − A), Z, K."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides G-BMO martingale theory, the G-Girsanov transformation, and quadratic G-BSDE well-posedness underlying the contraction argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines and solves the backward Skorokhod problem used to construct the deterministic correction A."},{"cited_title":"Mean Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Constraints","cited_arxiv_id":"2405.09103","evidence_quote":"Gives the double-mean-reflection G-BSDE framework under Lipschitz conditions and the comparison result used for uniqueness of A."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the θ-method and exponential-moment estimates for quadratic G-BSDEs with unbounded terminal conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Lemma 3.8, the exponential estimate used throughout the unbounded-terminal existence proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the mean-reflection representation on which the A-representation is based."}],"review_version":1}