{"id":"922477f1-573d-4ae8-b6ef-46d9cdeb42b5","arxiv_id":"1908.01188","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The random walk approximation for Markovian BSDEs converges in Wasserstein distance at rate n^{-(α∧ε/2)}, improving n^{-ε/4} and reaching the CLT-optimal n^{-1/2} for Lipschitz data.","lead":"This paper proves that the scaled random walk approximation of a Markovian backward stochastic differential equation converges in Wasserstein distance with rate n^{-(α∧ε/2)}, improving the previously known n^{-ε/4} bound. This is the first optimal-rate result for this Donsker-type scheme under Hölder regularity, and it matches the classical central limit theorem rate when the data are Lipschitz.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"I reviewed the proof of Theorem 10 and Proposition 11, with particular attention to the reader's identified weakest assumption, Lemma 6. Step 2 of Lemma 6 approximates the generator by inf-convolution fη; the key question is whether the gradient Hölder constant can be taken independent of η even though fη is Lipschitz in space with constant η. The Gronwall argument in Step 1 uses the Lipschitz-based bound (27) only to justify finiteness/integrability of the relevant functions; the final constant from Lemma 14 depends only on the structural constants C1 and ‖f_z‖_Lip, not on the x-Lipschitz constant. Thus the uniform gradient estimates (48) and (49) are justified. I also checked the flow of Proposition 11: the βn−γn system is closed through (41) and (44), the Volterra singular kernel is handled by Lemma 14, and the nonsingular Gronwall step then yields the claimed n^{-(α∧ε/2)} rates. The handling of grid points in Theorem 10 uses the càglàd extension of Zn and the identity Zn_s=Δn(s−h/2,B_{s−h/2}); with the paper's under/overline convention this is consistent. Several OCR-level ambiguities exist, notably the missing minus sign in (30) and the unhandled ns=nt case in Proposition 3, but neither affects the central argument once read with the intended notation. No circular step, missing support, or contradicting limitation statement was found. The central claim is new, the proof is self-contained, and the rate improvement from n^{-ε/4} to n^{-(α∧ε/2)} is credible.","tokens_in":26188,"tokens_out":41797,"duration_ms":396014,"concrete_test":"Verify Lemma 6 in the model case f=0, g(x)=|x|^ε: compute ∇u(t,x)=E[g(x+√(T−t)G)G]/√(T−t) and check |∇u(t,x)|≤C(T−t)^{−(1−ε)/2} and ‖∇u(t,·)‖_ε≤C/√(T−t) for ε∈(0,1), by direct Gaussian estimates; if both hold, the regularity that Proposition 11 leans on is credible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rate n^{-(α∧ε/2)} for Y and Z is supported by a coherent chain: Rio's Wasserstein bound, Corollary 4, the PDE regularity Lemma 6, and the discrete regularity Lemma 8. The most delicate point is Lemma 6: it asserts ∇u(t,·) is ε-Hölder with constant C/√(T−t) and |∇u| ≤ C/(T−t)^{(1−ε)/2} when f is only ε-Hölder in space. I traced Step 2's inf-convolution argument. The uniform-in-η estimates do not depend on the large x-Lipschitz constant of fη: that constant enters only through the a priori bound (27) used to justify Gronwall, not through the final Gronwall constant. The convergence ∇uη→v and dominated convergence to identify v=∇u is internally consistent. The only genuinely suspicious displayed formula is (30), whose exponent should be −(1−ε)/2 rather than (1−ε)/2; all subsequent applications use the negative exponent, so this is a typo, not a mathematical gap. The minor edge case in Proposition 3 when t<s but both points lie in one grid cell is handled by a direct bound and does not affect (22). I therefore find no load-bearing defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative Donsker-type theorem for the Briand-Delyon-Mémin random-walk approximation of Markovian BSDEs. Under Assumption (A1), where the terminal condition g is ε-Hölder and the generator f is α-Hölder in time, ε-Hölder in space, and Lipschitz in (y,z), the authors show that the random-walk-driven solution (Y^{n,t,x}, Z^{n,t,x}) converges in L^r-Wasserstein distance to the Brownian BSDE solution (Y^{t,x}, Z^{t,x}) at rate n^{-(α∧ε/2)} for Y, and with the additional factor (T-s)^{-1/2} for Z (Theorem 10). The proof uses Rio's Wasserstein central limit theorem, the Feynman-Kac/PDE representation, new regularity estimates for u and ∇u (Lemma 6, proved via inf-convolution in Appendix A.3), discrete BSDE a priori estimates (Lemma 12), and a two-stage Gronwall argument in Proposition 11.","tokens_in":26389,"tokens_out":27772,"duration_ms":266432,"significance":"If the result holds, it is a substantial improvement over the previous rate n^{-ε/4} obtained by Geiss-Labart-Luoto, and it matches the expected optimal rate n^{-ε/2}; for ε=1 and α≥1/2 it recovers the n^{-1/2} rate of Rio's Wasserstein CLT. The Wasserstein approach is well chosen for this problem and gives clean statements with explicit singular behavior of the Z-error near the terminal time. The proof is structurally self-contained: the essential inputs are Rio's theorem, Zhang's representation, and classical BSDE a priori estimates, and no constants are fitted or ad hoc. The main regularity lemma is delicate, but the inf-convolution argument in the appendix is a reasonable strategy and the overall chain of estimates is internally consistent.","major_comments":[],"minor_comments":[{"comment":"The exponent in the display should be −(1−ε)/2 rather than (1−ε)/2. The subsequent applications in Lemma 6(bii) and Proposition 9 consistently use the negative-exponent form, so I regard this as a typographical slip, but the displayed inequality as written is false and should be corrected.","section":"Section 4, Eq. (30)"},{"comment":"The claim that the inf-convolution fη satisfies (3) uniformly in η is not immediate for the ε-Hölder constant in x. The uniformity follows by combining the η-Lipschitz bound with the L∞-approximation error in (46); I recommend adding a short justification or a reference, since this uniformity is used to make the constants in (48) independent of η.","section":"Appendix A.3, Step 2"},{"comment":"The identity Z^{n,t,x}_s = Δ_n(s−h/2, B^{n,t,x}_{s−h/2}) relies on the convention that U_n is evaluated at the lower grid point; a brief reminder of this convention in the display would prevent confusion, since without it the time shift looks off by h/2.","section":"Section 5, proof of Theorem 10, grid-point case"},{"comment":"The inequality in (50) has the weakly singular kernel (s−r)^{−ε/2}, whereas Lemma 14 is stated for the kernel (s−r)^{−1/2}. The conclusion used in the text follows from a standard generalized Volterra Gronwall inequality, but the paper should either state that version or explain why Lemma 14 applies; as written this step is not immediate.","section":"Appendix A.3, use of Lemma 14"},{"comment":"The triangle-inequality step in the proof of Proposition 3 is hard to parse because the overline/underline notation for grid endpoints is not visible in the display; please ensure that the Brownian increments B_{\\underline{t}} and B_{\\overline{s}} are clearly distinguished from B_t and B_s.","section":"Section 3, Proposition 3"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well within the scope of the journal and the central claim is convincing. The issues I found are local presentation problems rather than obstacles to the main result. I do not see any citation or novelty concerns beyond the normal expectation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real improvement and the proof is sound. The paper moves the rate for the random-walk BSDE scheme from n^{-ε/4} to n^{-(α∧ε/2)} in Wasserstein distance, exactly the rate the authors' earlier work left as the expected optimum. The novelty is not the conjecture but the argument: Rio's Wasserstein CLT combined with PDE gradient regularity, with the Z-process error needing the (T-t)^{-1/2} singularity handled carefully. That is a substantial piece of work.\n\nWhat is done well: the proof is genuinely self-contained against standard external results (Rio, Zhang, El Karoui-Peng-Quenez), the hard regularity lemma (Lemma 6) is proved in the appendix by inf-convolution, and the discrete a priori estimates are included. The main theorem's statement is precise about the Z singularity. There are no fitted constants and no post-hoc calibration; the rate is derived, not assumed. The paper also honestly cites the previous n^{-ε/4} work and the Skorohod-embedding route, explaining why that route could not reach the optimal exponent.\n\nSoft spots are minor and typographical. Equation (23) in Proposition 3 has a repeated term on the right; it is clearly a typesetting slip because the next line treats the two terms separately. Equation (30) displays the exponent (1-ε)/2 where the calculation requires (ε-1)/2, and all later uses take the negative exponent; again a sign typo, not a gap. The edge case in Proposition 3 when t<s but both points lie in one grid cell is handled by a direct bound, though it is easy to misread. The proof is dense; a referee should ask for a few more explanatory sentences around Lemma 6's Step 2, but tracing that argument found no load-bearing defect.\n\nOne thing to note: the result is for Wasserstein distance rather than L^p convergence. That is the right metric for the Donsker approximation, but it means the comparison to L^2 results in earlier papers should be read carefully; they are not in competition, they measure different quantities.\n\nWho this is for: anyone working on numerical methods for BSDEs, especially Donsker-type and binomial-tree schemes, and people interested in weak convergence of semilinear PDEs. It deserves a serious referee. I would accept it for peer review and, if I were in the area, I would cite it.","headline":"The paper delivers the conjectured optimal Donsker rate n^{-(α∧ε/2)} for BSDEs in Wasserstein distance, and the proof survives a close read; the remaining issues are typos and density, not substance.","tokens_in":26969,"tokens_out":2471,"would_cite":true,"duration_ms":26157,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60H35","60F05","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random-walk driven BSDEs approximate Brownian BSDEs in Wasserstein distance at rate $n^{-(\\alpha\\wedge\\varepsilon/2)}$.","keywords":["Wasserstein distance","backward stochastic differential equations","random walk approximation","Donsker theorem","rate of convergence","semilinear heat equation","Hölder regularity","finite-difference scheme"],"falsifier":"Set $T=1$, take $f=0$ and $g(x)=|x|^\\varepsilon$ with $\\varepsilon\\in(0,1)$, and compute the random-walk scheme exactly for $n=2^m$. At a fixed time $s<1$, measure the Wasserstein-2 distance between $Z^{n,0,0}_s$ and $\\nabla u(s,B_s)$ for the heat-equation value function $u$; the theorem predicts decay like $n^{-\\varepsilon/2}$, so a slower observed exponent disproves it. One can also check Lemma 6 directly by testing whether $|\\nabla u(t,x)-\\nabla u(t,y)|\\le C(T-t)^{-1/2}|x-y|^\\varepsilon$ holds for a sequence $t\\uparrow T$.","tokens_in":25970,"feed_emoji":"🎲","tokens_out":14300,"duration_ms":130485,"temperature":0.7,"pith_summary":"This paper proves a rate of convergence in Wasserstein distance for approximating a Markovian backward stochastic differential equation (BSDE) by a BSDE driven by a scaled random walk. Under Assumption (A1), where the terminal condition $g$ is $\\varepsilon$-Hölder and the generator $f$ is $\\alpha$-Hölder in time and $\\varepsilon$-Hölder in space, the paper establishes $W_r(Y^{n,t,x}_s,Y^{t,x}_s)\\leq C_r(1+|x|)^\\varepsilon n^{-(\\alpha\\wedge\\varepsilon/2)}$ and $W_r(Z^{n,t,x}_s,Z^{t,x}_s)\\leq C_r(1+|x|)^\\varepsilon (T-s)^{-1/2}n^{-(\\alpha\\wedge\\varepsilon/2)}$. This confirms the expected improvement over the previous rate $n^{-\\varepsilon/4}$ and, for smooth data, reaches the $n^{-1/2}$ rate of the classical random-walk central limit theorem. The result matters because it identifies the exact weak error of a simple Donsker-type discretization of BSDEs with irregular data, which is the relevant regime for Monte Carlo simulation.","feed_headline":"Random-walk BSDEs converge at the n^{-(α∧ε/2)} rate","feed_subtitle":"Improves the prior n^{-ε/4} bound and matches the random-walk CLT speed for Hölder data.","key_machinery":"The load-bearing machinery consists of the value function $u$ of the semilinear heat equation, its discrete finite-difference analogue $U_n$, the discrete gradient $\\Delta_n(t,x)=h^{-1/2}(U_n(t+h,x+\\sqrt{h})-U_n(t+h,x-\\sqrt{h}))/2$, and the representation formulas that write $u$ and $\\nabla u$ as expectations against Brownian increments and $U_n$ and $\\Delta_n$ as expectations against random-walk increments. The rate transfer relies on Lemma 6, which states that $\\nabla u(t,\\cdot)$ is $\\varepsilon$-Hölder with constant $C(T-t)^{-1/2}$ even though $f$ is only Hölder in space, and on the Wasserstein central-limit bound $W_r(B^{n,t,x}_s,B^{t,x}_s)\\leq C n^{-1/2}$ for the scaled random walk. A Gronwall/Volterra argument then closes the estimate for $\\Delta_n$.","core_discovery":"The central claim, on the paper's own terms, is that the Wasserstein distance between the random-walk-driven BSDE solution and the Brownian BSDE solution decays at the rate $n^{-(\\alpha\\wedge\\varepsilon/2)}$, with the $Z$ component carrying the natural factor $(T-s)^{-1/2}$ that diverges near the terminal time. The proof works by reducing the stochastic approximation to a deterministic comparison: the random-walk BSDE is represented through a finite-difference value function $U_n$ and discrete gradient $\\Delta_n$, while the Brownian BSDE is represented through the solution $u$ of the associated semilinear heat equation. The paper proves the pointwise estimates $|u-U_n|\\leq C(1+|x|)^\\varepsilon n^{-(\\alpha\\wedge\\varepsilon/2)}$ and $|\\nabla u-\\Delta_n|\\leq C(1+|x|)^\\varepsilon (T-t)^{-1/2}n^{-(\\alpha\\wedge\\varepsilon/2)}$, and then transfers these bounds to Wasserstein distance using the Hölder regularity of $u$ and $\\nabla u$.","pith_inferences":["The rate $n^{-\\varepsilon/2}$ suggests that the dominant error is the weak approximation of Brownian motion by the random walk, with the nonlinearity contributing only through the Hölder modulus; one would expect the same rate for any CLT-scaled Markov chain satisfying a Wasserstein CLT bound with rate $n^{-1/2}$.","The paper leaves implicit that the same proof should extend to multidimensional Brownian motion and random walks with i.i.d. increments with exponential moments, provided the Wasserstein CLT estimate and the gradient regularity lemma have multidimensional analogues.","When the data are smoother than Hölder, the bound saturates at $n^{-1/2}$, so further improvement would require a higher-order weak scheme rather than the plain random walk; this is consistent with the known behaviour of binomial tree approximations."],"forward_implications":["For terminal data with $\\varepsilon=1$ and time regularity $\\alpha\\ge 1/2$, the approximation attains the $n^{-1/2}$ Wasserstein rate, matching the random-walk CLT; the discretization does not slow down the weak error.","The $Z$ error must be expected to grow like $(T-s)^{-1/2}$ near $T$, so any practical simulation based on this scheme needs to handle that singularity.","The pointwise finite-difference rates for $U_n$ and $\\Delta_n$ provide a standalone statement about semilinear heat equations with Hölder data.","The Wasserstein formulation is essential: the same quantities measured in strong $L^p$ norms would not show this clean rate, because the $Z$ processes are not close pathwise."],"supporting_citations":[{"why":"Supplies the Wasserstein central-limit bound $W_r\\leq Cn^{-1/2}$ for scaled random walks, the base rate of the entire argument.","marker":"[15]"},{"why":"Introduces the random-walk-driven BSDE and its explicit discrete solution formulas, the approximation scheme studied here.","marker":"[5]"},{"why":"Gives the representation theorems used to write $u$ and $\\nabla u$ as conditional expectations and to identify $Z$ with $\\nabla u$.","marker":"[13]"},{"why":"Provides the $C^{0,1}$ regularity of the PDE value function and the representation of $\\nabla u$, which Lemma 6 extends to Hölder generators.","marker":"[19]"},{"why":"Establishes existence and uniqueness of $L^p$ solutions for the Brownian BSDE under the Hölder assumptions, defining the target solution.","marker":"[4]"},{"why":"Supplies the a priori estimate for discrete BSDEs used in Lemma 12 to control $U_n$ and $\\Delta_n$.","marker":"[6]"}],"fun_headline_variants":["BSDE random-walk convergence hits n^{-(α∧ε/2)}","Improved BSDE rate: matches CLT speed for Hölder data","Random-walk BSDEs converge faster: new rate n^{-(α∧ε/2)}","Faster Wasserstein convergence for random-walk BSDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on a precise regularity estimate: the spatial derivative of the PDE solution must be Hölder continuous with a constant proportional to $(T-t)^{-1/2}$, and if that constant blew up any faster the integral estimates for the $Z$ error would diverge and the rate would fail.","fun_headline_variants_meta":{"raw":{"variants":["BSDE random-walk convergence hits n^{-(α∧ε/2)}","Improved BSDE rate: matches CLT speed for Hölder data","Random-walk BSDEs converge faster: new rate n^{-(α∧ε/2)}","Faster Wasserstein convergence for random-walk BSDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1448,"prompt_tokens":829,"completion_tokens":619,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":535}},"tokens_in":445,"tokens_out":619,"duration_ms":6072,"temperature":1.0,"reasoning_tokens":535,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:22:30.852900+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $T=1$, take $f=0$ and $g(x)=|x|^\\varepsilon$ with $\\varepsilon\\in(0,1)$, and compute the random-walk scheme exactly for $n=2^m$. At a fixed time $s<1$, measure the Wasserstein-2 distance between $Z^{n,0,0}_s$ and $\\nabla u(s,B_s)$ for the heat-equation value function $u$; the theorem predicts decay like $n^{-\\varepsilon/2}$, so a slower observed exponent disproves it. One can also check Lemma 6 directly by testing whether $|\\nabla u(t,x)-\\nabla u(t,y)|\\le C(T-t)^{-1/2}|x-y|^\\varepsilon$ holds for a sequence $t\\uparrow T$.","supporting_citations":[{"cited_title":"Bismut, Théorie probabiliste du contrôle des diffusions , Mem","cited_arxiv_id":null,"evidence_quote":"Supplies the Wasserstein central-limit bound $W_r\\leq Cn^{-1/2}$ for scaled random walks, the base rate of the entire argument."},{"cited_title":"In this section, we state the main result of this paper which g ives the rate of convergence in the Wasserstein distance between the solution to the BSDE (","cited_arxiv_id":null,"evidence_quote":"Introduces the random-walk-driven BSDE and its explicit discrete solution formulas, the approximation scheme studied here."},{"cited_title":"In view of the regularity of f in time, we have ⏐ ⏐ ⏐ ⏐ ⏐E [ ∫ T t ( f ( s, Θ n,t,x s ) −f ( s, Θ n,t,x s )) ds ] ⏐ ⏐ ⏐ ⏐ ⏐ ≤Cn −α","cited_arxiv_id":null,"evidence_quote":"Gives the representation theorems used to write $u$ and $\\nabla u$ as conditional expectations and to identify $Z$ with $\\nabla u$."},{"cited_title":"Briand, B","cited_arxiv_id":null,"evidence_quote":"Provides the $C^{0,1}$ regularity of the PDE value function and the representation of $\\nabla u$, which Lemma 6 extends to Hölder generators."},{"cited_title":"Let us start by known regularity properties of the function u that follow from classical a priori estimates for BSDEs","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of $L^p$ solutions for the Brownian BSDE under the Hölder assumptions, defining the target solution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the a priori estimate for discrete BSDEs used in Lemma 12 to control $U_n$ and $\\Delta_n$."}],"review_version":1}