Rough backward SDEs with discontinuous Young drivers

Dirk Becherer; Yuchen Sun

REVIEW 3 major objections 4 minor 1 cited by

Rough backward SDEs with discontinuous Young drivers

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Pith's one-line read 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.

desk verdict Genuinely new results on rough BSDEs with jumps, but the appendix Itô formula underpinning the main estimates is misstated and needs a real revision. read the letter →

arxiv 2505.20437 v1 pith:D6AHHXCQ submitted 2025-05-26 math.PR

Dirk Becherer , Yuchen Sun This is my paper · ORCID

classification math.PR MSC 60L9060J7660H2060H1537H30

keywords BSDEBDSDEroughpathswithjumpsSkorokhodtopologyMarcusintegrationWong-ZakaiapproximationbackwardYoungintegralfiniteq-variation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Second, 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.

The 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.

Other 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.

Who 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.

Referee Report

3 major / 4 minor

Summary. The paper studies backward stochastic differential equations driven by a Brownian motion and a deterministic càglàd path W of finite q-variation, q∈[1,2), in both forward- and Marcus-type jump conventions. The main results are: global existence and uniqueness in B_p × BMO under Assumption A (Theorem 3.4), global convergence of Picard iterations (Theorem 3.5), stability of solutions with respect to perturbations of the driver in a Skorokhod-type p-variation metric on decorated paths (Theorem 4.7), and well-posedness of a class of backward doubly stochastic differential equations driven by an independent finite-q-variation process (Theorem 5.10). The proofs proceed by apriori bounds, local contraction arguments, gluing across jumps, a double-indexed Picard scheme, and measurable selection results; the appendix extends Föllmer's pathwise Itô calculus to sums of continuous local martingales and càglàd finite-q-variation processes.

Significance. If correct, this is a substantial contribution: it provides the first global well-posedness theory for rough backward SDEs with discontinuous Young drivers, removes the one-dimensional restriction of earlier comparison-based arguments, introduces a stability theory in decorated-path Skorokhod topologies, and establishes a new BDSDE class beyond Brownian noise. The paper is carefully structured and contains detailed fixed-point estimates, explicit norm control, measurable-selection arguments, and self-contained appendix material. However, because the pathwise Itô formula in Theorem B.3 is central to the apriori bounds and contraction estimates, the correctness of the main theorems currently hinges on correcting that formula and re-verifying the estimates derived from it.

major comments (3)

[Appendix B, Theorem B.3] Theorem B.3 is not correct as stated. Take A to be the càglàd path with a single jump at t0: A_t=2 for t≤t0 and A_t=1 for t>t0, 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 3.2, Eq. (3.33)] 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 3.1, Eq. (3.15)] 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.

minor comments (4)

[Section 5.2, Corollary 5.11] 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 4.1, Lemma 4.5] In the statement of Lemma 4.5, the text reads 'q¿0' instead of 'q>0'; this should be corrected.
[Section 4.3, Theorem 4.7 proof] 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 3.1, Eq. (3.8)] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central well-posedness and stability theorems are proved from external Young integration and pathwise Itô calculus; the only self-reference is a non-load-bearing pointer to future work.

full rationale

The paper derives global well-posedness of forward- and Marcus-type RBSDEs by proving a priori bounds and local contraction estimates in B_p × BMO, using the backward Young integral of Appendix A and an Itô-type formula in Appendix B. These inputs are external calculus results or are proved in the appendices, not restatements of the solution concept; no parameter is fitted and no 'prediction' is renamed input. The BDSDE results in Section 5 are obtained by randomizing the already-established RBSDE solution and using measurable-selection lemmas to verify the integral equation, which is a genuine lifting rather than a definitional reduction. The only self-citation is [4], a pointer to future work on rough PDEs, which is nowhere used in a derivation. One caveat, which is a correctness/technical concern rather than a circularity, is that the displayed Itô formulas in Theorem B.3 appear to misspecify the jump correction: the stated second form uses f'(Y_s) where evaluation at the right endpoint appears needed, and the first form omits the jump sum that the proof's intermediate formula contains. Since the printed identities are used in deriving the estimates (3.9) and (3.33), the a priori bounds inherit a proof gap if the correction is substantive, but this is not a circular argument and does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper contains no empirical fitting; constants in proofs are chosen to satisfy inequalities. The central results rest on standard external tools (Young integration, Follmer pathwise Ito calculus, decorated path theory, martingale representation) and on explicitly stated regularity and scope assumptions (Assumptions A, B, and the finite q-variation requirement q < 2). No invented physical entities are introduced; the decorated path lift is a proven construction rather than an unverified postulate.

assumptions (4)

standard math Backward Young integral existence, stability, and sewing estimates (Appendix A, Propositions A.1, A.7) based on the rough path jump theory of Friz-Zhang [18].
All estimates of the Young integral term in the RBSDE equation use these external results.
standard math Pathwise Ito formula for Y = A + M with A caglad of finite q-variation q < 2 and M a continuous local martingale (Appendix B, Theorem B.3, Lemma B.1).
Used at (3.9) and (3.33) for a priori bounds and contraction; the proof adapts Follmer's pathwise calculus.
domain assumption Assumptions A and B: W in D_q, f Lipschitz, g in D_{p,2}C_b^2, xi in L-infinity, and for stability the stated convergence and uniform continuity conditions.
These explicitly define the class of equations for which Theorems 3.4, 3.5, 4.7, and 5.10 are claimed.
standard math Martingale representation and measurable selection on the product space, including the initially enlarged filtration result of Amendinger [2] and Stricker-Yor [32].
Needed to construct Z and to obtain the Prog tensor U-measurable versions in Section 5.

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Pith. "Pith review of Rough backward SDEs with discontinuous Young drivers." pith.science (2026). https://pith.science/paper/D6AHHXCQ

@misc{pith2026250520437,
  author       = {Pith},
  title        = {Pith review of: Rough backward SDEs with discontinuous Young drivers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6AHHXCQ}},
  note         = {Machine review of arXiv:2505.20437}
}

abstract

We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Allan and Jost Pieper, Rough Stochastic Analysis with Jumps , August 2024, arXiv:2408.06978

Andrew L. Allan and Jost Pieper, Rough Stochastic Analysis with Jumps , August 2024, arXiv:2408.06978

work page arXiv 2024

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J¨ urgen Amendinger, Martingale representation theorems for initially enlarged filtra- tions, Stochastic Processes and their Applications 89 (2000), no. 1, 101–116. MR 1775229

work page 2000

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104261, 37

Elena Bandini and Francesco Russo, Weak Dirichlet processes and generalized mar- tingale problems , Stochastic Processes and their Applications 170 (2024), Paper No. 104261, 37. MR 4682671 48

work page 2024

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Dirk Becherer, Peter Friz, and Yuchen Sun, Semilinear rough PDEs with discontinuous rough driver, in preparation, 2025+

work page 2025

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MR 1700749

Patrick Billingsley, Convergence of probability measures , second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. MR 1700749

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Ilya Chevyrev and Peter K. Friz, Canonical RDEs and general semimartingales as rough paths, The Annals of Probability 47 (2019), no. 1, 420–463. MR 3909973

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Friz, Alexey Korepanov, and Ian Melbourne, Superdiffusive limits for deterministic fast-slow dynamical systems , Probability Theory and Related Fields 178 (2020), no

Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, and Ian Melbourne, Superdiffusive limits for deterministic fast-slow dynamical systems , Probability Theory and Related Fields 178 (2020), no. 3-4, 735–770. MR 4168387

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MR 4840242

Ilya Chevyrev, Alexey Korepanov, and Ian Melbourne, Superdiffusive limits beyond the Marcus regime for deterministic fast-slow systems , Communications of the American Mathematical Society 4 (2024), 746–786. MR 4840242

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