{"id":"fb66ff2c-f987-455d-84fb-c7caa1c9bcd8","arxiv_id":"2505.20437","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves existence, uniqueness and stability for backward SDEs driven by Brownian motion and a discontinuous rough path of finite q-variation, lifting solution and driver into a Skorokhod-type decorated path space, and derives well-posedness for the corresponding backward doubly SDEs.","lead":"This paper proves that a large class of backward stochastic differential equations remain well-posed when the noise includes an irregular, jumping path alongside Brownian motion, and that solutions move continuously with the noise. It matters because such equations model hedging and nonlinear stochastic evolution under rough or jumpy randomness, and the result supplies a unified existence, uniqueness, and stability theory for them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem B.3's Itô formula is misstated: the jump correction is missing in the first form and uses the wrong evaluation point in the second form; the central estimates (3.9) and (3.33) depend on it.","rationale":"The reader's weakest-assumption pinpoints Appendix B as the background calculus on which the central estimates rest, and my reading confirms that this is the most load-bearing piece. However, rather than a vague possibility that the Föllmer extension might fail, I found a concrete, checkable error in the statement of Theorem B.3: the jump correction is missing or mis-evaluated in both displayed forms. A single-jump counterexample shows the printed second form violates the identity; the proof's intermediate formula is correct, so the error appears to be a typo in the theorem statement rather than a fatal flaw in the argument. The paper's applications, e.g. display (3.9), seem to use the correct combination in the specific RBSDE context, and my preliminary check with a trivial RBSDE indicates the apriori estimates survive. For this reason I do not recommend changing the reader's CONDITIONAL verdict, but the theorem statement must be corrected and the derivations of (3.9) and (3.33) re-verified with the corrected formula before the version of record. The concern is substantive enough to keep the verdict at CONDITIONAL rather than ACCEPT.","tokens_in":59129,"tokens_out":64886,"duration_ms":572319,"concrete_test":"Compute both sides of Theorem B.3 for the explicit càglàd path A = 2·1_{[0,t₀)} + 1_{[t₀,T]}, M ≡ 0, f(x) = x², using the paper's backward-Young-integral definition (right-point evaluation). Verify that the printed second form fails by the amount ∑ (f'(Y_{s+}) − f'(Y_s))ΔY_s, and that inserting f'(Y_{s+}) in the correction (resp. adding the missing jump sum in the first form) makes the identity hold. Then re-derive (3.9) for the trivial RBSDE with g ≡ 1, ξ = 1, W a single-jump càglàd path, to confirm the apriori bound estimates are unaffected by the corrected formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix B, Theorem B.3, states two equivalent Itô formulas for Y = A + M with A càglàd of finite q-variation. The second form writes the jump correction as sum_{0≤st0, so Δ⁺A_{t0}=-1, and take M=0 and f(x)=x². Then f(A_T)-f(A_0)=1-4=-3. The first displayed formula gives f(Y_T)=4+∫ f'(A+_s)dA_s=4-2=2, and the second displayed formula gives 4+∫ f'(A_s)d←-A_s + [f(1)-f(2)-f'(2)Δ⁺A_{t0}]=4-2+(1-4+4)=3. The jump correction must use f'(Y_{s+}) when the integral is the backward Young integral with right-endpoint evaluation; the corrected correction term in the example equals -1, yielding the correct value 1. This theorem is the basis for the displayed identities (3.9), (3.12), (3.33), and (3.38), so the central apriori bound and contraction estimates are not reliable as written. The estimates appear salvageable, since the erroneous sign enters through terms that can still be bounded by the same positive sums, but the proof must be re-derived with a corrected formula and the main estimates updated accordingly.","section":"Appendix B, Theorem B.3"},{"comment":"In the contraction proof, Itô's formula is applied to |\\bar Y^Δ_t|², so the quadratic variation term on the left should involve \\bar Z^Δ. As printed, Eq. (3.33) has E_t∫|Z^Δ_r|²dc_r, without the bar, while (3.30) and (3.39) correctly concern \\bar Z^Δ. The displayed inequality is therefore false as written and must be corrected; the surrounding estimates indicate the intended statement, so this is a local but necessary correction.","section":"Section 3.2, Eq. (3.33)"},{"comment":"Eq. (3.15) states ∥ΔY_{t_{i-1}}∥∞ = ∥φ(g_{t_{i-1}}ΔW_{t_{i-1}},Y_{t_{i-1}+})-Y_{t_{i-1}+}∥∞ ≤ C_g|ΔW_{t_{i-1}}|, but this is not the estimate supplied by Taylor's formula: Taylor yields |φ(V,x)-x| ≤ |g|∞|V| + O(|V|²), so the displayed bound with the bare constant C_g is false in general and can only be obtained after enlarging the constant using the local smallness of ∥W∥_{q;(t_{i-1},t_i]}. Two sentences later the forward-jump relation is printed as Y_{t_{i-1}}=-g_{t_{i-1}}(Y_{t_{i-1}+}), which is missing the factor ΔW; the correct reverse-time relation is Y_{t_{i-1}}=Y_{t_{i-1}+}+g_{t_{i-1}}(Y_{t_{i-1}+})ΔW_{t_{i-1}}. These are fixable, but as written the apriori bound proof contains false identities.","section":"Section 3.1, Eq. (3.15)"}],"minor_comments":[{"comment":"Corollary 5.11 is stated without proof, with only a reference to an analogous argument in [27]; given the two-sided filtration and the non-semimartingale finite-q-variation driver, a proof sketch or a precise statement of the analogy should be provided.","section":"Section 5.2, Corollary 5.11"},{"comment":"In the statement of Lemma 4.5, the text reads 'q¿0' instead of 'q>0'; this should be corrected.","section":"Section 4.1, Lemma 4.5"},{"comment":"The notation Y^{k,∞,δ_l} for the solution of the n-th Picard iterate is hard to parse because the superscript ∞ is used both for the limiting equation and for the Picard limit; consider a clearer notation such as Y^{k,δ_l} for the solution and Y^{k,n,δ_l} for the iterates.","section":"Section 4.3, Theorem 4.7 proof"},{"comment":"The displayed estimate (3.8) contains an unmatched parenthesis and an unclear line break in the term involving C_g(1+|Y|_{p,2;[T-ε,T]}); reformatting would improve readability.","section":"Section 3.1, Eq. (3.8)"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is Theorem B.3: the displayed Itô formula is wrong, and the stress-test note is accurate on this point. The main theorems are likely sound after a corrected formula is proved and the estimates (3.9), (3.12), (3.33), and (3.38) are re-verified; the other concerns are local. I would not reject on the present evidence, but the manuscript cannot be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a real advance: global well-posedness for forward- and Marcus-type RBSDEs with càglàd Young drivers, stability in a decorated-path Skorokhod metric, and a new BDSDE class are all genuinely new relative to the cited literature. The paper is self-contained and the strategy—apriori bounds, local fixed point, gluing, Picard iteration—is coherent. Credit where due: the measurable-selection section and the decorated-path stability proof are substantial pieces of work.\n\nSecond, the load-bearing Itô formula in Appendix B is misstated. Theorem B.3's first form drops the jump correction entirely; the second form writes the correction with f'(Y_s) while the backward Young integral evaluates at the right endpoint, so it should be f'(Y_{s+}). The concrete example in the stress note is exactly right: with A a single càglàd jump of size -1, M=0, f=x², the printed second form gives -1 where the true change is -3; the first form gives -2. Equations (3.9) and (3.33) are direct applications of this theorem, so the apriori bounds and the contraction estimates are not justified as written. This is not a typo in a peripheral lemma; it sits under the main theorems.\n\nThe good news is that the error looks repairable. Replacing the correction by the right-endpoint version changes signs in a favorable direction, and the extra jump terms are controlled by the q-variation of W. But the authors need to restate Theorem B.3 and rework the estimates in Section 3. Until that is done, I would not rely on the proofs.\n\nOther soft spots are minor by comparison: Corollary 5.11 is stated without a proof, and there are small presentation slips (the header date, some notation inconsistencies). The comparison with the literature is fair and the self-citation is not an issue.\n\nWho should read this: specialists in rough BSDEs and pathwise Itô calculus. It deserves a serious referee, and I would not desk-reject it. My recommendation: send it out, but tell the authors to fix the Itô formula and re-verify the central estimates before the version of record. If the fix is as local as it looks, the paper should be accepted after revision.","headline":"Genuinely new results on rough BSDEs with jumps, but the appendix Itô formula underpinning the main estimates is misstated and needs a real revision.","tokens_in":59946,"tokens_out":18410,"would_cite":false,"duration_ms":185873,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60L90","60J76","60H20","60H15","37H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that backward stochastic differential equations with discontinuous rough drivers, of both forward-jump and Marcus-jump type, have unique global solutions, and that the corresponding randomized equations are well posed.","keywords":["BSDE","BDSDE","rough paths with jumps","Skorokhod topology","Marcus integration","Wong-Zakai approximation","backward Young integral","finite q-variation"],"falsifier":"Take a càglàd path $A$ of finite $q$-variation with $q$ close to 2 and jumps placed at dyadic times, together with a Brownian motion $M$, and compute the cross-variation sums $\\sum (M_{t_{i+1}}-M_{t_i})(A_{t_{i+1}}-A_{t_i})$ along refined partitions. If for some such $A$ produced by the fixed-point map these sums do not vanish, Lemma B.1 and hence the a priori estimates (3.9) and (3.33) fail, and the well-posedness theorem would be false.","tokens_in":58896,"feed_emoji":"🎲","tokens_out":8052,"duration_ms":65610,"temperature":0.7,"pith_summary":"The paper proves global existence and uniqueness for backward stochastic differential equations driven jointly by Brownian motion and by a deterministic, possibly jumping path $W$ of finite $q$-variation with $q \\in [1,2)$. It treats two jump conventions: forward-type jumps, where the solution jumps in step with the driver, and Marcus-type jumps, where the jump follows the flow of the vector field. This extends rough-BSDE theory beyond continuous drivers, which is necessary for randomized versions of the equations in which the rough driver is itself a stochastic process. The paper also shows that solutions depend continuously on the driving path in a Skorokhod-type topology, and derives a new class of well-posed backward doubly stochastic differential equations.","feed_headline":"Backward SDEs survive discontinuous rough drivers","feed_subtitle":"Existence, uniqueness, and stability now hold for forward- and Marcus-type jumps; new BDSDEs follow.","key_machinery":"The load-bearing object is the backward Young integral $\\int_t^T g(r,Y_{r+})\\,(\\diamond)\\,dW$, whose definition (Appendix A) requires the integrand to have finite $p$-variation regularity; the fixed-point map of Theorem 3.2 is constructed so that this regularity is preserved inside a ball in $B_p \\times \\mathrm{BMO}$. The second mechanism is an extension of the pathwise Itô formula (Appendix B) to processes $Y = A + M$ with $A$ càglàd of finite $q$-variation for $q<2$ and $M$ a continuous local martingale; this yields the quadratic-variation estimates (3.9) and (3.33) that drive the a priori bounds. For stability, the driver $W$ and the solution $Y$ are embedded into the space of decorated paths, that is, rough paths augmented with extra information at each jump describing the trajectory used to cross the jump; this space is equipped with the Skorokhod-type $p$-variation metric $\\alpha_p$, whose $\\delta$-extensions encode jump excursions, so that convergence of drivers does not erase the jump dynamics.","core_discovery":"The central discovery, Theorem 3.4, is that under Assumption A the forward-type and Marcus-type rough backward SDEs each admit a unique solution $(Y,Z)$ in the space $B_p \\times \\mathrm{BMO}$ on the whole interval $[0,T]$, where $Y$ is a càglàd adapted process with conditional finite $p$-variation and $Z$ is a BMO integrand for the Brownian martingale. The proof derives global a priori bounds directly, without comparison theorems, so $Y$ need not be one-dimensional and the vector field $g$ may be nonlinear; local fixed-point contractions are then glued over a partition after prescribing the solution by hand at the large jumps of $W$. The paper further proves that the Picard iterates converge globally to the solution (Theorem 3.5), that the solution map is stable with respect to perturbations of $W$ measured in the Skorokhod-type $\\alpha_p$ metric on decorated paths (Theorem 4.7), and that the corresponding backward doubly stochastic SDEs driven by an independent finite-$q$-variation process $L$ are well posed (Theorem 5.10).","pith_inferences":["Reader inference: combining Theorem 4.7 with the time-stretching equivalence suggests a Wong–Zakai recipe — smooth the driver, solve the continuous RBSDE, and take the limit — with Marcus jumps recovered by linear excursions and forward jumps by constant excursions; the stability theorem is exactly the convergence ingredient such a recipe needs.","Reader inference: the measurable-selection results in Section 5.1 make the solution map a function of the driving path, so one could simulate a path of $L$, solve the deterministic RBSDE pathwise, and thereby approximate the BDSDE solution; this conditional-solution structure is a natural target for numerical schemes.","Reader inference: the direct a priori bound method is expected to degrade as $q$ approaches 2 because the local interval length depends on the $q$-variation of $W$; the theory should break down in the Brownian limit, which matches the paper's exclusion of $L$ being a Brownian motion."],"forward_implications":["Both forward-type and Marcus-type rough backward SDEs with Lipschitz coefficients are globally well posed, with no one-dimensionality restriction on $Y$.","The Picard iteration converges to the unique solution in $B_p \\times \\mathrm{BMO}$, giving a constructive approximation scheme.","If a sequence of rough drivers converges in the $\\alpha_q$ metric on decorated paths, the corresponding $Y$-solutions converge in $\\alpha_p$ in probability and the $Z$-components converge in $L^2(dt \\otimes P)$.","Marcus-type solutions coincide with the time-stretching solutions, so the classical Marcus jump intuition is validated in the backward setting.","Randomizing the rough driver yields a new well-posed class of backward doubly stochastic differential equations whose noise can be fractional Brownian motion with Hurst index $H>1/2$, pure-jump Lévy processes, or sums of such processes."],"supporting_citations":[{"why":"Supplies the $B_p \\times \\mathrm{BMO}$ solution spaces and the local fixed-point scheme for continuous Young drivers that this paper extends to discontinuous drivers.","marker":"[12]"},{"why":"Introduces rough backward SDEs and the formal limiting-equation problem that this paper resolves with an intrinsic integral equation.","marker":"[11]"},{"why":"Provides the distinction between forward-type and Marcus-type jumps and the flow $\\varphi$ used to define Marcus jumps.","marker":"[22]"},{"why":"Supplies the backward Young integral, its estimates, and the rough-differential-equations-with-jumps framework used throughout the proofs.","marker":"[18]"},{"why":"Provides the decorated path space and the Skorokhod-type metric $\\alpha_p$ on which the stability theorem is built.","marker":"[8]"},{"why":"Introduced backward doubly stochastic differential equations and the probabilistic representation for SPDEs that motivates Section 5.","marker":"[27]"},{"why":"Studies BDSDEs driven by fractional Brownian motion with Hurst parameter in $(1/2,1)$, the overlapping case of the new BDSDE class.","marker":"[21]"},{"why":"Gives the pathwise Itô formula that Appendix B extends to càglàd finite-$q$-variation processes, the backbone of the a priori estimates.","marker":"[19]"},{"why":"Provides the control-function estimates and $p$-variation inequalities used repeatedly in the fixed-point and stability arguments.","marker":"[17]"}],"fun_headline_variants":["Global well-posedness for RBSDEs with discontinuous Young drivers","Forward and Marcus rough backward SDEs tamed","Jumps in rough drivers no longer break backward SDEs","Stable solutions for rough backward SDEs with jumps","New BDSDE class emerges from rough backward SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on the extended Itô formula in Appendix B, which treats $Y = A + M$, a left-continuous finite-$q$-variation part plus a continuous martingale, as a pathwise quadratic-variation process; if that formula fails for the solution class, the a priori bounds—and therefore global existence—collapse.","fun_headline_variants_meta":{"raw":{"variants":["Global well-posedness for RBSDEs with discontinuous Young drivers","Forward and Marcus rough backward SDEs tamed","Jumps in rough drivers no longer break backward SDEs","Stable solutions for rough backward SDEs with jumps","New BDSDE class emerges from rough backward SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":3054,"prompt_tokens":992,"completion_tokens":2062,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":1977}},"tokens_in":608,"tokens_out":2062,"duration_ms":13620,"temperature":1.0,"reasoning_tokens":1977,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:57:04.239878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a càglàd path $A$ of finite $q$-variation with $q$ close to 2 and jumps placed at dyadic times, together with a Brownian motion $M$, and compute the cross-variation sums $\\sum (M_{t_{i+1}}-M_{t_i})(A_{t_{i+1}}-A_{t_i})$ along refined partitions. If for some such $A$ produced by the fixed-point map these sums do not vanish, Lemma B.1 and hence the a priori estimates (3.9) and (3.33) fail, and the well-posedness theorem would be false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $B_p \\times \\mathrm{BMO}$ solution spaces and the local fixed-point scheme for continuous Young drivers that this paper extends to discontinuous drivers."},{"cited_title":"4, 1715–1758","cited_arxiv_id":null,"evidence_quote":"Introduces rough backward SDEs and the formal limiting-equation problem that this paper resolves with an intrinsic integral equation."},{"cited_title":"Kurtz, ´Etienne Pardoux, and Philip Protter, Stratonovich stochastic differ- ential equations driven by general semimartingales, Annales de l’Institut Henri Poincar´ e","cited_arxiv_id":null,"evidence_quote":"Provides the distinction between forward-type and Marcus-type jumps and the flow $\\varphi$ used to define Marcus jumps."},{"cited_title":"Friz and Huilin Zhang, Differential equations driven by rough paths with jumps, Journal of Differential Equations 264 (2018), no","cited_arxiv_id":null,"evidence_quote":"Supplies the backward Young integral, its estimates, and the rough-differential-equations-with-jumps framework used throughout the proofs."},{"cited_title":"MR 4840242","cited_arxiv_id":null,"evidence_quote":"Provides the decorated path space and the Skorokhod-type metric $\\alpha_p$ on which the stability theorem is built."},{"cited_title":"2, 209–227","cited_arxiv_id":null,"evidence_quote":"Introduced backward doubly stochastic differential equations and the probabilistic representation for SPDEs that motivates Section 5."},{"cited_title":"5, 655–665","cited_arxiv_id":null,"evidence_quote":"Studies BDSDEs driven by fractional Brownian motion with Hurst parameter in $(1/2,1)$, the overlapping case of the new BDSDE class."},{"cited_title":"F¨ ollmer,Calcul d’Itˆ o sans probabilit´ es, Seminar on Probability, XV (Univ","cited_arxiv_id":null,"evidence_quote":"Gives the pathwise Itô formula that Appendix B extends to càglàd finite-$q$-variation processes, the backbone of the a priori estimates."},{"cited_title":"Friz and Nicolas B","cited_arxiv_id":null,"evidence_quote":"Provides the control-function estimates and $p$-variation inequalities used repeatedly in the fixed-point and stability arguments."}],"review_version":1}