Embedding martingale diffusions as binary posteriors in sequential inference

Karl Kristian Engelund; Steven Campbell

The paper proves that every time-homogeneous martingale diffusion on [0,1] with strictly positive smooth volatility and absorbing endpoints is the exact Bayesian posterior of an explicitly constructed binary diffusion experiment.

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2026-08-01 10:00 UTC pith:GARIHGZI

load-bearing objection A genuinely new embedding theorem: every autonomous win-martingale is the exact posterior of an explicit binary diffusion, with a sharp converse classification; the proofs are solid and the acknowledged one-dimensional limitation is the main caveat.

arxiv 2607.20373 v1 pith:GARIHGZI submitted 2026-07-22 math.PR math.STstat.TH

Embedding martingale diffusions as binary posteriors in sequential inference

Steven Campbell , Karl Kristian Engelund This is my paper

classification math.PR math.STstat.TH MSC 60G4460H1060J6062L1093E11

keywords posterior martingalesbinary sequential inferencestochastic filteringDoob h-transformLamperti transformRiccati equationmartingale diffusionsequential testing

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that every time-homogeneous martingale diffusion on [0,1] with strictly positive smooth volatility and absorbing endpoints—called an autonomous win-martingale—can be represented exactly as the Bayesian posterior belief of an observer who sees a scalar diffusion whose drift depends on a hidden binary state. The representation is constructive: place the martingale in Lamperti coordinates, condition its law on the two terminal outcomes, and the resulting conditional drifts define the experiment. Going the other way, the paper characterizes which binary diffusion experiments produce posteriors that are functions of the current time and observation alone: the difference between the two signal drifts must solve a Riccati equation, equivalently the likelihood ratio must be a Doob h-transform for the null dynamics. Autonomy of the posterior then forces either the spatially harmonic case or a constant drift difference, recovering the classical two-state sequential-testing filter. This gives concrete learning experiments for a broad family of bounded martingales, including absorbed Brownian motion, the Wright–Fisher diffusion, and the most-exciting-game martingale.

Core claim

On its own terms, the paper establishes an exact two-way correspondence between autonomous win-martingales (solutions dΠ_t = σ(Π_t) dB_t on [0,1], σ ∈ C^1((0,1);(0,∞)), absorbed at endpoints) and binary diffusion experiments dX_t = μ_θ(X_t) dt + dW_t. Theorem 3.2 gives the forward direction: with F(p)=∫_{1/2}^p du/σ(u) and its inverse G, setting Y_t=F(Π_t), the conditional laws of Y given Π_∞=1 and Π_∞=0 have explicit drifts ν_1 and ν_0, and the experiment with signal X and drifts ν_0,ν_1 has posterior P_t=G(X_t) with exactly the law of Π. The converse, Theorems 4.3 and 4.7, shows that the posterior is a smooth current-state function Γ(t,X_t) iff the drift gap δ=μ_1−μ_0 solves δ′+2μ_0δ+δ^2=−

What carries the argument

The central machinery is the Lamperti transform F(p)=∫_{1/2}^p du/σ(u), which maps an autonomous win-martingale to a unit-volatility diffusion Y=F(Π); conditioning on the terminal outcome then acts as a Doob h-transform with harmonic functions h_1(p)=p and h_0(p)=1−p, producing the two state-dependent signal drifts. In the converse, the Riccati identity δ′+2μ_0δ+δ^2=−κ serves as the compatibility condition, equivalent to the Girsanov density collapsing to an h-transform; it separates autonomy (κ=0) from forced constancy (κ≠0).

Load-bearing premise

The entire construction requires the volatility σ to be strictly positive on (0,1); if σ had an interior zero, the Lamperti transform would not be invertible and the explicit experiment drifts could not be defined.

What would settle it

Take a volatility σ in the class, compute δ(x)=σ(G(x))/(G(x)(1−G(x))) from the embedding drifts, and substitute it into the Riccati equation (4.1); a failure of the identity for some x would expose an inconsistency between the forward and converse directions. Alternatively, simulate the constructed experiment and compare the empirical law of the posterior path with the prescribed martingale law.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Every autonomous win-martingale, including absorbed Brownian motion, the Wright–Fisher diffusion, and the most-exciting-game martingale, comes with an explicit binary signal that a real observer could track.
The posterior-threshold formulas give closed-form error probabilities and expected decision times for any autonomous win-martingale decision rule.
Inverse design: choosing any null drift μ_0 and any constant C yields an alternative drift δ=s_0′/(C+s_0) whose experiment has autonomous posterior dynamics.
Outside the harmonic class κ=0, the only autonomous posterior is the classical two-state filter with volatility |Δ|p(1−p), so that filter is recovered rather than chosen.
Current-time-and-state sufficiency for the hidden state is equivalent to the Riccati condition, so any experiment violating it must have a posterior that depends on the path history.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The embedding suggests a converse design template for other scalar martingales: any bounded martingale with a positive volatility can be given a 'story' as a posterior, and one could ask which cost or optimality criteria select among the many possible experiments for the same posterior law.
Because the autonomy classification forces the classical filter for κ≠0, optimality criteria that reward belief movement will pick among the κ=0 family; the paper's examples hint that the most-exciting-game and a normal-mixture martingale are time-changed members of that family.
The threshold formulas imply that, for fixed thresholds, type I/II errors are volatility-independent, but optimizing thresholds reintroduces dependence on σ; a natural extension is to solve the optimal stopping problem for the autonomous posterior family.
The paper notes that higher-dimensional, multi-hypothesis extensions lack the scalar machinery; a concrete next step is to test whether any multidimensional bounded martingale posterior can be embedded in a similarly explicit multi-hypothesis diffusion experiment.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A genuinely new embedding theorem: every autonomous win-martingale is the exact posterior of an explicit binary diffusion, with a sharp converse classification; the proofs are solid and the acknowledged one-dimensional limitation is the main caveat.

read the letter

This paper deserves a serious referee. It proves a clean, constructive result: every time-homogeneous martingale diffusion on [0,1] with strictly positive C^1 volatility and absorbing endpoints is the exact Bayesian posterior of an explicit binary diffusion experiment. The converse is equally precise: under smoothness, the posterior is a function of current time and state exactly when the drift gap satisfies the Riccati equation (4.1), and then autonomy forces either the spatially harmonic (κ=0) case or constant drift, recovering the Shiryaev–Wonham posterior. The claims are stated carefully and the proofs hold up in the steps I checked: the Lamperti/Doob construction in Theorem 3.2, the support lemma used to pass from stochastic to pointwise identities, and the range-intersection argument in Theorem 4.7(ii) are all valid.

What is actually new is filling the gap left by Brigo–Vrins and Jeanblanc–Vrins: those papers represent bounded martingales but do not identify the sequential experiment that generates them. Here the experiment is explicit, with the conditional laws given by Doob h-transforms and the drifts in (3.5). The examples — absorbed Brownian motion, Wright–Fisher, the time-homogeneous Aldous martingale — are coherent and genuinely illustrative. Corollary 4.11 gives a nice inverse-design procedure. The paper does not overclaim: it repeatedly flags the one-dimensional binary setting as the boundary of the method.

Soft spots are minor. The regularity assumptions (σ>0 on the interior, δ∈C^1, Γ∈C^{1,2}) are explicit and standard; they are not hidden gaps. The posterior-threshold formulas in Section 5 are standard material, clearly presented but not new. There are a few typos, including the title's “Mar tingale,” and the self-citation to [11] is benign since it is not used as an input. The only real limitation is the scalar one-dimensional framework, which the authors acknowledge.

This is not a sweeping revolution, but it is a well-executed, self-contained paper that fills a real gap and gives useful tools. I would cite it and would welcome it at a journal specializing in stochastic filtering or probability. Send it to peer review.

Referee Report

0 major / 4 minor

Summary. This paper studies the relationship between time-homogeneous martingale diffusions on [0,1] and Bayesian posteriors in binary sequential experiments. It calls a process Π an autonomous win-martingale if it satisfies dΠ_t = σ(Π_t)dB_t with σ∈C^1((0,1);(0,∞)) and absorbing endpoints. The main positive result, Theorem 3.2, constructs, for every such Π, a binary diffusion experiment whose posterior is indistinguishable from Π: after the Lamperti transform F, the two state-dependent drifts are (3.5), and the posterior is P=G(X). The paper then characterizes the converse. Theorem 4.3 shows that the posterior depends on (t,X_t) only through a smooth function exactly when the drift gap δ satisfies the Riccati equation (4.1); Corollary 4.4 recasts this as a Doob h-transform of the null dynamics. Theorem 4.7 classifies autonomy: for κ=0 the posterior is a time-independent function of X_t, while for κ≠0 autonomy forces δ to be constant and recovers the classical volatility σ(p)=|Δ|p(1-p). Examples include absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale, with a table of explicit experiments. A short section gives posterior-threshold error probabilities and expected decision times.

Significance. If correct, the results provide a complete and elegant one-dimensional link between bounded martingale diffusions and Bayesian sequential experiments. The embedding theorem is fully constructive and parameter-free, and the converse gives a precise, checkable condition—Riccati equation (4.1), equivalently a space-time harmonic likelihood ratio—for current-state sufficiency. The classification of autonomy is a genuinely informative dichotomy: the κ=0 branch characterizes exactly the time-homogeneous posterior dynamics from the embedding, while the κ≠0 branch forces the classical Shiryaev–Wonham volatility. The paper is technically careful: proofs use standard tools (Engelbert–Schmidt, Girsanov, filtering theorems, support arguments) with proper localizations, and the examples are nontrivial and correctly computed. The one-dimensional binary setting is a limitation of scope, but it is explicitly acknowledged and does not undermine the paper's claims.

minor comments (4)

[§3, Eq. (3.5)] The drifts ν_i have denominators G(y) and 1−G(y), which vanish as y approaches the endpoints of the interval I. It would be helpful to state explicitly that continuity is claimed only on the open interval I, as required by the model in Section 2.1, and not up to the boundary; the current wording 'C(I;R)' is clear to specialists but could be misread.
[§4.1, proof of Theorem 4.7(i)] The use of Picard–Lindelöf for the ODE δ' = −2μ_0δ−δ^2 is slightly compressed. Since μ_0 is only assumed continuous, the standard local uniqueness theorem requires the right-hand side to be continuous in x and locally Lipschitz in δ; this holds here, and the argument then extends globally. A one-sentence explanation would make the step more transparent.
[§5, Table 1] For the first row, the sentence about the textbook presentation with drifts (0,1) changing κ from 0 to −1 is terse. Spelling out the shift X↦X+t/2 and how μ_0 changes from −1/2 to 0 would make the remark self-contained and avoid confusion about the sign convention in (4.1).
[Remark 4.5] The remark is correct but could be more reader-friendly: after defining S_t, a short computation showing that the posterior is C-S for S, e.g. logit P_t = logit p + Δ(S_t−S_0)−Δ^2 t/2, would make the point immediately verifiable.

Circularity Check

0 steps flagged

No significant circularity; the embedding is constructive and the converse is self-contained, with only non-load-bearing self-citations.

full rationale

The derivation chain is self-contained. Theorem 3.2 is a constructive embedding: from an autonomous win-martingale Pi, the paper defines Q_i by conditioning on Pi_infinity (eqs. 3.3-3.4), computes the Girsanov drifts nu_i of Y=F(Pi) under Q_i (eq. 3.5), and then takes these nu_i as the signal drifts in the experiment. The posterior identity P=G(X) is then proved via Bayes' formula and uniqueness in law, not assumed. The converse (Theorems 4.3 and 4.7) compares Ito expansions with the filtering equation (2.6) and uses the support lemma to pass to pointwise PDEs; no parameter is fitted and no data subset is predicted. The only reliance on the authors' own work is a non-load-bearing contextual citation [11] (Campbell-Zhang) in the introduction and a mention of forthcoming work after Corollary 5.1; neither is used as an input to any theorem. The strict positivity of sigma, which the skeptic flags, is an explicit standing assumption in Definition 2.1 rather than a hidden circular premise. All load-bearing citations are to external texts (Karatzas-Shreve, Liptser-Shiryaev, Bain-Crisan, Rogers-Williams). Hence no circular step is present.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The paper's central claims rest on standard probabilistic machinery (stochastic calculus, filtering theory, Doob transforms) plus the explicit domain assumptions of the model: binary state, scalar non-degenerate diffusion observation with smooth drifts, and win-martingales with strictly positive smooth volatility. No free parameters are fitted, and no new entities are postulated.

axioms (7)

standard math Engelbert–Schmidt and Yamada–Watanabe existence/uniqueness for SDEs (Lemmas 2.1, 2.2)
Justifies pathwise unique strong solutions for the win-martingale and the signal; standard probabilistic background.
standard math Girsanov theorem and Bayes/Kallianpur–Striebel formula for the filter (Proposition 2.3)
Used to derive the likelihood-ratio process, the innovation Brownian motion, and the filtering equation.
standard math Doob h-transform: positive space-time harmonic functions generate local changes of measure (Section 2.2)
Used to rephrase the Riccati condition in Corollary 4.4 and in the examples.
domain assumption Observation model (2.2): binary state, scalar diffusion with non-degenerate Brownian noise and drifts μ0, μ1 ∈ C(J;R); Section 4 assumes δ∈C^1(J)
This is the epistemic setup the whole paper operates in; it excludes degenerate/noise-free observations and non-smooth drift gaps.
domain assumption Win-martingale definition (2.1): time-homogeneous volatility σ∈C^1((0,1);(0,∞)), absorbing endpoints
Definition of autonomous win-martingale; strict positivity is needed for the Lamperti transform.
standard math Lévy's upward theorem and martingale convergence for P (Remarks 2.4, 4.8)
Used to justify posterior convergence to a terminal value in {0,1} and the absorption arguments.
standard math Picard–Lindelöf theorem for ODE uniqueness (Theorem 4.7)
Used to show δ either vanishes identically or has no zeros in the κ=0 case.

pith-pipeline@v1.3.0-alltime-deepseek · 16624 in / 20431 out tokens · 160362 ms · 2026-08-01T10:00:28.676217+00:00 · methodology

0 comments

read the original abstract

Posterior probabilities are bounded martingales, but this fact alone does not identify the experiment that generates them. We show that every time-homogeneous martingale diffusion on $[0,1]$ with strictly positive $C^1$ volatility on the interior and absorbing endpoints, which we call an autonomous win-martingale, can be realized as the Bayesian posterior of an explicit binary sequential experiment observed through a scalar diffusion. Conversely, in a general binary diffusion experiment, the current time and observation are jointly sufficient for the hidden state precisely when the difference between the signal drifts under the two hypotheses satisfies a Riccati equation. Equivalently, this holds when the likelihood ratio is generated by a Doob $h$-transform associated with a positive space-time harmonic function for the null dynamics. Within this class, when that function is spatially harmonic, the posterior is an autonomous martingale. Outside this harmonic case, autonomy forces the drift difference to be constant, recovering the classical two-state Shiryaev-Wonham filter from binary testing. These results give concrete sequential inference interpretations to a broad class of bounded martingale diffusions, including absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale of the "most exciting game."

discussion (0)

Reference graph

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