{"id":"f37440bf-df2f-4140-96c5-77c0777698f3","arxiv_id":"2506.00762","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a fully general existence theorem for Markovian projections of Itô semimartingales with jumps, including projections that match the marginals of updating functionals of the process.","lead":"This paper proves that any jump diffusion can be replaced by a simpler Markovian-type process with the same probability distribution at each fixed time, even when tracking functions such as the running maximum or total integral. It completes a program in stochastic analysis, giving a general existence theorem useful for simplifying models in finance and for studying path-dependent quantities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6's proof is not literally verbatim from [4]: the concatenation construction needs X closed under pasting, which Δ-stability alone does not imply; the specific X* used in Theorem 2.12 is pasting-closed, so the central claim is likely repairable but the proof as written is incomplete.","rationale":"The reader correctly identifies Theorem 3.6 as the load-bearing imported step. My stress-test sharpens the concern: the missing property is not merely 'Polishness and Δ-stability' but closure under the pasting operation used in the Brunick–Shreve proof. The paper states the proof is verbatim, but the cited construction requires a path-space operation that is not available from Δ-stability alone. For the canonical space X* actually used in the proof of Theorem 2.12, pasting closure does hold, so the central claim is credible and likely correct after a repair of Theorem 3.6. I therefore recommend a conditional verdict rather than outright rejection: the main theorem should be accepted once the authors either add the missing closure hypothesis to Theorem 3.6 and prove it for X*, or give a direct proof of the concatenation theorem for the specific canonical space used. The secondary import from [11] Lemma 2.5 appears standard and is not a serious concern. The proof has no obvious circularity or internal inconsistency beyond the delegated Theorem 3.6, and the new jump-specific components are otherwise handled with care.","tokens_in":29030,"tokens_out":40637,"duration_ms":397323,"concrete_test":"Verify that the concatenation map used in [4] Theorem 4.3 is Borel and maps X* × X* into X*, then reproduce the measure-pasting construction on Ω_{E*,X*} and check properties (i)–(ii) of Theorem 3.6. Equivalently, test the claimed generality: find a closed, Δ-stable X that is not closed under pasting and show Theorem 3.6 fails for it; if such an X exists, Theorem 3.6 needs the extra closure hypothesis, while Theorem 2.12 may still hold because its X* satisfies the hypothesis.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 2.12 depends on Theorem 3.6, which asserts that the Brunick–Shreve concatenated-measure construction transfers verbatim to any Δ-stable closed X⊂D^{E'}_0. The verbatim transfer is not automatic: [4] constructs the concatenated measure by pasting a past path with a future increment path via the operation (x,y,t) ↦ x_t ⊕_t y, where (x_t ⊕_t y)(s)=x(s) for s≤t and =x(t)+y(s−t) for s≥t. For the pasted path to remain in the canonical space, X must be closed under this operation (and under stopping); this closure is not implied by Δ-stability alone. The statement of Theorem 3.6 therefore has a missing hypothesis. In the concrete case used to prove Theorem 2.12, X* = D^d_0 × C^d_0 × C^{d^2}_0 × C^{M+,d}_{0,i} is closed under pasting, so the main theorem is probably salvageable. But the assertion that the proof is 'verbatim the same' does not establish Theorem 3.6 in the stated generality, and the central existence proof rests on that gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an extension of the Brunick-Shreve mimicking theorem to Itô semimartingales with jumps. Theorem 2.12 states that, under the integrability condition (2.4), for any Polish space E, any E-valued F_0-measurable Z_0, any R^d-valued Itô semimartingale Y with differential characteristics (b,c,κ), and any continuous updating function Φ, there exist measurable Markovian coefficients (b̂,ĉ,κ̂) satisfying the conditional-expectation identities (2.5), together with a filtered probability space supporting (bZ_0,bY) whose differential characteristics are given by (2.6), such that bZ = Φ(bZ_0,bY) has the same fixed-time marginals as Z = Φ(Z_0,Y). Corollary 2.15 gives a truncation-free version for special semimartingales with canonical characteristics, and Example 2.17 shows preservation of iterated-integral structure. The proof builds a canonical space Ω* = [0,1]×E×D^d_0×C^d_0×C^{d^2}_0×C^{M+,d}_{0,i}, constructs concatenated probability measures à la Brunick-Shreve, recovers the jump compensator from a measure-valued coordinate via Lemma 3.15, proves tightness, and identifies the limiting process's characteristics in Step 6.","tokens_in":29325,"tokens_out":7825,"duration_ms":74725,"significance":"If correct, this is a definitive existence result for Markovian projections in the jump setting: it removes the boundedness, continuity, and growth conditions present in earlier work of Bentata-Cont and in the authors' previous paper, and it extends the mimicking result to continuous updating functionals. The choice of the canonical space for the third characteristic, using the space C^{M+,d}_{0,i} of increasing continuous measure-valued paths and recovering the compensator through Lemma 3.15, is a genuine technical contribution. The paper also contains full proofs of several new lemmas (3.15, 3.17, 3.18) rather than mere citations, and the overall structure of the proof is transparent enough to audit. However, one load-bearing transfer step, Theorem 3.6, is asserted rather than proved in the stated generality, so the manuscript is not yet complete as written even though the main theorem appears repairable.","major_comments":[{"comment":"The theorem asserts that the Brunick-Shreve concatenated-measure construction extends verbatim to every Δ-stable closed X ⊂ D^{E'}_0, with the proof reduced to the Polishness of Ω_{E,X} and the Δ-stability of X. The construction in [4] pastes a past path with a future increment path via (x,y,t) ↦ x_t ⊕_t y, and it also uses stopped paths; for the pasted path to remain in X, X must be closed under this pasting operation and under stopping. Δ-stability alone gives x(t+·)-x(t) ∈ X but does not imply x^t ∈ X or x_t ⊕_t y ∈ X. Thus Theorem 3.6 is not proved as stated. This is load-bearing because Theorem 2.12's existence proof invokes Theorem 3.6 on X* = D^d_0 × C^d_0 × C^{d^2}_0 × C^{M+,d}_{0,i}; that particular X* is closed under pasting and stopping, so the main theorem is likely salvageable, but the missing hypothesis must be added and verified, or Theorem 3.6 must be proved from Δ-stability alone.","section":"§3.2, Theorem 3.6"},{"comment":"The existence of b̂, ĉ, and κ̂ satisfying (2.5) is attributed to [4, Proposition 5.1] and [11, Lemma 2.5] with an 'obvious extension' to E-valued processes and transition kernels from R_+ × E to R^d. This is a nontrivial measurable-selection step: the conditional expectations in (2.5) are taken with respect to an E-valued random variable Z_t, and κ_t is a transition kernel, not merely a real-valued process. Since (2.5) is the only link between the original characteristics and the candidate coefficients used in Step 6, the extension should be stated as a lemma and proved, or the exact statement from [11] should be reproduced so the reader can verify the extension.","section":"Proof of Theorem 2.12, first sentence"}],"minor_comments":[{"comment":"The notation F_t := σ(E, X_t) uses the stopped-path convention x^t = x(t∧·) introduced in Section 2.1, but this convention is not recalled in Section 3.1; it should be restated there to avoid confusion with the coordinate at time t.","section":"§3.1"},{"comment":"After Lemma 3.15, the statement that λ* is a random measure is justified by a Dynkin π-λ argument, but the measurability of the Carathéodory extension from the algebra A is not shown in detail; a brief argument that the set function on A is measurable in ω would improve readability.","section":"§4, Step 1"},{"comment":"The continuity proof for the maximal jump-to-date updating function is terse. The inequality max_{s≤t}|y(s)-y(s-)| ≤ 2 sup_{s≤t}|y(s)| is correct, but the passage from Skorokhod convergence to convergence of the running maximum of jumps would benefit from an explicit time-change argument, as in Example 2.4.","section":"Example 2.5"},{"comment":"There are numerous typesetting artifacts, such as missing spaces in 'LetP', 'Q m', and 'bP', and inconsistent spacing around operators. These do not affect the mathematics but should be cleaned before publication.","section":"Throughout"},{"comment":"The discussion of the two failed attempts at choosing a canonical space for the third characteristic is helpful, but it would be even more useful if it explicitly stated why the chosen space C^{M+,d}_{0,i} is closed under the pasting operation used in the concatenated-measure construction, since this is exactly the property whose absence makes Theorem 3.6's generality problematic.","section":"Remark 4.1"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a likely correct main theorem, but the proof currently contains a real gap in the stated generality of Theorem 3.6. I recommend major revision rather than rejection because the concrete canonical space X* used in Theorem 2.12 appears to satisfy the missing closure properties, so the gap is repairable within the manuscript's scope. The authors should also replace the 'obvious extension' of [11, Lemma 2.5] with a self-contained statement, as this is a load-bearing import."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the right generalization of Brunick–Shreve to full Itô semimartingales with jumps and updating functionals, and it mostly delivers. The main theorem removes the growth condition from the authors' earlier [11] and the boundedness/continuity assumptions of Bentata–Cont, under only the natural integrability condition (2.4). The new machinery is real: the space C^{M+,d}_{0,i} of increasing measure-valued paths for the third characteristic, Lemma 3.15 recovering the compensator ν from M pathwise, and the tightness step in Step 4. The examples are useful, especially the doubly stochastic compound Poisson process that satisfies (2.4) but violates the growth condition of [11].\n\nThe soft spot is Theorem 3.6. The claim that the Brunick–Shreve concatenated-measure construction transfers verbatim to any Δ-stable closed X is not supported by the two facts named (Polishness and Δ-stability). The construction pastes a past path with a future increment path via (x,y,t) ↦ x_t ⊕_t y, and for the concatenated measure to live on Ω_{E,X}, X must be closed under that pasting operation (and under stopping). Δ-stability alone does not give this, so Theorem 3.6 as stated has a missing hypothesis. The specific X* used in Theorem 2.12 is D^d_0 × C^d_0 × C^{d^2}_0 × C^{M+,d}_{0,i}, which is pasting-closed; hence the main theorem is likely salvageable by adding the closure condition or by proving Theorem 3.6 only for X*. But as written, the proof of Theorem 2.12 relies on a theorem whose stated hypotheses do not cover the application. This should be fixed before publication.\n\nThe other imports are less concerning. The use of [11, Lemma 2.5] with an 'obvious extension' to E-valued kernels is standard conditional-expectation machinery and should check out. The heavy reliance on [4] for parts of Sections 3.2–3.3 is honest delegation rather than hand-waving.\n\nWho this is for: anyone working on mimicking theorems, Markovian projections, or path-dependent SDEs with jumps. The paper deserves a serious referee. I would send it to a specialist with an explicit instruction to verify Theorem 3.6 and the closure properties of X*. If the authors patch that gap, this is a strong paper.","headline":"Significant generalization of Brunick–Shreve to jumps that mostly works, but Theorem 3.6 has a missing pasting-closure hypothesis that is repairable for the paper's own canonical space.","tokens_in":29836,"tokens_out":2778,"would_cite":true,"duration_ms":26337,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G44","60J76","60G57","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every Itô semimartingale with jumps, a Markovian-type process can match the fixed-time marginals of any continuous updating functional.","keywords":["Markovian projection","mimicking process","Itô semimartingale","jump process","updating function","semimartingale characteristics","concatenated probability measure","one-dimensional marginals"],"falsifier":"Find an Itô semimartingale with jumps satisfying (2.4) and a continuous updating function whose fixed-time marginals cannot be matched by any process with characteristics $\\hat{b}(s,\\hat{Z}_s)$, $\\hat{c}(s,\\hat{Z}_s)$, $\\hat{\\kappa}(s,\\hat{Z}_s,d\\xi)$; more directly, produce a $\\Delta$-stable closed Polish subset $X$ of $D_0^{E'}$ and an extended partition for which the concatenated measure does not exist or fails its defining conditional-expectation property.","tokens_in":28837,"feed_emoji":"⚡","tokens_out":13643,"duration_ms":117761,"temperature":0.7,"pith_summary":"The paper proves that every Itô semimartingale with jumps has a Markovian projection that can also match the fixed-time marginals of any continuous updating functional of it, such as running maximum, integral-to-date, or maximal jump-to-date. The only assumption is an integrated integrability condition on the differential characteristics: over finite time horizons, the drift, the diffusion matrix, and the jump measure must have finite expected integrals with a quadratic weight on small jumps. This extends the continuous-process mimicking theorem to the full jump setting and removes the boundedness, non-degeneracy, and continuity conditions required by earlier jump mimicking results. If the theorem is correct, complicated path-dependent jump processes can be replaced, for fixed-time marginal questions, by processes whose differential characteristics are deterministic functions of the current state.","feed_headline":"Markovian projection theorem extends to jump processes","feed_subtitle":"Continuous updating functionals such as running maxima keep fixed-time marginals under a mild integrability condition.","key_machinery":"The machinery has three parts. An updating function is a continuous map $\\Phi:E\\times D_0^d\\to D^E$ that starts at the initial value, respects nonanticipativity, and satisfies a Markov-type semigroup relation; it packages running maxima, integrals, maximal jumps, and the process itself. The concatenated probability measure construction, taken from the continuous case, takes a probability measure on the canonical space $\\Omega_{E,X}=E\\times X$ with $X$ a $\\Delta$-stable closed subset of $D_0^{E'}$ and an extended partition of time, and rebuilds the conditional law of increments given the current value of $Z$. The jump-specific addition is the coordinate $M_t(A)=\\int_0^t\\int_A(1\\wedge|\\xi|^2)\\kappa_s(d\\xi)ds$, an increasing continuous measure-valued path in $C_{0,i}^{M+,d}$; Lemma 3.15 recovers the third characteristic's random measure from $M$ in a measure-theoretic, probability-free way, which lets the concatenation argument control the jump compensator.","core_discovery":"The central claim is that, under the condition $\\mathbb{E}\\int_0^t(|b_s|+|c_s|+\\int_{\\mathbb{R}^d}(1\\wedge|\\xi|^2)\\kappa_s(d\\xi))ds<\\infty$ for every $t>0$, any continuous updating function $\\Phi$ admits a mimicking triplet $(\\hat{b},\\hat{c},\\hat{\\kappa})$ and a filtered probability space carrying $(\\hat{Z}_0,\\hat{Y})$ such that $\\hat{Y}$ is an Itô semimartingale with characteristics $\\hat{b}(s,\\hat{Z}_s)$, $\\hat{c}(s,\\hat{Z}_s)$, $\\hat{\\kappa}(s,\\hat{Z}_s,d\\xi)$, and $\\hat{Z}=\\Phi(\\hat{Z}_0,\\hat{Y})$ has the same fixed-time marginal laws as $Z=\\Phi(Z_0,Y)$. The coefficients are conditional expectations of the original characteristics given $Z_t$, so the identities (2.5) are both the definition of the projected dynamics and the mechanism that matches marginals. The proof constructs a canonical space carrying $(Z_0,Y,B,C,M)$ with $M_t(A)=\\int_0^t\\int_A(1\\wedge|\\xi|^2)\\kappa_s(d\\xi)ds$, uses randomized time discretizations and concatenated probability measures to rebuild conditional laws of increments, proves tightness of the resulting processes, and identifies the weak limit's characteristics.","pith_inferences":["Not claimed in the paper, but a natural numerical test would be to implement the conditional-expectation coefficients for a stochastic-volatility model with jumps: the formula is explicit only if the conditional law of $(b,c,\\kappa)$ given $Z_t$ is known, so a practical scheme would have to estimate those conditional expectations.","Because the chosen canonical space for the third characteristic is Polish and closed under increments, similar concatenation arguments may work for other measure-valued path functionals of the jump measure, provided their path space has those two properties; functionals that break closure under increments would need a different carrier.","Combining the existence result with regularity conditions on $\\hat{b}$, $\\hat{c}$, $\\hat{\\kappa}$ that guarantee uniqueness of martingale solutions would upgrade the projection from existence to full Markovian dynamics; the paper only notes that such conditions cannot be read off from assumptions on the original characteristics.","The same concatenation mechanism might be adaptable to conditional McKean–Vlasov equations with jumps, where each particle's coefficient depends on the common law; this connection is not explored in the paper."],"forward_implications":["Under only the integrated condition (2.4), running maxima, integrated values, maximal jumps, and other continuous updating functionals of a jump Itô process can be mimicked jointly with the process itself at fixed times by a Markovian-type pair $(\\hat{Y},\\hat{Z})$.","Earlier jump mimicking theorems required boundedness, non-degeneracy, decay of the jump kernel, or growth conditions on the projected coefficients; for existence questions, the present result removes those requirements.","For special Itô semimartingales, a truncation-free version holds with canonical characteristics under the slightly stronger condition $\\mathbb{E}\\int_0^t(|b_s|+|c_s|+\\int_{\\mathbb{R}^d}(|\\xi|\\wedge|\\xi|^2)\\kappa_s(d\\xi))ds<\\infty$.","The mimicking process need not be Markov or unique in law; the theorem guarantees existence only, and additional regularity of $\\hat{b}$, $\\hat{c}$, $\\hat{\\kappa}$ would be needed for those properties.","Iterated-integral structures such as $Y=\\int X_-\\,dX$ are preserved: the projection of $(X,Y)$ again satisfies the same relation, as shown in Example 2.17."],"supporting_citations":[{"why":"Supplies the updating-function framework, the concatenated probability measure construction, and the approximation lemmas that the proof extends from continuous processes to jump semimartingales.","marker":"[4]"},{"why":"The authors' earlier jump-case mimicking theorem provides the conditional-expectation lemma used to obtain the projected kernel for the third characteristic.","marker":"[11]"},{"why":"Provides the semimartingale characteristics formalism and the weak-convergence theorems used to identify the limit process's characteristics.","marker":"[7]"},{"why":"It is the original Markovian projection theorem for continuous Itô processes, whose conditional-expectation coefficient formula the paper generalizes.","marker":"[6]"},{"why":"It is an earlier mimicking theorem for semimartingales with jumps whose boundedness, non-degeneracy, and continuity assumptions the present result removes.","marker":"[2]"},{"why":"It is the superposition principle for non-local generators, the tool used in the authors' earlier paper and cited as unavailable in path-dependent updating-function settings.","marker":"[16]"},{"why":"It is the first use of a superposition principle in a mimicking theorem, providing the alternative approach the paper does not follow.","marker":"[10]"}],"fun_headline_variants":["Markovian projections now cover jump semimartingales","Jump processes gain full Markovian projection theorem","Mimicking jump semimartingales via Markovian projections","Extending Brunick-Shreve to Itô jumps and functionals","Fixed-time marginals matched for jump processes with Markovian updates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the concatenated-measure construction from the continuous case transfers verbatim to the jump canonical space solely because that space is Polish and $\\Delta$-stable (closed under taking increments); this transfer is asserted without proof, and the extension of a conditional-kernel lemma from the authors' earlier paper to general state spaces is likewise imported.","fun_headline_variants_meta":{"raw":{"variants":["Markovian projections now cover jump semimartingales","Jump processes gain full Markovian projection theorem","Mimicking jump semimartingales via Markovian projections","Extending Brunick-Shreve to Itô jumps and functionals","Fixed-time marginals matched for jump processes with Markovian updates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1191,"prompt_tokens":956,"completion_tokens":235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":151}},"tokens_in":572,"tokens_out":235,"duration_ms":2963,"temperature":1.0,"reasoning_tokens":151,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:58:22.670971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an Itô semimartingale with jumps satisfying (2.4) and a continuous updating function whose fixed-time marginals cannot be matched by any process with characteristics $\\hat{b}(s,\\hat{Z}_s)$, $\\hat{c}(s,\\hat{Z}_s)$, $\\hat{\\kappa}(s,\\hat{Z}_s,d\\xi)$; more directly, produce a $\\Delta$-stable closed Polish subset $X$ of $D_0^{E'}$ and an extended partition for which the concatenated measure does not exist or fails its defining conditional-expectation property.","supporting_citations":[{"cited_title":"Brunick and S","cited_arxiv_id":null,"evidence_quote":"Supplies the updating-function framework, the concatenated probability measure construction, and the approximation lemmas that the proof extends from continuous processes to jump semimartingales."},{"cited_title":"Larsson and S","cited_arxiv_id":null,"evidence_quote":"The authors' earlier jump-case mimicking theorem provides the conditional-expectation lemma used to obtain the projected kernel for the third characteristic."},{"cited_title":"Jacod and A","cited_arxiv_id":null,"evidence_quote":"Provides the semimartingale characteristics formalism and the weak-convergence theorems used to identify the limit process's characteristics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the original Markovian projection theorem for continuous Itô processes, whose conditional-expectation coefficient formula the paper generalizes."},{"cited_title":"Röckner, L","cited_arxiv_id":null,"evidence_quote":"It is the superposition principle for non-local generators, the tool used in the authors' earlier paper and cited as unavailable in path-dependent updating-function settings."},{"cited_title":"Lacker, M","cited_arxiv_id":null,"evidence_quote":"It is the first use of a superposition principle in a mimicking theorem, providing the alternative approach the paper does not follow."}],"review_version":1}