Existence of $q$-Bass martingales in the semidiscrete setting

Antonio Marini; Beatrice Acciaio

Existence of $q$-Bass martingales in the semidiscrete setting

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Pith's one-line read For finitely supported initial marginals and irreducible pairs, the paper proves existence of q-Bass martingales for any absolutely continuous reference measure q, and uniqueness of the Bass measure up to translation under regularity assump

desk verdict A genuinely new existence theorem for q-Bass martingales in the semidiscrete setting, with a coherent geometric proof and an honest but real limitation to irreducible pairs. read the letter →

arxiv 2607.15872 v1 pith:TJXDTUS2 submitted 2026-07-17 math.PR q-fin.MF

Beatrice Acciaio , Antonio Marini This is my paper · ORCID

classification math.PRq-fin.MF MSC 60G4249Q2260E05

keywords q-BassmartingalesemidiscreteoptimaltransportconvexorderintegratedquantilefunctionsBassmeasurepolygonalchainsuniquenessfixed-pointequation

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 asks when a martingale with prescribed initial and terminal marginals can be made to have transition kernels as close as possible to a given absolutely continuous reference measure q, in the sense of a q-Bass martingale. It proves that if the initial marginal sits on finitely many atoms and the pair of marginals is irreducible, such a martingale always exists, for every absolutely continuous q. Under additional regularity assumptions, it also proves that the underlying Bass measure is unique up to an additive translation constant. The argument is geometric: solutions are encoded by an n-atomic q-Bass map whose image coincides with the family of convex polygonal chains strictly dominating the graph of the terminal integrated quantile, so existence reduces to matching a convex chain. The result matters because it singles out a canonical closest-to-q martingale coupling in a natural semidiscrete calibration problem, without requiring finite second moments.

What carries the argument

The n-atomic q-Bass map f: R^{n−1}_{≥0} → R^{n−1}, defined componentwise by f_i(h) = ∫ Q_ν(∑_{j=1}^n p_j F_q(z − ∑_{k<j} h_k)) ∑_{j=1}^i p_j ρ_q(z − ∑_{k<j} h_k) dz, is the central object. Solving f_i(h) = x^*_i for all i is exactly the reduced quantile equation for the Bass measure. The proof's geometric machinery compares the family Λ_ν of convex polygonal chains strictly above U_ν with the family Λ_f of chains generated by f; the equality of these families under (A1)–(A2) is what turns existence into a chain-matching problem. The technical engine is an induction that corrects one vertex at a time using minimal diagonal gaps and L-approximations, with a matrix adjugate-ratio estimate (Lemm

What would settle it

For a concrete irreducible atomic µ and absolutely continuous ν and q satisfying (A1)–(A2), compute the image of the n-atomic q-Bass map f on [0, diam(supp q))^{n−1} and check whether every strictly convex chain in Λ_ν is attained. In the two-atom case, this reduces to verifying whether the scalar equation ∫ Q_ν(p F_q(z) + (1−p)F_q(z−h)) p ρ_q(z) dz = p x_1 has a solution h in (0, diam(supp q)); a single irreducible example where the image misses a strictly dominating chain vertex vector would falsify Theorem 3.5.

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Extended reading notes

Core claim

The central claim is that the semidiscrete existence question is equivalent to solving the finite-dimensional system f_i(h) = x^*_i for i = 1,...,n−1, where f is the n-atomic q-Bass map constructed from the atom weights and the terminal quantile function, and h collects the spacings of the Bass atoms. Theorem 3.5 identifies the image of f with the family of convex polygonal chains that strictly dominate the integrated quantile U_ν while keeping prescribed abscissas; under assumptions (A1)–(A2) the two families coincide exactly. Hence, when U_μ itself is such a chain — which irreducibility guarantees — the system has a solution h in (0, diam(supp q))^{n−1}. The paper's induction shows that an

Load-bearing premise

The proof requires the pair of marginals to be irreducible: the integrated quantile of the initial law lies strictly above that of the terminal law at every interior point; if the two curves merely touch, the geometric chain-matching argument is not available and existence is left open.

Editorial extensions

If this is right

For any irreducible finitely supported initial law and any absolutely continuous reference measure q, a q-Bass martingale exists; no second-moment condition is needed.
When the paper's regularity assumptions hold, the Bass measure realizing the closest-to-q coupling is unique up to an additive constant.
Existence can be checked by solving the finite system f_i(h) = x^*_i, and the convex-chain identity gives a concrete target for numerical algorithms.
The result covers scenario-based initial marginals with finitely many outcomes, such as event-date distributions, while matching a continuous terminal law in a q-close way.
Because the proof does not rely on optimality, it identifies a canonical coupling even when the martingale-transport value is infinite.

Reading between the lines

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

The strict-dominance requirement suggests that reducible pairs will need a separate treatment, likely by decomposing into irreducible components; the paper explicitly leaves that case open.
If the same geometric mechanism can be pushed through an approximation argument for non-atomic initial laws, q-Bass martingales would exist for essentially all convex-ordered marginals; the paper flags this as forthcoming work.
The contraction machinery behind the induction suggests an alternating-coordinate numerical scheme with geometric error decay for semidiscrete calibration; this is an algorithmic consequence not proved in the paper.
Allowing q to be fat-tailed or compactly supported rather than Gaussian would make the canonical coupling more flexible in practical calibration; this follows from the theorem but is not a claim the paper makes.

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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

The paper proves that q-Bass martingales exist when the initial law is finitely supported, the pair is irreducible, and q is any absolutely continuous reference measure. That is a real advance: prior existence results, such as Hasenbichler et al., only covered Gaussian q, and Tschiderer had the concept but not existence. The uniqueness result for the Bass measure up to translation, under Assumption 1.10, is also new.

What is genuinely new here is the geometric machinery: the n-atomic q-Bass map, the convex-polygonal-chain families Λν and Λf, and the surjectivity theorem Λν ⊆ Λf. The quantile characterization in Theorem 1.4 is proved in both directions, and the potential/concavity analysis in Proposition 2.9 is careful and explicit. I checked several of the internal steps and found them coherent. The approximation argument that removes Assumption 1.10 for existence is a nice touch and makes the main existence theorem pleasingly general.

The soft spots are real but not fatal. The irreducibility assumption is load-bearing: it forces Uμ to lie strictly above Uν, and the whole geometric construction collapses without that strictness. The paper is honest about this and defers the reducible case to the authors' forthcoming paper, but that means the semidiscrete problem is not fully closed. The uniqueness half needs Assumption 1.10, which is a genuine regularity restriction; that is fine as a theorem, but it should not be oversold as a completely general uniqueness result. The most intricate part is Lemma 3.15, whose inductive casework I could not fully certify by reading. The stress-test note found no load-bearing flaw, and the overall strategy is convincing, but this is where a referee should spend time. One minor point: the fixed-point characterization leading to equation (1.6) is used only as context and is not proved; that is fine, but the wording in Section 1 could make clearer that it is not needed for the main results.

The citation pattern is reasonable. The paper cites its own [1] and [2], but the central claims do not rest on those; the main proofs are self-contained modulo standard results.

This paper deserves a serious referee. It solves a genuine open question, introduces a reusable geometric method, and is mostly well written. I would send it to peer review and ask the referee to focus on Lemma 3.15 and the approximation step. For my own work, I would cite it, and I would bring it to a reading group for the geometric proof.

Referee Report

0 major / 4 minor

Summary. The paper studies q-Bass martingales in the semidiscrete setting (µ finitely atomic). It proves two main results: Theorem 1.9, existence of a q-Bass martingale from µ to ν for any absolutely continuous reference measure q, provided the pair (µ, ν) is irreducible; and Theorem 1.12, uniqueness of the Bass measure up to an additive translation under additional regularity assumptions (Assumption 1.10). The proofs are geometric. The n-atomic problem is reduced to solving a system of n−1 equations (2.5) for the n-atomic q-Bass map f. The paper shows that f is the gradient of a strictly concave C² potential (Proposition 2.9), satisfies a non-expansiveness property via adjugate minors (Proposition 2.11), and that the family of convex polygonal chains strictly dominating U_ν coincides with the family generated by f (Theorem 3.5). The key surjectivity result (Theorem 3.7) is proved by an intricate induction involving L-approximations of a target chain and a geometric-series error control (Lemma 3.15), then extended from regular approximating measures to the general case by approximation arguments in Appendices A and B.

Significance. If the results hold, the paper makes a substantial contribution to martingale optimal transport. It establishes existence of the canonical q-Bass martingale for essentially arbitrary absolutely continuous reference measures when the initial law is atomic and the convex-order pair is irreducible, thereby going beyond the Gaussian case treated in earlier work. The geometric approach via convex polygonal chains is original and is explicitly positioned as the foundation for a forthcoming general existence result. The paper is methodologically strong: the quantile characterization (Theorem 1.4) is proved in both directions; the potential structure of the n-atomic Bass map (Proposition 2.9) is derived with explicit Hessian formulas; the non-expansiveness bound (Proposition 2.11) rests on a careful Cauchy–Binet/adjugate argument; and the approximation step (Section B) is fully written. The main theorems are stated with precise hypotheses, and the paper is honest about the irreducible-pair assumption and about deferring the general case to a companion paper.

minor comments (4)

[§2, Proposition 2.6] The statement 'y solves (2.2) if and only if h solves (2.5)' is not literally true unless the mean equality mean(µ)=mean(ν) is included as a standing hypothesis: if h solves (2.5) but the means differ, the n-th equation in (2.2) fails. In every application in the paper, irreducibility supplies this equality, so the issue is purely one of statement precision. Please add 'assuming mean(µ)=mean(ν)' to the equivalence, or rephrase the proposition to separate the necessary and sufficient parts.
[Definition 1.1 and Remark 1.2] The condition 'q ⋆ T is increasing α-a.s.' should be stated as 'strictly increasing' to match the strict convexity interpretation and the construction, where S_α(y_i)=x_i with x_1<...<x_n. As written, the non-strict reading conflicts with the example in Remark 1.2 and with the later use of strict monotonicity.
[Lemma 3.15, induction step] In the induction step, Proposition 2.11 is applied to h and h^(1) with I_dom=eI∪{i*}. This requires that the i*-th coordinate actually changes, i.e., h_{i*}≠h^(1)_{i*}. The constructive proof of the induction hypothesis presumably yields this property when E_{i*}(h)≠0, but the statement of the lemma does not record it. Please add a sentence explaining that the induction output can be chosen with h^(1)_{i*}≠h_{i*} when the initial error at i* is nonzero (or handle the case h_{i*}=h^(1)_{i*} separately).
[Throughout] There are a few small slips, e.g., 'ef is the is the one-dimensional map' in the proof of Lemma 3.15, and the notation µ is reused in the approximation part of Theorem 3.7 for the distribution whose graph is C. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the geometric existence proof is self-contained.

full rationale

The central derivation chain is not circular. The paper constructs the Bass measure through the quantile equation (1.7) and reduces the existence problem to solving the finite system f(h) = x* for the n-atomic q-Bass map f, defined directly from q, ν, and the weights. The load-bearing inclusion Λν ⊆ Λf, proved as Theorem 3.7 via the induction in Lemma 3.15, is a nontrivial surjectivity statement; it does not assume the existence of a Bass measure or of a solution to (2.5). Uniqueness in Theorem 1.12 follows from Proposition 2.9, where V is shown, under the stated Assumption 1.10, to be strictly concave with ∇V = f, so gradient injectivity forces h = h′. No fitted parameter is renamed as a prediction, and no result invoked by definition is claimed as a consequence. The self-citations [1] and [2] are used only to cite the origin of the quantile formulation and to defer the general (non-semidiscrete) case; the main theorem is proved in-text, and the approximation step removing Assumption 1.10 uses independent approximation lemmas (Propositions B.1 and B.2) together with the stability result Proposition 2.12. The explicit deferral to [1] for the reducible/general case is a stated limitation, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data anywhere in the paper: this is a pure existence/uniqueness theorem. The internal constant L ∈ (0,1) in Lemma 3.15 is a proof artifact — a uniform bound on adjugate ratios over a compact cube, guaranteed < 1 by Lemma A.2 and compactness — not a model parameter, so it is not listed as a free parameter. The n-atomic q-Bass map f is a new mathematical device with no empirical content; the Bass measure α is not invented by this paper (it is defined by Tschiderer [13], reformulated in Definition 1.1). The axioms above are the explicit domain assumptions (irreducibility, absolute continuity, regularity conditions) and the standard external results (Strassen, the quantile convex-order characterization, absolute continuity of L^p translations) on which the proofs rest.

assumptions (6)

domain assumption Irreducibility of (μ,ν): U_μ(p) > U_ν(p) for all p ∈ (0,1); equivalently {C_μ < C_ν} is an interval with μ-mass 1
Definition 1.5; used in Lemma 3.3 to place the graph of U_μ in the strictly-dominating chain family Λ_ν, the entry point of the geometric existence proof (Theorem 1.9). Reducible pairs are out of scope.
standard math Convex-order characterization via integrated quantiles: μ ⪯_c ν iff means equal and U_μ ≥ U_ν pointwise
Proposition 1.8, proved in the paper, relying on [11, Theorem 3.A.5]. Used throughout the geometric arguments to translate convex order into comparisons of convex polygonal chains.
standard math Strassen's theorem: M(μ,ν) is non-empty iff μ ⪯_c ν
Cited [12]; frames the necessary condition for existence of any martingale coupling and justifies the standing convex-order hypothesis.
domain assumption q ≪ λ (reference measure absolutely continuous w.r.t. Lebesgue)
Definition 1.1 and all theorems; needed for α∗q to be atomless, so that T_α = Q_ν ∘ F_{α∗q} pushes α∗q onto ν (Theorem 1.4 converse).
domain assumption Assumption 1.10 (A1)–(A4): ν, q absolutely continuous with interval supports and strictly positive densities on interiors; ν ∈ P^a, q ∈ P^b with b > a/(a−1); Q'_ν ∈ L^ς(0,1); ρ_q ∈ L^{max((2ς−1)/(ς−1), a/(a−1))}
Postulated regularity for the uniqueness theorem (Theorem 1.12) and for Proposition 2.9 (strict concavity and C² regularity of the potential V, with ∇V = f). These are explicit stated conditions, not fitted values.
domain assumption No-three-consecutive-collinear-vertices condition for chains in Λ_ν (Definition 3.2(ii))
Guarantees the constructed increment vector h lies in the open domain (0, diam(supp q))^{n−1} (Corollary 3.8); automatically satisfied by chains coming from an atomic law with distinct atoms x_1 < ... < x_n.

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Pith. "Pith review of Existence of $q$-Bass martingales in the semidiscrete setting." pith.science (2026). https://pith.science/paper/TJXDTUS2

@misc{pith2026260715872,
  author       = {Pith},
  title        = {Pith review of: Existence of $q$-Bass martingales in the semidiscrete setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJXDTUS2}},
  note         = {Machine review of arXiv:2607.15872}
}

abstract

The class of $q$-Bass martingales provides a natural answer to a central question in martingale optimal transport: how to construct martingales with prescribed initial and terminal marginals whose transition kernel remains as close as possible to a given reference measure $q$. We prove the existence of $q$-Bass martingales when the initial marginal is supported on finitely many atoms, and establish uniqueness, up to an additive translation constant, of the associated Bass measure. Our approach is geometric and relies on the analysis of a suitable parametrization of convex polygonal chains.

Figures

Figures reproduced from arXiv: 2607.15872 by the authors.

**Figure 2.** Minimal diagonal gap and L-approximations of a convex polygonal chain C. Definition 3.10 (Minimal diagonal gap). Let C ∈ Λ p1,...,pn ν . For every i ∈ {1, . . . , n − 1}, define δi := dist (p ∗ i , Ci), [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Illustration of the algorithm in Theorem 3.7 and Lemma 3.15. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Example with i = 2. Definition 3.10 gives δ2 < δ. Therefore, by Remark 2.4 and Proposition 3.6, f2(eh1, 0, eh3) ≥ C2. On the other hand, Proposition 3.6 and Definition 3.2 imply lim t→∞ E (r) i (eh1, . . . , ehi−1, t, ehi+1, . . . , ehn−1) = Uν(p ∗ i ) − C(r) i ≤ Uν(p …

**Figure 5.** Figure 5: In this example, C (r) is shown in red and iδ = 2. Since E (r) iδ+1(h (2)) ≥ 0, we also have E (r) iδ (h ′ ) ≥ 0, where h ′ is the vector obtained from h (2) by setting its iδ-th component equal to 0. Hence, the iδ-th component can be adjusted again using the standard …

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

13 extracted references · 5 canonical work pages

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Fixed Points for theq-Bass Martingale: Existence, Stability, and Convergence

Beatrice Acciaio and Antonio Marini. “Fixed Points for theq-Bass Martingale: Existence, Stability, and Convergence”. 2026. Forthcoming

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Calibration of the Bass Local Volatility Model

Beatrice Acciaio, Antonio Marini, and Gudmund Pammer. “Calibration of the Bass Local Volatility Model”. In:SIAM Journal on Financial Mathematics16.3 (2025), pp. 703–833. doi:10.1137/23M1622660

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Fixed Points for theq-Bass Martingale: Existence, Stability, and Convergence

Beatrice Acciaio and Antonio Marini. “Fixed Points for theq-Bass Martingale: Existence, Stability, and Convergence”. 2026. Forthcoming

2026

[2] [2]

Calibration of the Bass Local Volatility Model

Beatrice Acciaio, Antonio Marini, and Gudmund Pammer. “Calibration of the Bass Local Volatility Model”. In:SIAM Journal on Financial Mathematics16.3 (2025), pp. 703–833. doi:10.1137/23M1622660

work page doi:10.1137/23m1622660 2025

[3] [3]

Mar- tingaleBenamou-Brenier:aprobabilisticperspective

JulioBackhoff-Veraguas,MathiasBeiglböck,MartinHuesmann,andSigridKällblad.“Mar- tingaleBenamou-Brenier:aprobabilisticperspective”.In:Ann. Probab.48.5(2020),pp.2258– 2289.issn: 0091-1798.doi:10.1214/20-AOP1422.url:https://doi.org/10.1214/20- AOP1422

work page doi:10.1214/20-aop1422.url:https://doi.org/10.1214/20- 2020

[4] [4]

Existence of Bass martingales and the martingale Benamou Brenier problem inRd

JulioBackhoff-Veraguas,MathiasBeiglböck,WalterSchachermayer,andBertramTschiderer. “Existence of Bass martingales and the martingale Benamou Brenier problem inRd”. In: ArXiv e-prints(2023)

2023

[5] [5]

The Bass func- tionalofmartingaletransport

Julio Backhoff-Veraguas, Walter Schachermayer, and Bertram Tschiderer. “The Bass func- tionalofmartingaletransport”.In:The Annals of Applied Probability35.6(2025),pp.4282– 4301

2025

[6] [6]

Skorokhod embedding via stochastic integrals

Richard F. Bass. “Skorokhod embedding via stochastic integrals”. en. In:Séminaire de Probabilités XVII 1981/82. Ed. by Jacques Azéma and Marc Yor. Lecture Notes in Math- ematics986.Springer,1983,pp.221–224.isbn:978-3-540-12289-0,978-3-540-39614-7.url: http://link.springer.com/chapter/10.1007/BFb0068318

work page doi:10.1007/bfb0068318 1981

[7] [7]

On a problem of optimal transport under marginal martingale constraints

Mathias Beiglböck and Nicolas Juillet. “On a problem of optimal transport under marginal martingale constraints”. In:Ann. Probab.44.1 (2016), pp. 42–106.issn: 0091-1798.doi: 10.1214/14-AOP966.url:http://dx.doi.org/10.1214/14-AOP966

work page doi:10.1214/14-aop966.url:http://dx.doi.org/10.1214/14-aop966 2016

[8] [8]

Bass Construction with Multi-Marginals: LightspeedComputationinaNewLocalVolatilityModel

Antoine Conze and Pierre Henry-Labordere. “Bass Construction with Multi-Marginals: LightspeedComputationinaNewLocalVolatilityModel”.en.In:SSRN Electronic Journal (2021).issn: 1556-5068.doi:10.2139/ssrn.3853085.url:https://www.ssrn.com/ abstract=3853085(visited on 07/25/2023)

work page doi:10.2139/ssrn.3853085.url:https://www.ssrn.com/ 2021