{"id":"40cba9bd-e149-4c85-9a58-66729910590c","arxiv_id":"1908.08076","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Doubly reflected BSDEs with predictable barriers and non-quasi-left-continuous filtration have a unique solution under Mokobodzki's condition.","lead":"This paper proves that a type of equation used for pricing and stopping problems, a doubly reflected backward stochastic differential equation, has a unique solution when the information flow can jump at announced times and the barriers are predictable. It is a technical generalization of earlier results, relevant to stochastic analysis and mathematical finance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The terminal jumps of B and B′ are unconstrained: equation (2.6) only uses B_{T−}, and condition (2.8) at τ=T is vacuous because Y_T=ξ_T, so B_T can be changed without affecting the solution; this breaks the uniqueness asserted in Theorem 4.1.","rationale":"The reader's concern about the external one-barrier operator Pre is legitimate, but the more immediate load-bearing issue is internal to the present paper. The definition of solution uses only left limits of B and B′ in the equation, while the Skorohod condition at the terminal time is vacuous because Y_T=ξ_T=ζ_T. Thus a terminal jump in B or B′ can be added arbitrarily, giving distinct tuples that all satisfy Definition 2.6. The explicit ξ=ζ=0, g=0 example settles this: the zero solution and any solution with a terminal jump in B are different but satisfy every stated condition, so the uniqueness assertion in Theorem 4.1 is false as written. The proof can likely be repaired by normalizing B_T=B_{T−} and B′_T=B′_{T−}, which is why a conditional verdict rather than outright rejection seems appropriate. The external dependency on the prior paper [2] remains a secondary concern, but it is not the weak point that makes the stated theorem incorrect.","tokens_in":22488,"tokens_out":20908,"duration_ms":204648,"concrete_test":"Verify the counterexample with T=1, ξ_t=ζ_t=0, g=0. Take the zero solution (Y,Z,M,A,A′,B,B′)=(0,0,0,0,0,0,0). For a nonzero F_{T−}-measurable η≥0 in L², define B_t=0 for t<T, B_1=η, and keep B′=0 and all other components zero. Check Definition 2.6: equation (2.6) is unchanged because B_{1−}=0; condition (2.8) at τ=1 is vacuous since Y_1=ξ_1=0; and dB is singular to dB′=0. If this tuple is accepted as a second solution, the uniqueness claim in Theorem 4.1 is falsified exactly as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Definition 2.6 the unknown B enters the DRBSDE only through B_{T−}−B_{τ−}; the value B_T never appears. The Skorohod condition (2.8) at τ=T is (Y_T−ξ_T)(B_T−B_{T−})=0, which is automatic because the terminal condition forces Y_T=ξ_T. The same holds for B′ and for the one-barrier Definition 5.1. Consequently B_T and B′_T are free parameters, subject only to dB⊥dB′ and square integrability. This is not a harmless convention: with ξ=ζ=0, g=0, the zero tuple is a solution; if η is any nonzero F_{T−}-measurable nonnegative square-integrable random variable, setting B_t=0 for t<T and B_T=η, with all other components unchanged, still satisfies (2.6), (2.7), (2.8), and dA⊥dA′, dB⊥dB′. Theorem 4.1 therefore asserts uniqueness of an object that is not unique. The gap is fixable by imposing B_T=B_{T−} and B′_T=B′_{T−}, or by defining the increasing processes on [0,T), but as written the central claim is false.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of doubly reflected backward stochastic differential equations (DRBSDEs) on a filtered probability space with a general, not necessarily quasi-left-continuous, filtration, where the two barriers are predictable processes. Under a Mokobodzki-type condition, the authors prove existence and uniqueness of a solution (Y,Z,M,A,B,A',B') in S^{2,p} x H^2 x M^{2,perp} x (S^{2,p})^2 x (S^{2,p})^2. The proof proceeds by reducing the (y,z)-independent case to a coupled system of one-barrier predictable RBSDEs solved by Picard iteration, deriving a priori estimates via the Gal'chouk-Lenglart change-of-variable formula, and then using a fixed-point argument for Lipschitz drivers. The main results are Theorem 3.1 and Theorem 3.3 for driver processes and Theorem 4.1 for general Lipschitz drivers.","tokens_in":22788,"tokens_out":9806,"duration_ms":88839,"significance":"If the terminal-jump issue identified below is repaired, the paper is a substantive extension of the DRBSDE theory from the optional/right-continuous setting to predictable obstacles in a general filtration, complementing Grigorova et al. (2018). The use of the Gal'chouk-Lenglart formula to handle both left and right jumps is appropriate, and the a priori estimates in Lemma 3.3 are well suited for the fixed-point step. The proofs are detailed and, apart from the issues listed below and the reliance on the external operator Pre from the authors' prior work [2], internally coherent; the analytic arguments are written in a checkable form. A correct version of the main theorem would be valuable for applications to game options and optimal stopping with non-right-continuous information.","major_comments":[{"comment":"The terminal values B_T and B'_T are unconstrained. The equation (2.6) only involves B_{T-} and B'_{T-}, and at tau=T the Skorohod condition (2.8) is automatic because the terminal condition forces Y_T=xi_T. Consequently, with xi=zeta=0 and g=0, the zero tuple is a solution, but so is any tuple with B_t=0 for t<T, B_T=eta, B' identical to 0, and all other components zero, where eta is any non-zero F_{T-}-measurable non-negative square-integrable random variable. This tuple satisfies (2.6), (2.7), (2.8), and dA perp dA', dB perp dB'. Theorem 4.1 therefore asserts uniqueness of an object that is not unique. The gap is readily repaired by requiring B_T=B_{T-} and B'_T=B'_{T-} (or by writing the equation with B_T and B'_T); the existence proof already leaves the value of B_T unused, so the normalization does not affect the constructed solution.","section":"Definition 2.6, Eq. (2.6), condition (2.8), Theorem 4.1"},{"comment":"The contraction estimate as printed is miscalculated. From the Lipschitz property of g one obtains |g(t,U_t,V_t)-g(t,U'_t,V'_t)|^2 <= 2K^2(|U_t-U'_t|^2+|V_t-V'_t|^2), so the factor in (4.2) should be 2 K^2 (3+16c^2) (for the sum of the two norms) rather than 2 K (3+16c^2). As written, inequality (4.2) is false for drivers with K>1, although the contraction argument still goes through after replacing K by K^2 and choosing epsilon sufficiently small.","section":"Section 4, inequality (4.2)"},{"comment":"The Lipschitz condition is printed as |g(t,y1,z1)-g(t,y2,z2)| <= K(|y1-y2| - |z1-z2|). The minus sign is a typo; with the minus sign the right-hand side is not a metric and the condition is impossible for non-constant g. The proof of Theorem 4.1 relies on the reversed triangle inequality with a plus sign, so the definition must be corrected to K(|y1-y2| + |z1-z2|).","section":"Definition 2.3"}],"minor_comments":[{"comment":"The space H^{2,p} is used without definition; presumably it is the same as H^2 or a localized version, but it should be defined to avoid ambiguity.","section":"Section 3, Theorem 3.1 and Lemma 3.3"},{"comment":"The text after equation (3.25) says 'Since beta < 1/epsilon^2', but the lemma assumes beta > 1/epsilon^2; the inequality sign is reversed and should be corrected.","section":"Section 3.2, proof of Lemma 3.3"},{"comment":"The phrase 'mutually singularity constrain t (2.6)' should refer to condition (iv) of Definition 2.6, not to equation (2.6).","section":"Remark 2.4"},{"comment":"The phrase 'where the last inequality follows' should be 'where the last equality follows', since the preceding display contains an equality.","section":"Proposition 2.1, proof"},{"comment":"The heading 'PROOF OF LEMMA 3.16' is a typo; it should read 'PROOF OF LEMMA 3.2'.","section":"Appendix, proof of Lemma 3.2"},{"comment":"The paper repeatedly uses 'essentially non quasi-left continuous' in the abstract and introduction; the precise assumption (that the filtration is not quasi-left-continuous, or is arbitrary) should be stated unambiguously.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is not acceptable as stated because of the unconstrained terminal jumps of B and B', which makes the claimed uniqueness false. However, the flaw is local and repairable by adding the normalization B_T=B_{T-} and B'_T=B'_{T-}, and the rest of the proof structure appears sound. I would also ask the editor to ensure that the heavy reliance on Proposition 5.1 from the authors' own preprint [2] for the operator Pre is acceptable; the present paper does not reprove that result, and the correctness of the base case of the induction depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the main theorem is false as stated. In Definition 2.6, the processes B and B′ appear only through B_{T-}-B_{τ-} and B′_{T-}-B′_{τ-}, and the Skorohod condition at τ=T is vacuous because Y_T=ξ_T. So B_T and B′_T are free: shifting B_T by any nonnegative F_{T-}-measurable square-integrable random variable leaves every condition satisfied, and uniqueness in Theorem 4.1 fails. The same issue appears in the one-barrier Definition 5.1 that underlies the operator Pre. The fix is straightforward—require B_T=B_{T-} and B′_T=B′_{T-}, or define the increasing processes on [0,T)—so the paper's core program is salvageable, but the advertised result is not correct as written.\n\nNow the positive side. The paper targets a real gap: doubly reflected BSDEs with predictable (not just optional) barriers in a non-quasi-left-continuous filtration. The proof strategy is coherent. The driver-independent case is rewritten as a coupled pair of one-barrier predictable RBSDEs, solved by Picard iteration; then Gal'chouk–Lenglart gives a priori estimates; then a fixed point argument handles Lipschitz drivers. The authors are transparent that the method follows Grigorova et al. [16] and that the one-barrier operator comes from their own prior paper [2]. The a priori estimates in Lemma 3.3 look plausible, though the details are compressed and several terms are asserted non-positive without full justification.\n\nMinor but real: Definition 2.3's Lipschitz condition has a sign typo—the minus must be a plus. The contraction bound in Theorem 4.1 writes 2ε²K where the square on K is missing, and the \"for all ε≥0\" in Lemma 3.3 is incompatible with \"β>1/ε².\" These are typos, not conceptual errors.\n\nThe most serious soft spot beyond the terminal jump issue is the dependency on [2]. If [2] does not include the same normalization, the induction base is unsound. The authors should either prove the one-barrier result with a clean convention or restate the import from [2] explicitly.\n\nWho is this for? People working on reflected BSDEs, game options, or optimal stopping in filtrations with jumps. The contribution is modest but real. I would send it to a serious referee, because the fix is small and the setting is useful, but the referee should flag the uniqueness gap and require a revision that normalizes the terminal jumps.","headline":"The main theorem is false as stated because B_T and B′_T are unconstrained, but a one-line normalization fix restores it; the paper fills a genuine gap in the predictable-barrier DRBSDE literature.","tokens_in":23302,"tokens_out":7449,"would_cite":false,"duration_ms":64978,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H20","60H30","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Doubly reflected BSDEs with predictable barriers have unique solutions","keywords":["predictable DRBSDEs","doubly reflected backward stochastic differential equations","non quasi-left continuous filtrations","Mokobodzki condition","Picard iteration method","Banach fixed point theorem","predictable strong supermartingales","Skorohod conditions"],"falsifier":"Find a pair of predictable barriers satisfying Mokobodzki's condition for which the Picard sequence (3.15) fails to converge in $S^{{2,p}}$ to a pair solving the coupled system (3.11), or for which the two limits do not satisfy the system; such an example would disprove Lemma 3.2 and hence Theorem 4.1. A concrete check in a simple filtration with one or two predictable jump times would settle it.","tokens_in":22313,"feed_emoji":"📐","tokens_out":5426,"duration_ms":57040,"temperature":0.7,"pith_summary":"This paper establishes existence and uniqueness for doubly reflected backward stochastic differential equations (DRBSDEs) when the two reflecting barriers are predictable processes and the filtration need not be quasi-left-continuous. In that setting martingales can jump at predictable times, so the solution must be kept between the barriers by four increasing processes: two acting at left jumps and two at right jumps. The paper shows that the natural analogue of Mokobodzki's condition, namely the existence of two nonnegative predictable strong supermartingales whose difference lies between the barriers, is necessary and sufficient for a solution. The proof first solves the case where the driver does not depend on the solution by rewriting the problem as a coupled pair of one-barrier predictable reflected BSDEs and applying a Picard iteration, then obtains the general Lipschitz-driver case as the fixed point of a contraction. Because the filtration is general, the argument uses the Gal'chouk–Lenglart change-of-variables formula, which handles processes that are neither right- nor left-continuous.","feed_headline":"Doubly reflected BSDEs with predictable barriers have unique solutions","feed_subtitle":"The proof extends to filtrations where martingales can jump at predictable times, using a fixed-point argument.","key_machinery":"The load-bearing object is the operator $\\mathrm{Pre}$, defined as the first component of the unique solution of the one-barrier predictable reflected BSDE with zero driver. Equivalently, $\\mathrm{Pre}[\\xi]$ is the predictable Snell envelope of $\\xi$: the smallest predictable strong supermartingale dominating $\\xi$. The paper rewrites the two-barrier problem as a coupled system $J=\\mathrm{Pre}[(\\bar J+\\tilde\\xi^{g,p})1_{[0,T)}]$ and $\\bar J=\\mathrm{Pre}[(J-\\tilde\\zeta^{g,p})1_{[0,T)}]$, and solves it by monotone Picard iteration. The Gal'chouk–Lenglart formula provides the a priori estimates that give uniqueness and make the Lipschitz case a contraction in a weighted norm, while Mertens decomposition identifies and makes unique the increasing processes $A,B,A',B'$.","core_discovery":"For any Lipschitz driver $g$ and any pair of predictable admissible barriers $\\xi,\\zeta$ satisfying Mokobodzki's condition, there is a unique tuple $(Y,Z,M,A,B,A',B')$ in $S^{2,p}\\times H^2\\times M^{2,\\perp}\\times (S^{2,p})^2\\times (S^{2,p})^2$ solving equation (2.6), respecting $\\xi\\leq Y\\leq \\zeta$ at all predictable stopping times, and satisfying the Skorohod minimality conditions (2.7)–(2.8) together with mutual singularity of the pairs $(A,A')$ and $(B,B')$. The pairs deal with the two kinds of jumps: $A$ and $A'$ act only when $Y$ hits a barrier from the left, while $B$ and $B'$ account for right jumps, with $\\Delta B=(pY_+-Y)^-$ and $\\Delta B'=(pY_+-Y)^+$. Mokobodzki's condition is shown to be necessary for existence in the driver-process case and sufficient under the stated hypotheses. The proof first solves the case where $g$ does not depend on $(y,z)$ by identifying $Y$ with a difference of two predictable strong supermartingales coupled through the one-barrier operator $\\mathrm{Pre}$, and then applies a Banach fixed point argument for Lipschitz drivers.","pith_inferences":["If the theorem is correct, game options with predictable rather than optional exercise times become tractable in filtrations with predictable jumps, a direction the paper's motivation points toward but does not develop.","A concrete testable extension would be to write down an explicit two-jump or finite-jump filtration example and verify directly that the solution satisfies $Y=(pY_+\\vee \\xi)\\wedge\\zeta$, with the jump sizes of $A,B,A',B'$ matching the stated left- and right-jump formulas.","The same coupled-system and contraction machinery may adapt to drivers with jumps or to weaker integrability assumptions on the barriers, though the paper itself does not claim those extensions.","The monotone Picard construction suggests that the solution is the minimal pair of predictable strong supermartingales dominating the shifted barriers, which could yield comparison results or a predictable version of the Dynkin-game value beyond what the paper explicitly states."],"forward_implications":["If the driver is a square-integrable process independent of $(y,z)$, the solution exists exactly when Mokobodzki's condition holds, and its first component is given explicitly as $J^p_t-\\bar J^p_t+E[\\xi_T+\\int_t^T g_s\\,ds\\,|\\,\\mathcal{F}_{t-}]$.","When the lower barrier is right-continuous, the right-jump pushing process $B$ vanishes; when the upper barrier is right-continuous, $B'$ vanishes; and when the barriers are suitably semicontinuous along predictable stopping times, $A$ and $A'$ are continuous.","Mokobodzki's condition is a necessary condition for existence, so the result pins down exactly which barrier pairs can support a solution in the predictable setting.","The mutual singularity conditions on $(A,A')$ and $(B,B')$ yield uniqueness of the four pushing processes without requiring the usual strict separation $\\xi<\\zeta$.","The contraction argument works in the Banach space $S^{2,p}\\times H^2$ with an exponentially weighted norm, so the uniqueness and stability estimates hold uniformly over the whole time horizon."],"supporting_citations":[{"why":"Supplies the existence, uniqueness, and monotonicity of the one-barrier predictable reflected BSDE operator Pre that the coupled system and Picard iteration are built on.","marker":"[2]"},{"why":"Provides the two-barrier existence strategy for non-right-continuous obstacles in general filtrations that the present coupled-system and Picard proof follows.","marker":"[16]"},{"why":"Gives the Gal'chouk optional change-of-variables formula used to establish the a priori estimates.","marker":"[14]"},{"why":"Gives the Lenglart form of the change-of-variables formula for optional semimartingales used in the uniqueness estimates.","marker":"[21]"},{"why":"Provides Mertens decomposition of predictable strong supermartingales, used to identify and prove uniqueness of A,B,A',B'.","marker":"[22]"},{"why":"Supplies the canonical decomposition and mutual-singularity framework used to ensure uniqueness of the increasing processes without strict separation of the barriers.","marker":"[6]"}],"fun_headline_variants":["Doubly reflected BSDEs with predictable jumps solved uniquely","Predictable barriers ensure unique solutions for doubly reflected BSDEs","Unique solutions for doubly reflected BSDEs with predictable barriers","Doubly reflected BSDEs with predictable barriers: existence and uniqueness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument takes as given that the operator Pre from the authors' earlier one-barrier paper exists, is monotone, and maps $S^{{2,p}}$ into itself; the present paper does not reprove or relax that external theorem.","fun_headline_variants_meta":{"raw":{"variants":["Doubly reflected BSDEs with predictable jumps solved uniquely","Predictable barriers ensure unique solutions for doubly reflected BSDEs","Unique solutions for doubly reflected BSDEs with predictable barriers","Doubly reflected BSDEs with predictable barriers: existence and uniqueness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000917,"raw_usage":{"total_tokens":3938,"prompt_tokens":947,"completion_tokens":2991,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":2921}},"tokens_in":563,"tokens_out":2991,"duration_ms":20568,"temperature":1.0,"reasoning_tokens":2921,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:50:36.403717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a pair of predictable barriers satisfying Mokobodzki's condition for which the Picard sequence (3.15) fails to converge in $S^{{2,p}}$ to a pair solving the coupled system (3.11), or for which the two limits do not satisfy the system; such an example would disprove Lemma 3.2 and hence Theorem 4.1. A concrete check in a simple filtration with one or two predictable jump times would settle it.","supporting_citations":[{"cited_title":"Non linear optimal stopping problem and Reflected BSDEs in the predictable setting","cited_arxiv_id":"1811.00695","evidence_quote":"Supplies the existence, uniqueness, and monotonicity of the one-barrier predictable reflected BSDE operator Pre that the coupled system and Picard iteration are built on."},{"cited_title":"(2018): Doubly Reflected BSDEs and ε f -Dynkin games: beyond the right-continuous case, Electron ic Journal of Probability, V olume 23, paper no","cited_arxiv_id":null,"evidence_quote":"Provides the two-barrier existence strategy for non-right-continuous obstacles in general filtrations that the present coupled-system and Picard proof follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Gal'chouk optional change-of-variables formula used to establish the a priori estimates."},{"cited_title":"(1980): Tribus de Meyer et théorie des proce ssus, Sèminaire de probabil- ités de Strasbourg XIV 1978/79, Lecture Notes in Mathematic s V ol","cited_arxiv_id":null,"evidence_quote":"Gives the Lenglart form of the change-of-variables formula for optional semimartingales used in the uniqueness estimates."},{"cited_title":"(1976): Un cours sur les intégrales stochas tiques (exposés 1 à 6), Sémi- naire de probabilités de Strasbourg X, pp","cited_arxiv_id":null,"evidence_quote":"Provides Mertens decomposition of predictable strong supermartingales, used to identify and prove uniqueness of A,B,A',B'."},{"cited_title":"(2016): Gen eralized Dynkin Games and Doubly reflected BSDEs with jumps, Electronic Journal of Probability, V olume 21, paper no","cited_arxiv_id":null,"evidence_quote":"Supplies the canonical decomposition and mutual-singularity framework used to ensure uniqueness of the increasing processes without strict separation of the barriers."}],"review_version":1}