{"id":"7e2c8619-1d52-427a-81c1-04fba2a1cb72","arxiv_id":"2507.01767","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Semi-martingale second-order BSDEs with jumps are proved well-posed over a unified class of diffusions, pure-jump processes, and discrete-time processes, while the jump-measure integrands resist model-independent aggregation.","lead":"This paper builds a unified mathematical framework for second-order backward stochastic differential equations (2BSDEs) that describe stochastic control and pricing problems when the underlying random processes can jump, not just diffuse continuously. It proves the value process has a unique aggregated characterization and exposes an obstruction: jump-related integrands cannot generally be glued together across probability models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3.3(iv) is unverified and fails for a simple Poisson-control generator, so the advertised pure-jump/discrete-time scope of Theorem 3.13 is not established.","rationale":"The reader's weakest assumption is exactly the load-bearing one: Assumption 3.3(iv) is used to construct the equivalent measure Q via E(η·X^{c,P}+ρ†∗µ-tilde) in the down-crossing argument that yields the path regularization (Theorem 3.5(i)), which in turn feeds the decomposition (Theorem 3.10) and the extrinsic 2BSDE characterization (Theorem 3.13). The paper itself flags the reversed quantifiers as crucial (Remark 3.4(iii)) but supplies no verification for pure-jump or discrete-time generators. My check goes further: for a simple Poisson-control generator f^P(u)=-u(1)^+, the required uniform lower bound over all U,U' forces ρ†(1)≤-1 and ρ†(1)≥-1/2 simultaneously, so Assumption 3.3(iv) fails in a natural class the abstract claims to cover. This does not make the theorems internally inconsistent; they remain valid conditional statements. But it does narrow the advertised scope substantially. The intrinsic characterization has an additional quasi-left-continuity restriction (Assumption 3.16(iv)), which separately excludes discrete-time processes, but the primary concern is 3.3(iv). I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":84941,"tokens_out":8782,"duration_ms":103214,"concrete_test":"Analytical check: fix d=1, C_t=t, K(dx)=δ_1(dx), P the law of a compensated Poisson process with compensator K(dx)dC, and generator f^P(y,y-,z,u)=inf_{α∈[0,1]}∫u(x)(α-1)δ_1(dx). Test solvability of Assumption 3.3(iv) in the one-dimensional variable ρ:=ρ†(1): the pair (u=1_{x=1},u'=0) requires -1 ≥ ρ, while (u=-1_{x=1},u'=1_{x=1}) requires 1 ≥ -2ρ, i.e. ρ ≥ -1/2. Since no scalar ρ satisfies both, Assumption 3.3(iv) fails. If pure-jump control generators of this form are intended, the authors must either weaken 3.3(iv) or add structural hypotheses (e.g., linearity or concavity of f in u) making the uniform lower bound hold; otherwise Theorem 3.13 does not cover the advertised pure-jump regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main construction (Theorem 3.5(i), and hence the 2BSDE characterization Theorem 3.13) rests on Assumption 3.3(iv): for each P∈P0 a single ρ†∈H^2_T(µ^X;F^P_+,P) with E(ρ†∗µ-tilde^{X,P})>-1, ⟨ρ†∗µ-tilde^{X,P}⟩≤θ^µ dC, and f^P(Y,Y-,Z,U)-f^P(Y,Y-,Z,U') ≥ d⟨ρ†∗µ-tilde^{X,P},(U-U')∗µ-tilde^{X,P}⟩/dC for all Y,Z,U,U'. Remark 3.4(iii) calls the reversed quantifiers crucial, but no pure-jump or discrete-time example is verified. The condition is not merely unverified: it fails for a standard jump-control generator. Take C_t=t, ν^P(dt,dx)=dt δ_1(dx) (compensated Poisson), and f^P(y,y-,z,u)=inf_{α∈[0,1]}∫u(x)(α-1)δ_1(dx) = -u(1)^+. With y,z frozen, the required lower bound becomes f(U)-f(U') ≥ ρ†(1)(U(1)-U'(1)) (for continuous C the cross-variation term is the L^2(K) inner product). For U=1_{x=1}, U'=0 this gives -1 ≥ ρ†(1). For U=-1_{x=1}, U'=1_{x=1} it gives 1 ≥ -2ρ†(1), i.e. ρ†(1)≥-1/2. Contradiction. Thus no ρ† exists, even though this f is Lipschitz and arises from a pure-jump weak-control problem (controlling jump intensity). Consequently the measure-change argument in Theorem 3.5(i) cannot be run for this generator, and the abstract's unified-treatment claim for pure-jump processes is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for second-order backward stochastic differential equations (2BSDEs) driven by general semimartingales with jumps on the canonical Skorokhod space. It constructs an aggregated value process from a family of BSDEs indexed by a set P0 of possibly non-dominated semimartingale laws, proves a path-regularisation result, derives a semimartingale decomposition via reflected BSDEs, and characterises the regularised value process as the unique solution of an extrinsic and an intrinsic 2BSDE system. The paper claims that the framework unifies controlled diffusions, pure-jump processes, and discrete-time processes, and it explicitly discusses why the integrands of stochastic integrals with respect to compensated jump measures cannot in general be aggregated across the probability measures.","tokens_in":85240,"tokens_out":7675,"duration_ms":91667,"significance":"If the central construction is correct, this would be a substantial contribution: it would extend the 2BSDE theory well beyond the Brownian and Brownian--Poisson settings, provide a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space, and correct known gaps in earlier down-crossing and aggregation arguments. The paper is unusually careful in its proofs: Section 7 and Appendices A--C contain detailed arguments, and Remark 7.9 explicitly identifies and repairs an erroneous application of Doob's down-crossing inequality in [141, Lemma 3.2]. The sharp discussion of the non-aggregation of jump integrands, including the criticism of claims in [85] and [47], is also valuable. However, the advertised pure-jump and discrete-time scope is not established, because a central assumption used in the path-regularisation proof fails for a natural pure-jump weak-control generator.","major_comments":[{"comment":"Assumption 3.3(iv) is load-bearing for the entire construction: in the proof of Theorem 3.5(i) (Section 7.2, around Eq. (7.17)--(7.19)), the existence of a single process rho-dagger with a uniform lower bound on generator differences is used to build the equivalent measures that make the down-crossing argument work. The assumption is not merely unverified; it fails for a standard pure-jump Lipschitz generator. Take d=1, C_t=t, and let P be the law of a compensated Poisson process with compensator dt delta_1(dx). Let f^P(y,y-,z,u) = inf_{alpha in [0,1]} int_{R} u(x)(alpha-1) delta_1(dx) = -u(1)^+. This generator is 1-Lipschitz in the relevant \\hat L^2 norm and independent of (y,z), so it satisfies the other structural assumptions. If Assumption 3.3(iv) held, there would exist a predictable rho-dagger with rho-dagger(t,1) > -1 on the jump times (from Delta(rho-dagger * tilde-mu^{X,P}) > -1) and, since dC=dt, with the required lower bound reducing to f(U)-f(U') >= rho-dagger(1)(U(1)-U'(1)) for all U,U'. Choosing U=1_{x=1} and U'=0 gives -1 >= rho-dagger(1), contradicting rho-dagger(1) > -1. Thus no such rho-dagger exists. Consequently, the measure-change/down-crossing proof of Theorem 3.5(i) cannot be run for this generator, and the claimed unified treatment of pure-jump processes in Theorem 3.13 is not obtained. The paper notes in Remark 3.4(iii) that the reversed quantifiers are crucial, but it supplies no verification for any pure-jump or discrete-time class; the example above shows the condition is not a harmless technical restriction.","section":"Assumption 3.3(iv), Remark 3.4(iii), and Theorem 3.5(i)"},{"comment":"The intrinsic characterisation relies on the same type of existential condition in Assumption 3.16(i), which requires a pair (rho_1,rho_2) of integrands satisfying two-sided cross-variation bounds for all Y,Z,U,U'. The same Poisson-control generator f^P(y,y-,z,u)=-u(1)^+ shows this assumption also fails: taking U=1_{x=1} and U'=0, the lower bound f(U)-f(U') >= d<rho_1 * tilde-mu, (U-U') * tilde-mu>/dC = rho_1(1) together with Delta(rho_1 * tilde-mu) > -1 gives the contradiction -1 >= rho_1(1) > -1. Hence Theorem 3.20 is not available for this natural pure-jump generator either.","section":"Assumption 3.16(i) and Theorem 3.20"},{"comment":"The paper's stated goal of a unified treatment of 'controlled diffusions, pure-jump processes, and discrete-time processes' is not supported by the results as they stand. Section 5.2 suggests that extending the framework to other decompositions of X would involve only technical challenges, but the failure of Assumption 3.3(iv) for the elementary pure-jump generator above is a substantive obstacle, not a mere technicality. The authors should either prove that a nontrivial class of pure-jump or discrete-time generators satisfies Assumption 3.3(iv), or explicitly restrict the main theorems to a class for which the condition can be verified.","section":"Introduction and Section 5.2"}],"minor_comments":[{"comment":"The intersection '\\bigcap_{\\beta'(0,\\hat\\beta)}' is missing the membership symbol; it should read '\\bigcap_{\\beta'\\in(0,\\hat\\beta)}'.","section":"Theorem 3.20(i)"},{"comment":"The definition of the stochastic exponential E(\\hat\\beta A)_r uses the convention an integral over (0,r], but the factor '(T-s\\wedge T)^{s,\\omega}' appears in the integrability condition in Assumption 2.20(iv) without clear bracketing; please clarify the intended stopping time in that display.","section":"Section 2.4, display before (2.5)"},{"comment":"In the displayed equation following (7.24), the expression 'E(\\hat\\beta A)^{1/2}_{t\\wedge T}/E(\\hat\\beta A)^{1/2}_{t_n\\wedge T}' is written with inconsistent placement of the exponent; this should be cleaned up to avoid confusion.","section":"Section 7.2, proof of Theorem 3.5(ii)"},{"comment":"There are several small typographical issues, such as 'P0–q.s.' and 'Theorem 2.5' being used where 'Remark 2.5' is meant, and the notation 'L^2_{T,\\beta}(P0)' in Section 3.4 is used before the space is formally introduced. These do not affect the mathematics but should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The counterexample in the major comments appears decisive: Assumption 3.3(iv) fails for a natural Lipschitz pure-jump weak-control generator, so the paper's advertised pure-jump and discrete-time scope is not established. The authors should either verify the assumption for a nontrivial class of jump generators, modify the proof so that such generators are covered, or explicitly narrow the main claims. The paper otherwise contains substantial and careful work, and the correction of the down-crossing gap in [141] is a useful contribution in its own right."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a serious and unusually candid paper: the authors correct a genuine error in the down-crossing argument of [141, Lemma 3.2], repair a gap in [19, Lemma A.1], and prove new measurability and stability results that are likely to be of independent use. Second, the headline claim—a unified treatment of diffusions, pure-jump processes, and discrete-time processes—is not actually delivered. The path-regularization and the 2BSDE characterization rest on Assumption 3.3(iv), an existential condition on a process rho-dagger that is never verified for a single pure-jump or discrete-time generator. In fact, it fails in a very simple case: take C_t=t, nu(dt,dx)=dt delta_1(dx), and f(y,y-,z,u)=-u(1)^+. The required inequality forces rho-dagger(1) <= -1 from one pair of integrands and rho-dagger(1) >= -1/2 from another, while the stochastic exponential condition forces rho-dagger(1) > -1. No such rho-dagger exists, so the main construction cannot run for this generator.\n\nThis matters because the abstract promises pure-jump and discrete-time applications. The intrinsic characterization additionally imposes quasi-left-continuity (Assumption 3.16(iv)), which excludes discrete-time. The authors are honest that the aggregation of the jump integrands (U^P) remains open, and they do not claim a counterexample. But the gap between what is proved and what is advertised is real.\n\nWhere the assumptions do hold—for example, continuous diffusions or jump-diffusions with monotone dependence on the u-variable—the framework is coherent and the proofs are unusually detailed (roughly 70 pages plus appendices). The measurability results for integrands depending on the probability P (Theorem 3.1, Lemma 7.2) and the new stability results in Appendix C are genuine contributions, and the paper's corrections to [141] and [19] are valuable on their own.\n\nWho should read this: specialists in BSDEs and 2BSDEs, stochastic control under model uncertainty, and anyone relying on the corrected results from [141]. It deserves peer review—the technical work is serious and the errors it fixes are real—but the referee should require the authors to either verify Assumption 3.3(iv) on concrete pure-jump generators, give sufficient conditions for it, or scale back the advertised scope. I would not accept the current version unconditionally; the pure-jump and discrete-time claims need support or qualification.","headline":"A serious, honest paper that fixes real errors in the 2BSDE literature, but the advertised pure-jump/discrete-time scope is not delivered: a key assumption fails for a simple Poisson control generator.","tokens_in":85968,"tokens_out":4134,"would_cite":true,"duration_ms":47765,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60G44","93E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs one path-regularised process aggregating BSDE values under every probability, and proves it solves a second-order BSDE system with jump integrands indexed by the measure.","keywords":["second-order BSDEs","semi-martingales","jump processes","aggregation of value processes","stochastic control","model uncertainty","nonlinear expectations","reflected BSDEs"],"falsifier":"Test Assumption 3.3(iv) on a pure-jump generator: take $X$ Poisson-like and $f^P$ depending on $U$ through $\\|U(\\cdot)\\|_{\\hat L^2(K)}$, and attempt to exhibit $Y,Z,U,U'$ for which no $\\rho^\\dagger$ satisfies the cross-variation bound (3.12); if such a pair is found, the change-of-measure step in the proof of Theorem 3.5 cannot be run, so the aggregated 2BSDE solution need not exist for that generator class.","tokens_in":84519,"feed_emoji":"🎲","tokens_out":15879,"duration_ms":154970,"temperature":0.7,"pith_summary":"The paper aims to give a unified aggregation theorem for stochastic control problems whose payoff is the solution of a backward stochastic differential equation (BSDE) driven by a general semi-martingale with jumps. It builds a single path-regularised process $\\hat{Y}^+$, obtained as the right limit along dyadic times of the pointwise supremum over probability measures of the BSDE values, and shows that for each measure $P$ this process coincides with the essential supremum of the BSDE values over all measures equal to $P$ on the current information. The central result is that $\\hat{Y}^+$ carries a semi-martingale decomposition under every $P$ and that this decomposition is the unique solution of an extrinsic second-order BSDE (2BSDE) system. A key structural finding is that the jump integrands $(\\hat{U}^P)_{P\\in\\mathcal P_0}$ must be indexed by the probability measure: unlike the diffusion integrand, they cannot in general be aggregated into a single process. If the construction is correct, 2BSDE techniques become available for controlled diffusions with jumps, pure-jump processes, and discrete-time processes under non-dominated model uncertainty.","feed_headline":"Jump integrands resist aggregation in second-order BSDEs","feed_subtitle":"One system now covers diffusions, pure-jump and discrete-time control, with jump integrands indexed by the measure.","key_machinery":"The central object is the path-regularised value function $\\hat{Y}^+(T,\\xi)$, defined as the right limit along dyadic times of $\\hat{Y}_s(T,\\xi):=\\sup_{P\\in\\mathcal P(s,\\omega)} E^P[Y^{s,\\omega,P}_0((T-s\\wedge T)^{s,\\omega},\\xi^{s,\\omega})]$. Here a second-order BSDE (2BSDE) is a backward equation that must hold under a family of probability measures at once, with the second-order feature being the nonlinear dependence on the diffusion coefficient. The argument is carried by three mechanisms: measurable selection of semi-martingale characteristics in the probability parameter, a corrected down-crossing inequality for nonlinear super-martingales that yields the càdlàg regularisation, and the well-posedness of reflected BSDEs in weighted spaces, which produces the decomposition of $\\hat{Y}^+$ into $(\\hat{Z}, (\\hat{U}^P, \\hat{N}^P, \\hat{K}^P)_{P\\in\\mathcal P_0})$. Uniqueness is enforced by the extrinsic condition (2B3), $\\hat{Y}^+_t = \\operatorname{ess\\,sup}_{\\bar P\\in\\mathcal P_0(G_{t+},P)} Y^{\\bar P}_t(T,\\xi)$, rather than by a minimality condition on $\\hat{K}^P$, which only works under extra assumptions.","core_discovery":"The discovery is that the aggregated value process has a semi-martingale decomposition simultaneously under all measures, and that this decomposition is characterised as the unique solution of the extrinsic 2BSDE system. In its concrete form, for every $P\\in\\mathcal P_0$, $$\\hat{Y}^+_t = \\xi + \\int_t^T f^P_r(\\hat{Y}^+_r,\\hat{Y}^+_{r-},\\hat{Z}_r,\\hat{U}^P_r(\\cdot))\\,dC_r - \\left(\\int_t^T \\hat{Z}_r\\,$dX^{{c,P}}$_r\\right)^{(P)} - \\left(\\int_t^T\\int_{\\mathbb R^d} \\hat{U}^P_r(x)\\,\\tilde{\\mu}^{X,P}(dr,dx)\\right)^{(P)} - \\int_t^T d\\hat{N}^P_r + \\hat{K}^P_T - \\hat{K}^P_t,\\quad P\\text{-a.s.}$$ with $\\hat{K}^P$ non-decreasing and predictable, $\\hat{N}^P$ orthogonal to the continuous local martingale part and the compensated jump measure, and $\\hat{Z}$ a single integrand common to all $P$. The extrinsic system (2B1)-(2B3) uses the representation of $\\hat{Y}^+$ as an essential supremum of the underlying BSDE values as its uniqueness condition; an intrinsic characterisation replaces that by a minimality condition on $\\hat{K}^P$ under additional assumptions. The paper further argues that the family $(\\hat{U}^P)_{P\\in\\mathcal P_0}$ cannot generally be collapsed to one integrand, and that the same obstruction persists when the BSDE is driven directly by a jumping martingale.","pith_inferences":["If the non-aggregation claim is correct, applications that rely on measure-independent 2BSDE solutions, such as principal-agent contracting and superhedging under volatility and jump uncertainty, will need either a structural restriction on the generator or an explicitly law-dependent solution when jumps are present.","A direct stress test is to check Assumption 3.3(iv) for pure-jump generators, for example Poisson-driven generators depending on $\\|U(\\cdot)\\|_{\\hat L^2}$; the paper notes the reversed quantifier is crucial and gives no verification there, so a counterexample would delimit the theory.","The path-regularisation method may extend to aggregated reflected BSDEs on Skorokhod space, where the extrinsic condition would supply uniqueness without the minimality condition that fails for reflected problems.","Numerical implementations of these 2BSDEs would need to represent the whole family $P\\mapsto \\hat{U}^P$, a higher-dimensional object than a single integrand, suggesting approximation schemes based on parameterising the law set."],"forward_implications":["If the hypotheses hold, the value process of a stochastic control problem with BSDE payoff and controlled semi-martingale characteristics is the first component of a unique 2BSDE solution, so 2BSDE well-posedness extends to controlled diffusions with jumps, pure-jump processes, and discrete-time processes.","Because the jump integrands stay indexed by $P$, a control that acts on the compensator of the jump measure will produce controls that depend on the probability law, so the usual worst-case-control interpretation fails in that regime; this is a direct consequence of the non-aggregation argument.","The path-regularised value process satisfies an invariance principle: shifting the path and restarting the system gives the solution of the 2BSDE with shifted data, which is a pathwise dynamic programming identity.","The corrected down-crossing lemma gives a sound path-regularisation step for nonlinear super-martingales in this generality, closing a gap in earlier arguments.","Norm estimates and a comparison principle for the 2BSDE solution follow from the same construction, giving stability of the value process in the data of the problem."],"supporting_citations":[{"why":"Establishes the analytic-selection framework for conditional sublinear expectations that the value-function construction generalises.","marker":"[115]"},{"why":"Gives the continuous-process 2BSDE well-posedness whose path-regularisation programme is extended and whose down-crossing proof is repaired.","marker":"[141]"},{"why":"Supplies well-posedness and stability for the BSDEs and reflected BSDEs used to obtain the decomposition of the regularised value process.","marker":"[137]"},{"why":"Provides measurable versions of semi-martingale characteristics in the probability parameter, used throughout for measurability and for aggregating the diffusion integrand.","marker":"[107]"},{"why":"Gives conditioning and concatenation of semi-martingale laws, which underlies the stability properties of the family $\\mathcal P(s,\\omega)$.","marker":"[108]"},{"why":"States the earlier claimed aggregation of jump integrands in a Brownian-Poisson 2BSDE setting, which the present paper addresses.","marker":"[84; 85]"},{"why":"Introduces the original 2BSDE well-posedness and nonlinear Doob-Meyer ideas behind the intrinsic characterisation.","marker":"[153; 154]"},{"why":"Contains the nonlinear super-martingale crossing lemma whose proof is corrected in this paper's regularisation step.","marker":"[19]"}],"fun_headline_variants":["Jump integrands stall in second-order BSDE aggregation","Aggregated value solves 2BSDEs, but jump parts split","One system for diffusions, jumps, and discrete time, but jumps resist","Second-order BSDEs: unified control, but jump integrands stay plural","When jumps break the one-integrand rule in 2BSDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 3.3(iv): for every probability measure there must be one process $\\rho^\\dagger$ whose jumps never hit $-1$, whose quadratic variation is controlled by the reference clock $C$, and whose covariation with jump-integrator differences lower-bounds the generator's dependence on the jump integrand; if this single process does not exist, the linearisation and down-crossing argument that builds the regularised value function breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Jump integrands stall in second-order BSDE aggregation","Aggregated value solves 2BSDEs, but jump parts split","One system for diffusions, jumps, and discrete time, but jumps resist","Second-order BSDEs: unified control, but jump integrands stay plural","When jumps break the one-integrand rule in 2BSDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000544,"raw_usage":{"total_tokens":2649,"prompt_tokens":1034,"completion_tokens":1615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":1520}},"tokens_in":650,"tokens_out":1615,"duration_ms":13182,"temperature":1.0,"reasoning_tokens":1520,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:45:25.732776+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Assumption 3.3(iv) on a pure-jump generator: take $X$ Poisson-like and $f^P$ depending on $U$ through $\\|U(\\cdot)\\|_{\\hat L^2(K)}$, and attempt to exhibit $Y,Z,U,U'$ for which no $\\rho^\\dagger$ satisfies the cross-variation bound (3.12); if such a pair is found, the change-of-measure step in the proof of Theorem 3.5 cannot be run, so the aggregated 2BSDE solution need not exist for that generator class.","supporting_citations":[{"cited_title":"Nutz and R","cited_arxiv_id":null,"evidence_quote":"Establishes the analytic-selection framework for conditional sublinear expectations that the value-function construction generalises."},{"cited_title":"Possamaï, X","cited_arxiv_id":null,"evidence_quote":"Gives the continuous-process 2BSDE well-posedness whose path-regularisation programme is extended and whose down-crossing proof is repaired."},{"cited_title":"Possamaï and M","cited_arxiv_id":null,"evidence_quote":"Supplies well-posedness and stability for the BSDEs and reflected BSDEs used to obtain the decomposition of the regularised value process."},{"cited_title":"Neufeld and M","cited_arxiv_id":null,"evidence_quote":"Provides measurable versions of semi-martingale characteristics in the probability parameter, used throughout for measurability and for aggregating the diffusion integrand."},{"cited_title":"Neufeld and M","cited_arxiv_id":null,"evidence_quote":"Gives conditioning and concatenation of semi-martingale laws, which underlies the stability properties of the family $\\mathcal P(s,\\omega)$."}],"review_version":1}