{"id":"b534624b-de58-481b-8821-ae702f4b9ab0","arxiv_id":"2605.25719","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves O(1/sqrt(penalty)) convergence rate between reflected quadratic-generator BSDEs and penalized BSDEs via BMO martingales, with application to Euler polygonal approximation for sub-quadratic generators.","lead":"The paper proves that reflected BSDEs with quadratic generators converge to their penalized versions at a rate of order 1/2 in the penalty parameter, using BMO martingale techniques. This rate supports numerical approximations via Euler schemes for related sub-quadratic cases.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"BMO control on the difference process may fail to close uniformly in the penalty parameter without explicit a priori bounds on the penalized solution's quadratic variation.","rationale":"The reader's weakest_assumption already isolates the precise point where the BMO technique must succeed; the above makes that assumption concrete by identifying the potential n-dependence inside the BMO norm itself. No other internal inconsistency appears from the abstract and the stated application.","tokens_in":1558,"tokens_out":342,"duration_ms":20475,"concrete_test":"Extract the precise BMO-norm estimate for the difference process from the proof of the main theorem (likely around the application of the BMO inequality to the penalized BSDE); recompute the bound replacing the generic constant C by an explicit function of n and verify whether sup_n ||M^n||_BMO remains finite; if it diverges, the 1/2-rate derivation does not hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that BMO-martingale estimates applied to the difference (Y - Y^n, Z - Z^n) between the reflected solution and the penalized approximation yield an O(1/sqrt(n)) bound. For quadratic generators the penalized equation introduces an extra drift term n(Y^n - L)^- whose size grows with n; the standard BMO comparison (via the exponential martingale or the quadratic variation estimate) implicitly needs the BMO norm of the difference driver to remain controlled independently of n. If this norm grows even logarithmically with n, the resulting Gronwall-type inequality only recovers a slower rate or a rate that depends on additional constants that may diverge.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies reflected BSDEs with quadratic generators and their penalized approximations. It claims to prove, via BMO-martingale techniques, that the convergence rate between the reflected solution and the penalized solution is of order 1/2 in the penalty parameter. The result is then applied to obtain a numerical approximation result for reflected BSDEs with sub-quadratic generators using the Euler polygonal-line method.","tokens_in":1676,"tokens_out":447,"duration_ms":18674,"significance":"If the rate holds under the stated conditions, the quantitative error bound would be useful for justifying penalization as a practical approximation tool and for error analysis in numerical schemes for reflected BSDEs. The application to Euler discretization adds a concrete consequence. The abstract, however, supplies neither the precise assumption list nor any derivation steps, so the significance cannot be fully assessed from the given information.","major_comments":[{"comment":"The central claim (abstract) asserts that BMO-martingale estimates applied to the difference process between the reflected and penalized solutions close at rate 1/2 uniformly in the penalty parameter n. The penalized driver contains the term n(Y^n - L)^- whose size grows linearly with n. Without an explicit a priori bound showing that the BMO norm of this difference driver (or its quadratic variation) remains controlled independently of n, the standard exponential-martingale or quadratic-variation comparison may produce a Gronwall factor that diverges with n, preventing the claimed rate. The manuscript must supply the missing uniform estimate or show why it is unnecessary.","section":"Proof of the main convergence-rate result (as described in the abstract)"}],"minor_comments":[{"comment":"The abstract states the result but lists neither the precise integrability/growth assumptions on the generator and terminal condition nor the definition of the penalty term, making it impossible to verify applicability of the BMO estimates from the abstract alone.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"The abstract is unusually terse for a convergence-rate claim; the full manuscript should be checked to confirm whether the uniform BMO bound is actually derived or merely asserted."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and insightful comments on our manuscript. We address the major comment below.","responses":[{"response":"We thank the referee for highlighting this potential issue in the application of BMO estimates. The current proof applies BMO-martingale techniques to the difference process but does not isolate an explicit uniform-in-n bound on the BMO norm of the penalized driver term as a separate step. We agree that making this control explicit would strengthen the argument and prevent any concern about n-dependent factors in the estimates. We will revise the manuscript by adding a preliminary lemma that establishes the required uniform BMO bound on the difference driver, derived from the quadratic growth condition, comparison principles, and a priori L^2 estimates on the solutions that hold independently of n. This addition will be incorporated into Section 3 without changing the main result or assumptions.","revision_made":"yes","referee_comment":"[Proof of the main convergence-rate result (as described in the abstract)] The central claim (abstract) asserts that BMO-martingale estimates applied to the difference process between the reflected and penalized solutions close at rate 1/2 uniformly in the penalty parameter n. The penalized driver contains the term n(Y^n - L)^- whose size grows linearly with n. Without an explicit a priori bound showing that the BMO norm of this difference driver (or its quadratic variation) remains controlled independently of n, the standard exponential-martingale or quadratic-variation comparison may produce a Gronwall factor that diverges with n, preventing the claimed rate. The manuscript must supply the missing uniform estimate or show why it is unnecessary."}],"tokens_in":1204,"tokens_out":340,"duration_ms":31334,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central claim is a convergence rate of order 1/2 between the reflected BSDE and its penalized version when the generator is quadratic, obtained by controlling the difference process with BMO-martingale inequalities. They then use the rate to justify an Euler polygonal approximation for the sub-quadratic case.\n\nThat rate is the concrete new piece; most prior work on reflected BSDEs with quadratic drivers gives existence or qualitative convergence but not this explicit dependence on the penalty parameter. The application to numerical schemes is a reasonable follow-on and could be useful for people who actually compute these things.\n\nThe soft spot is exactly the one flagged in the stress-test note. The penalized driver contains the term n(Y^n - L)^-, whose size grows with n. For the BMO comparison on the difference to deliver a uniform-in-n bound, the quadratic variation or exponential-martingale estimates have to remain controlled without constants that blow up. The abstract states the result but does not show the a priori bounds or the n-independent estimates that would close the argument. If the full proof supplies those, the claim holds; if it relies on implicit uniformity that is not verified, the rate could degrade. The rest of the assumptions look standard for the area.\n\nThis is narrow-scope work inside stochastic analysis. Specialists who need rates for reflected BSDE numerics will find it worth reading. It is coherent on its own terms and engages the existing literature, so it deserves a serious referee rather than a desk reject. Referees should be asked to verify the BMO uniformity step in detail.","headline":"The paper gives an explicit 1/2 rate for penalization of reflected quadratic BSDEs via BMO estimates on the difference process, then applies it to Euler schemes, but the uniformity of those estimates in the penalty parameter is the part that needs checking.","tokens_in":2126,"tokens_out":412,"would_cite":false,"duration_ms":16609,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Reflected BSDEs with quadratic generators converge to their penalized counterparts at rate 1/2.","keywords":["reflected BSDEs","quadratic generators","penalization","convergence rate","BMO martingales","numerical approximation","Euler method"],"falsifier":"Finding a quadratic generator and terminal condition where the L^p norm of the difference between reflected and penalized solutions fails to be O(ε^{1/2}) for small penalty parameter ε.","tokens_in":2451,"feed_emoji":"","tokens_out":590,"duration_ms":25262,"temperature":0.7,"pith_summary":"This paper proves that the solutions to reflected backward stochastic differential equations with quadratic generators and the solutions to their penalized versions differ by an amount that shrinks like the square root of the penalty parameter. The proof relies on estimates from BMO martingales applied to the difference processes. A reader might care because the explicit rate gives a concrete way to balance penalty size against approximation error when solving these equations numerically. The result is then used to justify an Euler polygonal line scheme for reflected BSDEs that have only sub-quadratic generators.","feed_headline":"Reflected BSDEs converge at order 1/2 to penalized versions","feed_subtitle":"BMO estimates give an explicit rate in the penalty parameter that supports error analysis for numerical schemes.","key_machinery":"BMO martingale techniques controlling the difference process between the reflected and penalized equations.","core_discovery":"Using techniques of BMO martingales, we prove the convergence rate is at order 1/2 as a function of the penalty parameter between reflected BSDEs with quadratic generators and their penalized BSDEs. The result is applied to study numerical approximation of reflected BSDEs with sub-quadratic generators by the Euler's polygonal line method.","pith_inferences":["Whether the 1/2 rate is optimal could be checked by constructing an example where the error decays exactly like sqrt(ε).","Similar BMO estimates might yield rates for other approximation schemes such as time-discretization of the reflected equations.","Connections to optimal stopping problems could follow since reflected BSDEs often represent solutions to those."],"forward_implications":["The error between reflected and penalized solutions is bounded by C times the square root of the penalty parameter.","This bound holds for quadratic generators under standard integrability conditions.","The rate extends the applicability of penalization to numerical methods for reflected BSDEs.","The same penalization approach works for sub-quadratic generators in the Euler scheme context."],"fun_headline_variants":["Reflected quadratic BSDEs converge at 1/2 rate to penalized ones","BMO martingales establish 1/2 order convergence in reflected BSDEs","Convergence rate of 1/2 for reflected BSDEs by penalization method","1/2 convergence between reflected BSDEs and penalized versions shown","Quadratic reflected BSDEs achieve 1/2 penalization convergence rate"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The generators and terminal conditions admit BMO-martingale estimates that control the difference between the reflected equation and the penalized equation.","fun_headline_variants_meta":{"raw":{"variants":["Reflected quadratic BSDEs converge at 1/2 rate to penalized ones","BMO martingales establish 1/2 order convergence in reflected BSDEs","Convergence rate of 1/2 for reflected BSDEs by penalization method","1/2 convergence between reflected BSDEs and penalized versions shown","Quadratic reflected BSDEs achieve 1/2 penalization convergence rate"]},"model":"grok-4.3","cost_usd":0.004891,"raw_usage":{"total_tokens":2314,"prompt_tokens":500,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":48912000,"prompt_tokens_details":{"text_tokens":500,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1724,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":500,"tokens_out":90,"duration_ms":12795,"temperature":1.0,"reasoning_tokens":1724,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T20:45:33.442689+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a quadratic generator and terminal condition where the L^p norm of the difference between reflected and penalized solutions fails to be O(ε^{1/2}) for small penalty parameter ε.","supporting_citations":[],"review_version":1}