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Markovian projections for functionals of It\^o semimartingales with jumps

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Pith's one-line read For every Itô semimartingale with jumps, a Markovian-type process can match the fixed-time marginals of any continuous updating functional.

desk verdict 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. read the letter →

arxiv 2506.00762 v2 pith:Q4FCOOTL submitted 2025-06-01 math.PR q-fin.MF

Martin Larsson , Shukun Long This is my paper
classification math.PRq-fin.MF MSC 60G4460J7660G5760H10
keywords MarkovianprojectionmimickingprocessItôsemimartingalejumpupdatingfunctioncharacteristicsconcatenatedprobabilitymeasureone-dimensionalmarginals
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [§3.2, Theorem 3.6] 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.
  2. [Proof of Theorem 2.12, first sentence] 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.
minor comments (5)
  1. [§3.1] 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.
  2. [§4, Step 1] 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.
  3. [Example 2.5] 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.
  4. [Throughout] 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.
  5. [Remark 4.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: mimicking coefficients are conditional expectations of the original characteristics, and the marginal-law matching is derived, not assumed.

full rationale

No circularity is present. The central coefficients in Theorem 2.12 are defined by (2.5) as conditional expectations of the original differential characteristics (b,c,κ) given Z_t; the matching of the one-dimensional laws of Z-hat is not assumed but derived through the Brunick–Shreve concatenated-measure construction and a compactness/limit argument. Thus no fitted input is renamed as a prediction, and the target does not enter the construction by definition. The only self-citation is [11, Lemma 2.5], used to obtain measurable versions of the kernel κ-hat; this is a measure-theoretic lemma from the authors' earlier paper, not a result that assumes Theorem 2.12 or the existence of the mimicking process, so it is not load-bearing in a circular sense. The caveat about Theorem 3.6—that the 'verbatim' transfer of [4, Theorem 4.3] may require pasting-closure of X beyond Δ-stability, while the concrete X* used in Theorem 2.12 is pasting-closed—is a proof-completeness and correctness issue, not a circularity. The paper is otherwise self-contained against the external benchmark [4], and no step reduces by construction to its own conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted or hand-chosen constants appear. The projected coefficients (b̂, ĉ, κ̂) are defined as conditional expectations in (2.5), determined by the original characteristics and the marginal conditioning, not by fitting to a target output. The paper introduces no new physical or mathematical objects beyond the canonical space construction; the measure-valued process M is a technical device built from the given compensator.

assumptions (5)
  • domain assumption The Brunick-Shreve concatenated measure theorem (Theorem 3.6) extends verbatim to the jump canonical space Ω_{E,X}; only Polishness and Δ-stability are needed.
    Section 3.2, proof of Theorem 3.6: 'The proof is verbatim the same as that of [4], Theorem 4.3... We only need to replace their canonical space by ours.' This is the main structural import from [4] and is load-bearing for the concatenation step.
  • domain assumption Existence of measurable kernels b̂, ĉ, κ̂ satisfying (2.5) follows from [4] Proposition 5.1 and [11] Lemma 2.5, which 'obviously extends' to E-valued processes.
    Step 1 of the proof of Theorem 2.12 states this without proof. The extension to E-valued processes and transition kernels from R_+ × E to R^d is asserted.
  • domain assumption The integrability condition (2.4) is the sole standing assumption on the differential characteristics.
    Statement of Theorem 2.12. This is the explicit assumption that makes the conditional expectations finite and the construction work.
  • standard math The class C^1(R^d) convergence-determining functions from Jacod-Shiryaev exists and can be chosen countable.
    Definition 3.21 and Remark 3.22, used for measure-determining properties of the third characteristic.
  • standard math Standard semimartingale characteristic theory (Jacod and Shiryaev, Limit Theorems for Stochastic Processes) is taken as background.
    Used throughout for the equivalence of local martingale properties and characteristics, e.g., Theorem II.2.21 and IX.2.4.

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Pith. "Pith review of Markovian projections for functionals of It\^o semimartingales with jumps." pith.science (2026). https://pith.science/paper/Q4FCOOTL

@misc{pith2026250600762,
  author       = {Pith},
  title        = {Pith review of: Markovian projections for functionals of It\^o semimartingales with jumps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q4FCOOTL}},
  note         = {Machine review of arXiv:2506.00762}
}
abstract

Given an It\^o semimartingale $X$, its Markovian projection is an It\^o semimartingale $\widehat{X}$, with Markovian differential characteristics, that matches the one-dimensional marginal laws of $X$. One may even require certain functionals of the two processes to have the same fixed-time marginals, at the cost of enhancing the differential characteristics of $\widehat{X}$ but still in a Markovian sense. In the continuous case, the definitive result on existence of Markovian projections was obtained by Brunick and Shreve~\cite{MR3098443}. In this paper, we extend their result to the fully general setting of It\^o semimartingales with jumps.

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