Executable Robust Arbitrage Certificates on Prediction Sets: Attainment and Faithful Discretization

Keywords. Robust superhedging; prediction sets; bid–ask spreads; arbitrage certificates; optional decomposition; martingale cubature; discretization of martingale classes.
JEL classification. G13, G12, C61.
MSC 2020. 91G20, 60G42, 90C05, 49N15, 65D32.

1Introduction

When market quotes cannot be matched by any model in a chosen library, the quotes have rejected the library; they have not yet produced an arbitrage. This distinction is the starting point of robust pricing on prediction sets, a formulation that descends from statistical prediction regions (Mykland, 2000, 2003), through the beliefs of Hou and Obłój (2018), to pathwise duality on prediction sets (Bartl et al., 2020), and, alongside it, the model-free fundamental theorem of Acciaio et al. (2016) for markets with statically traded options. One declares the set of price paths one takes seriously and asks what the quotes imply for every martingale law on it. Turning a rejection into an arbitrage then needs two further steps: the inconsistency must hold against every martingale law on the declared set, not only against the preferred models, and it must be converted into an explicit trade, a finite static portfolio and a predictable strategy whose realized profit is positive on every declared path.

This paper sets out to supply both steps in the following setting. Fix finitely many trading dates, a prediction set $\Omega$, a finite vector $X$ of executable common-numeraire prices, and statically quoted claims with bid and ask sides; the same price vector is written $S$ in the Bellman formulation below. The object sought is

where $a^\pm$ are finite static positions bought at ask and sold at bid, $\pi$ is their actual acquisition cost, and $H$ is a predictable strategy in the traded menu. Inequality (1) is a booked trade, not a separating functional or a positive expected-value gap.

Two settings are treated. On a finite prediction tree the problem is a linear programme, and the certificate comes straight out of a Farkas alternative. On a compact continuum of states, duality still gives the right value, but a value is not a hedge: nothing yet guarantees that the optimal hedge exists as a bounded strategy. What guarantees it is a uniform relative-interior radius: for every date $n$ there is $\kappa _n>0$ such that

where $D_n$ is the active increment space. Under this condition an optimal one-step hedge can be chosen with slope at most

This bound is the engine of both the attainment theorem and the discretization analysis. The paper’s central computational finding is that martingale cubature can reproduce every robust value while failing to preserve (2): prices converge, yet the optimal tree hedges blow up. A finite nodewise augmentation restores a uniform radius, and with an off-grid shadow estimate a positive buffered margin then yields an executable continuum certificate.

The main theorem, heuristically. Take a compact prediction set with continuously varying successor sets, a finite traded menu, and, as dual class, the full set of martingale laws on the declared set. Under the radius condition (2) — at every history the market can move at least $\kappa _n$ in every active direction — robust pricing on the declared set is exact and executable. Superhedging and subhedging prices are attained by bounded hedges, not only by laws; a quote book inconsistent with every martingale law, finite or not, yields a finite trade with negative cost and positive payoff on every declared path; and computation is faithful once the cubature trees are augmented to keep the radius — values and margins converge, tree hedges can be kept bounded, and, under an off-grid shadow estimate, a tree certificate whose margin clears an explicit buffer is a continuum certificate. One bound carries all of it: the radius limits every one-step hedge to (3). The precise statements are Corollary 4.3 and Theorem 7.10.

1.1The radius, attainment, and its precise scope

The mechanism is a one-step fact (Lemma 6.1). On a compact successor set, the ball (2) makes the one-step hedging problem dual to a martingale-kernel problem with both sides attained, and it caps the optimal slope by (3). Everything else follows by iterating this through the Bellman recursion: the value functions are continuous, the optimizers are bounded and Borel, super- and subhedging duality holds pointwise with all extrema attained, the price process has a pathwise optional decomposition, and continuous $\varepsilon$-hedges exist (Theorem 6.6). Because the ball lets every successor carry positive mass under some martingale kernel, every declared path is charged by some law in $\mathcal M_\Omega$ (Lemma 5.3), so pointwise statements need no efficient-set qualification; the gain cone of continuous payoffs is closed, with $\mathcal M_\Omega$ as its exact polar (Corollary 6.7); and compactness reduces an infeasible quote book of any size to a finite infeasible sub-book, hence to a finite certificate (Theorem 6.8). The decomposition is the compact-state descendant of the classical one (Kramkov, 1996; Föllmer and Kramkov, 1997); for the dominated discrete-time theory see Föllmer and Schied (2025).

How much is the radius really buying? Two things, best kept apart. It is sufficient and cannot be dropped: with zero on the boundary of the increment hull, the one-period payoff $f(y)=-\sqrt y$ has a finite value but no finite attaining slope (Section 8.2). It is necessary only up to balance: for families of one-step geometries in which opposite moves come in comparable size, uniformly bounded least-norm hedges force a common positive radius (Theorem 6.4), while without such balance an asymmetric two-point family has collapsing radii and bounded hedges, and the two-sided quantity of Proposition 6.2 is what actually controls them. Inside a single compact market with continuous successor sets none of this subtlety arises, because pointwise interiority already makes the radius continuous and hence uniformly positive (Remark 6.5); uniformity is a genuine extra requirement only across a family of markets, which is exactly the situation of a discretization sequence.

This sharpens the existence theory rather than replacing it. Robust duality with measurable strategies holds under far weaker analytic hypotheses (Bouchard and Nutz, 2015; Nutz, 2014; Burzoni et al., 2017, 2019; Obłój and Wiesel, 2021; Cheridito et al., 2017); dynamic characterizations of the robust price are given by Carassus et al. (2019); that dual attainment can fail, or needs relaxation, is shown in (Beiglböck et al., 2013, 2017); full support and efficient sets are studied by Burzoni et al. (2016); and Riedel (2015) attains both sides in one period on a compact space. What the radius adds is a dynamic completion that is bounded, uniform over histories, and valid on every path, and that can be tracked through finite-tree computation.

1.2Faithful discretization

In practice a robust certificate is computed on a finite tree, so the question is what survives the passage to the tree. Raw-history martingale cubature (Lemma 7.1), a Tchakaloff-type construction (Tchakaloff, 1957; Bayer and Teichmann, 2006), produces nested finite subtrees $\Omega _m\subseteq \Omega$ whose full martingale classes $\mathcal M_m$ converge to $\mathcal M_\Omega$ in the weak Hausdorff metric (Proposition 7.2); every robust value therefore converges, in line with the value-level discretization literature (Guo and Obłój, 2019; Dolinsky and Soner, 2013, 2014a).

Values, however, are not hedges. Proposition 7.3 gives a two-step market with continuum radius one and cubature trees that recover the whole martingale class while some of their local radii shrink like $2^{-2j}$; for one fixed continuous payoff the tree-optimal slopes then diverge. Law and value convergence thus guarantee neither bounded tree hedges nor validity off the grid: the lattice prices correctly and hedges wrongly.

The repair is local and geometric. At each node of active dimension $d_n\ge 1$, adjoining at most $2d_n(d_n+1)$ allowed successors puts a cross-polytope, hence a ball of radius $\kappa _n/\sqrt {d_n}$, inside the increment hull (Lemma 7.4 and Proposition 7.5). On the augmented trees, values converge for every continuous payoff; for each fixed noncrossed finite quote list the separation margins converge and bounded Bellman-optimal holdings can be selected (Theorem 7.6 and Corollary 7.9); the value and margin errors are $O(1/m)$ on a finite span of the metrizing test functions, with approximation bounds for general continuous payoffs (Proposition 7.7); and under the prefix-compatible shadow estimates of Proposition 7.8 a finite-tree certificate lifts to every continuum path with the explicit buffer $\gamma _m-\sqrt q\,\xi _m-2NL\eta _m$, so it is a continuum certificate whenever that buffer is positive. Convergence of the hedge ratios themselves is not claimed; what is claimed is that the certificate — its margin, its cost, and its validity on every declared path — survives.

1.3The executable certificate

On a finite tree no radius condition is needed: with finitely many successors the local envelope is polyhedral, so whenever the barycentric constraints are feasible a finite supporting slope exists. The Farkas alternative (Theorem 3.4) then says that exactly one of two things exists, a calibrated martingale law in the bid–ask polytope or an executable certificate. Its nonnegative dual variables are the long-at-ask and short-at-bid quantities, and its free node multipliers are a predictable strategy. The certificate can be normalized, the compact margin programme (25) attains a numerical inconsistency margin, and when the martingale polytope is nonempty finite-tree subhedging is attained on both sides (Proposition 3.5).

Executability also requires trading the contractual payoff rather than an equal-expectation mark. The finite-leg carry identity (Lemma 4.1) and its invariance (Proposition 4.2) show when fixing-date marks and payment ledgers have identical feasibility and static cost, with an explicit adjustment of the predictable strategy; likewise, a separating potential must be reduced to finitely many actually quoted legs, whether masses, forwards, calls, or puts. These are the ingredients that turn a separating value into the trade (1).

The transaction-cost and robust-friction literature establishes consistent price systems and fundamental alternatives in quasi-sure, pointwise, and model-free forms (Bouchard and Nutz, 2016; Bayraktar and Zhang, 2016; Bouchard et al., 2019; Bayraktar and Burzoni, 2020; Burzoni, 2016; Dolinsky and Soner, 2014b), building on the classical theory (Jouini and Kallal, 1995; Schachermayer, 2004; Kabanov and Safarian, 2009; Pennanen, 2011); finite-tree pricing and hedging algorithms are studied by Roux and Zastawniak (2009); static quote consistency and numerical repair are treated in (Davis and Hobson, 2007; Cousot, 2007; Gerhold and Gülüm, 2020; Cohen et al., 2020). What is new here is the finite, sign-structured, pathwise certificate with its attained margin and contractual carry, and its survival under discretization.

1.4Contributions, scope, and organization

The tools used below are standard: finite-dimensional linear duality, the maximum theorem, measurable selection, Carathéodory and Tchakaloff cubature, and Hausdorff convergence. The contributions are three.

1.

The uniform interior radius is identified as the mechanism behind bounded, attained, pointwise hedging, and its scope is made precise: across families of one-step geometries a common balance constant makes it equivalent to uniformly bounded hedges, and without balance a two-sided slope quantity takes over (Section 6).

2.

Martingale cubature is shown to recover the full martingale class while letting local radii collapse; a finite nodewise augmentation restores the radius and yields uniform hedge bounds together with quantitative value, margin, and conditional certificate-lifting guarantees (Section 7).

3.

Finite-tree Farkas duality, compact-state robust subhedging, finite-sub-book extraction, and exact fixing-to-payment carry are assembled into a pipeline from quote infeasibility to an executable certificate (Sections 3 to 6).

The regime is compact and finite-calendar, with a finite dynamically traded menu; unbounded prices need a separate localization or truncation argument. Rectangularity is assumed rather than inferred, since nonrectangular restrictions generate untraded dual multipliers, the market analogue of the robust-control distinction between a law class and its rectangular hull (Epstein and Schneider, 2003; Iyengar, 2005; Nilim and El Ghaoui, 2005). Continuous-time and growing-menu extensions need additional closedness and stochastic-integral stability; single-reference-law stability results include (Kardaras, 2024; De Donno and Pratelli, 2005; Alberti and Karbach, 2026), and the nondominated growing-menu problem is open. The results are stated abstractly, with interest-rate contracts entering only as worked instances of the carry identity, and the examples in Section 8 delimit the present claims.

Section 2 fixes the market and certificate conventions. Sections 3 and 4 give the finite-tree alternative and carry identity. Sections 5 and 6 establish the compact-state martingale class, attained duality, optional decomposition, and finite executable certificates. Section 7 proves cubature recovery, radius collapse, augmentation, and lifting, and includes the finite-level separator-to-trade principle. Section 8 records the sharp boundaries and open questions, and Section 9 closes the argument. Contract-level carry examples are collected in appendix A.

2Market, trading, and certificate conventions

This section fixes, once and exactly, the objects every later statement is made of: the finite master calendar and the canonical gains of a finite executable menu; the statically quoted claims with their bid–ask accounting and the definition of an executable robust arbitrage certificate; and the carry interface through which fixing-date expressions are attached to payment-date ledgers. Nothing here is deep; the point is that every later claim of executability is checked against these conventions and no others.

2.1Finite master calendar and canonical gains

Fix deterministic dates

Every fixing publication, exercise, contractual carry, delivery, and payment date used by the finite certificate belongs to this calendar. The state at $t_i$ is a history $h_i=(x_0,\ldots ,x_i,a_0,\ldots ,a_i)$, where $a$ is the revealed shared fixing/settlement tape and $x_i\in \R ^d$ is a finite vector of common-terminal-numeraire primitive prices. It contains every state-price bond, gross-coupon price, state-price annuity, or linear switch claim that the certificate trades. Redundant nonlinear rates are derived from these primitive coordinates and are not traded as separate assets.

At a publication time, the history used for a trade is the marked right-hand (post-publication) state. Equivalently, a market with both a pre-publication and a post-publication decision splits them into two ordered nodes. Thus every fixing-dependent coefficient used in a holding selected at $t_i$ is already $\mathcal F_{t_i}$-measurable; the strategy never sees an unpublished fixing.

For a predictable strategy $H=(H_i)_{i=0}^{N-1}$, with $H_i(h_i)\in \R ^d$, define the discounted gain

The reference asset corresponding to one unit of the selected terminal numeraire is the constant one after normalization. It need not be the money-market account. Thus a cash coordinate $z$ and the gain (4) are already expressed in common terminal-numeraire units. On a finite tree every predictable strategy is admissible. On a compact continuum-state prediction set we shall use the bounded Borel class, and its continuous subclass, declared in Section 5; no unstated doubling strategy is allowed.

2.2Static claims and bid–ask accounting

Let $f^1,\ldots ,f^q$ be the terminal-numeraire payoffs of the finitely many vanilla or linear-switch claims actually used in a certificate. Their market bid and ask prices are finite:

No ordering is imposed in the finite-tree fundamental alternative: if $\ell _k>u_k$, the quote is crossed and the alternative returns the direct buy-at-ask/sell-at-bid round trip. Results that use a nonempty quote box, including the compact-state separation theorem, explicitly impose $\ell _k\le u_k$. These are normalized prices. If the currency quotes are $\ell _k^{\rm ccy},u_k^{\rm ccy}$ and the common initial state-price scale is $\zeta _0$, then $\ell _k=\zeta _0\ell _k^{\rm ccy}$ and $u_k=\zeta _0u_k^{\rm ccy}$. The long and short quantities are separate vectors $a^+,a^-\in \R _+^q$. The initial acquisition cost and terminal payoff are

This convention is load-bearing: buying costs the ask and shorting receives the bid. Replacing $a^+,a^-$ by one signed coefficient and pricing it at a single mid would prove a different and economically invalid theorem.

If $Q\in \cM (\ell ,u)$, then for every bounded predictable $H$,

Indeed $E_Q(H\mathbin {\cdot }X)_N=0$, while each long expectation is at most its ask and each short expectation is at least its bid. Consequently, the following pathwise object rules out every calibrated law.

2.3The carry interface

Expectation-level separators of the kind discussed in Section 7.6 produce finite piecewise-linear fixing-date expressions of the form

where every $\phi _r$ is a finite linear combination of constants, the identity, and call payoffs $(y-K)^+$. The corresponding market cost is the same linear combination of observed caplet/swaption prices, using ask for positive holdings and bid for negative holdings.

The caplet, physical swaption, declared linear cross-CSA switch, and declared cash-settlement instances are proved in Appendix A. The abstract definition is retained because no pathwise trading theorem can manufacture a missing settlement asset or turn an untraded continuum strike potential into a finite portfolio.

3Exact finite-tree trading certificates

This section turns finite quote infeasibility into an executable trading statement. There is no limiting argument here: the prediction tree, the traded gain coordinates, and the quote panel are all finite. The result is therefore a theorem of linear alternatives. The economically important point is that its dual variables have exactly the signs of a bid–ask portfolio and exactly the measurability of a predictable strategy.

3.1The canonical finite prediction tree

Fix a master calendar

containing every determination, exercise, carry, and payment date in the finite panel under consideration. Let $\mathbb T=\bigsqcup _{i=0}^N\mathbb T_i$ be a finite rooted tree. Its root is $o\in \mathbb T_0$, every $v\in \mathbb T_i$, $i<N$, has a nonempty finite child set $\operatorname {ch}(v)\subset \mathbb T_{i+1}$, and every nonroot node has one predecessor. The leaves

are the declared prediction paths. If $\omega \in \Omega _{\mathbb T}$, write $v_i(\omega )$ for its unique level-$i$ ancestor.

Let

be a finite panel of executable carried gain coordinates, expressed in one common reference state-price unit. A matured coordinate is stopped, so keeping it in the vector only adds zero future increments. The constant unit account is denoted by $S^0\equiv 1$ and is handled as the cash coordinate below. Merely being a canonical state variable is not enough for inclusion in $S$: each component of $S$ must be the value process of a traded linear claim, including its contractual collateral gain stream.

A predictable simple strategy is a family

Its terminal gain on a leaf is

All such strategies are bounded because the node set is finite. In particular, there is no hidden admissibility or stochastic-integrability issue in the finite-tree setting.

For $v\in \mathbb T_i$, $i<N$, and $1\le k\le d$, define the row payoff

After choosing an ordering of nodes, assets, and leaves, let $D$ be the matrix with entries $D_{(v,k),\omega }=d_{v,k}(\omega )$. If $h$ is the vector formed by the node holdings $H_i^k(v)$, then

A law is a vector $q=(q_\omega )_{\omega \in \Omega _{\mathbb T}}$. Put

For a node $v$, let $q(v)$ be the mass of its descendant leaves. If $q(v)>0$, the row equations belonging to $v$ say precisely

If $q(v)=0$, they reduce to $0=0$. Thus (15) is equivalent to martingality of the traded panel in its raw tree filtration, without imposing an artificial conditional law at a null node.

A failed barycentric condition at a nonroot node is only a conditional arbitrage unless reaching that node is itself forced or hedgeable. A global martingale law may assign that node zero mass. The compact-state theory of Section 5 therefore either prunes such nodes or assumes local barycentric feasibility at every history in its rectangular prediction set.

Let $g^1,\ldots ,g^J:\Omega _{\mathbb T}\to \mathbb R$ be the actual common-unit terminal ledger payoffs of the quoted claims. Write

for their bid and ask vectors, respectively. The Farkas theorem does not require them to be noncrossed: if some $\underline p_j>\overline p_j$, it returns the immediate buy-at-ask, sell-at-bid round trip. The calibrated-law polytope is

All inequalities between vectors are componentwise.

For long and short quantities $\vartheta ^+,\vartheta ^-\in \mathbb R_+^J$, the executable static payoff and its acquisition cost are

A positive coefficient is bought at the ask; a negative coefficient is shorted for the bid. This sign convention will be recovered, rather than inserted after the duality calculation.

This is equivalent to the profit-after-cost convention in (9). Indeed, if $\mathsf C_0=c+\pi ^{\mathrm {ba}}<0$, then

Conversely, if the left side is at least $\gamma >0$, choosing $c=-\pi ^{\mathrm {ba}}-\gamma /2$ gives initial cost $-\gamma /2$ and terminal payoff at least $\gamma /2$.

The certificate already is an arbitrage under the negative-cost convention. Under a zero-initial-capital convention, buy $-\mathsf C_0>0$ units of the constant account with the initial receipt. The resulting terminal payoff is $\mathsf V_N-\mathsf C_0>0$ on every prediction path.

3.2The exact LP/Farkas alternative

With the tree, the executable panel, and the certificate class fixed, the fundamental alternative is a statement about one finite linear system. We first record the mixed Farkas lemma in exactly the form needed — equality rows for the martingale constraints, inequality rows for the bid–ask sides — and then read its dual variables as a trade.

If the quotes are not crossed, simultaneous long and short positions in one claim are unnecessary. Indeed, subtracting $m_j=\min (\vartheta _j^+,\vartheta _j^-)$ from both sides leaves the payoff unchanged and changes the cost by $-m_j(\overline p_j-\underline p_j)\le 0$. A certificate can therefore be chosen with $\vartheta _j^+\vartheta _j^-=0$ for every $j$.

3.3Finite robust subhedging and the constructive separator route

The preceding theorem also decomposes into the two operations needed to turn an expectation-level separation into a trade: static separation followed by dynamic subhedging.

At a fixed finite node, no relative-interior hypothesis is needed for this duality. Indeed, if the successor values are $s_1,\ldots ,s_m$, the local lower-envelope problem

has the dual

Whenever the barycentric constraint is feasible, both finite LPs attain. Equivalently, the lower convex envelope is polyhedral and has at least one finite supporting slope at $x$, even when $x$ lies on the relative boundary of the successor hull. The continuum difficulty is not existence of a slope at one node; it is obtaining a measurable, uniformly admissible selection as the history and successor set vary. That is why the compact-state theory needs additional continuity and slope-control hypotheses.

Assume now that $\mathcal Q_{\mathbb T}\ne \varnothing$, every quote is noncrossed, and put

Both are compact convex sets, and $\mathcal Q_{\mathbb T}^{\mathrm {ba}}\ne \varnothing$ exactly when $\mathcal K_{\mathbb T}\cap \mathcal B\ne \varnothing$. If they are disjoint, strict finite-dimensional separation, with its sign reversed if necessary, gives a vector $\vartheta \in \mathbb R^J$ such that

The displayed support-function identity fixes the orientation:

Thus the separated claim is acquired long at asks and short at bids. Choose $z$ strictly between the two sides. By Proposition 3.5, an attained predictable $H$ satisfies

Buy the static portfolio and short this subhedge. Its initial cost is

while its terminal payoff is

This is the conversion of a separating value into a trade described in the introduction. If $\mathcal Q_{\mathbb T}=\varnothing$, the direct Farkas theorem remains valid with $J=0$ and gives a purely cash/dynamic global certificate; the robust lower value in (27) should not be used with an empty model set.

The theorem uses the full martingale class on the declared tree. A restriction to a preferred subclass introduces dual multipliers which need not be traded claims, and almost-sure nonnegativity under one law does not replace nonnegativity on every declared leaf. Likewise, a state coordinate cannot be placed in the gain vector merely because candidate laws make it a martingale: it must be executable. The explicit failures are collected in Section 8.

4Exact carry from fixing marks to payment ledgers

The finite potentials produced by an expectation-level separation are written naturally as fixing- or exercise-date state-price marks. A traded option, however, delivers its contractual payment ledger. Equality of their expectations under the model class would not by itself make one a hedge for the other. This section supplies the required pathwise self-financing identity.

4.1Normalization and a finite-leg carry lemma

Fix a common initial state-price scale $\zeta _0>0$. For each declared linear collateral convention $c$ and payment date $\tau =t_{n(\tau )}$ in the master calendar, let $\mathsf Z_t^c(\tau )$ be the common-terminal-numeraire price at $t\le \tau$ of one unit of payout currency at $\tau$, and put $\zeta _t^c:=\mathsf Z_t^c(t)$. We assume $\zeta _0^c=\zeta _0$ and, whenever this payment claim is used in a carry, the stopped process

is a component of the executable panel $S$. These are pathwise state and tradability requirements; they do not select a reference law.

A currency bid or ask $p^{\mathrm {ccy}}$ is placed in the finite LP as

Thus the model value of a common-unit payoff $g$ is $E_Qg$, whereas its time-zero currency value is $\zeta _0^{-1}E_Qg$. For a payment $C$ at date $p$ under linear collateral convention $c$, the common-unit ledger payoff is

This is one common state-price convention. For $c\ne 0$, it retains the relative collateral kernel and must not be replaced by the reference-collateral deflator. This normalization assumes the one-payout- currency setup. A foreign-currency payment is covered only after its executable FX conversion claim has been added to the traded state.

The following elementary telescoping identity is the load-bearing carry result.

The lemma allows different payment dates and signed coefficients. A contract with optional collateral, nonlinear funding, or an untraded conversion factor is outside it unless that extra state is itself supplied with an executable linear carry decomposition.

Representative caplet, cash-settled swaption, physical-settlement, and cross-CSA instances are collected in appendix A; they are applications of the pathwise identity, not additional hypotheses for the general theory.

4.2Finite-strike executability

A finite piecewise-linear smile potential is executable through the same carry, but only once it has been reduced to actually quoted legs. Let $\phi :\mathbb R\to \mathbb R$ be continuous and piecewise affine, and let $K_1<\cdots <K_m$, $m\ge 0$, contain all slope changes. If $m\ge 1$, choose an anchor $K_0<K_1$ in the first affine interval, let $s_0$ be the slope to the left of $K_1$, and let $s_r$ be the slope immediately to the right of $K_r$. If $m=0$, choose any $K_0$ and let $s_0$ be the global affine slope. Then, with the sum interpreted as empty when $m=0$,

Therefore

The mass claim $W$, weighted-forward claim $WY$, and finitely many calls on the right must each be available with their own bid and ask and their declared carry. A quoted put may equivalently be retained as its own leg or reduced by put–call parity using these same mass and forward claims. For a variable-mass smile, omitting $W$ would make the constant part of the potential nontradeable. If $\phi$ is constant in both tails, then $s_0=0$ and $\sum _r(s_r-s_{r-1})=0$, recovering the bounded-potential special case. The acquisition cost is the sum of the leg-level bid–ask costs; it is not a fictitious linear price assigned directly to $\phi$.

Equation (34) is first an identity in the units of the potential’s affine primitive. Before execution, every term must be converted to the corresponding actual quote leg with the same deterministic notional, accrual, initial state-price or annuity mass, and $\zeta _0$ scale. For example, if $W=\overline A_e/\overline A_0$, then the common-unit mark of a unit physical option is $\overline A_0W(S_e-K)^+$; the traded quantity and its bid–ask cost must carry that factor. A fixed deterministic scale may equivalently be absorbed into the portfolio coefficient. A model-dependent mass may not: in that case the potential and the certificate use the unnormalized finite-measure weight and require the mass claim itself to be traded or finitely synthesized.

An equality of expectations without the pathwise identity is insufficient. On a two-leaf one-step space with no risky traded asset, let $f\equiv 0$ and $g(-)=-1,g(+)=1$. They have equal expectation under the equal-weight law, but no self-financing strategy converts $f$ into $g$. Replacing an actual ledger by a fixing mark solely on the strength of this expectation identity would therefore manufacture a false trading certificate.

The hypotheses that must survive the compact-state lift are now visible. The continuum prediction set must be rectangular, or every nonrectangular restriction must have a traded dual; local martingale successor kernels must be nonempty; the value recursion and its argmin correspondence must be measurable; predictable slopes must obey an admissibility bound; actual contract ledgers must retain their pathwise finite-leg carries; and the terminal inequality must hold on the declared prediction set rather than only under one law. Compactness and continuity replace finite polyhedrality in Section 5, while a uniform barycentric interior condition replaces automatic finite-dimensional slope control.

5The compact continuum-state prediction market

An expectation-level separation places a target quote field outside the price range of a compact set of pricing laws. Such a separation is not, by itself, a trading theorem. This section specifies the additional market object whose polar consists exactly of tradeable dynamic gains. The state and successor sets may be continua, but the trading calendar and the dynamically traded menu are finite. Continuous-time cylindrical curve trading is deliberately not treated here; under one law it is available in the literature (Kardaras, 2024; De Donno and Pratelli, 2005), and under model uncertainty it remains open.

5.1Histories, successors, and common-numeraire prices

Fix dates

Let $\mathsf H_0=\{h_0\}$. Recursively, let $E_{n+1}$ be a compact metric space, let

be nonempty, compact-valued, and continuous (both upper and lower hemicontinuous), and put

Thus every $\mathsf H_n$ is compact metric. The prediction set is $\Omega =\mathsf H_N$, equipped with its raw prefix filtration $\mathbb F=(\mathcal F_n)_{n=0}^N$. A history includes both the primitive price field and the progressively revealed fixing tape. In particular, $\Gamma _n(h)$ is allowed to depend only on the tape already present in $h$; it cannot inspect a future fixing.

Choose one strictly positive reference numeraire and express every traded asset in its units. Let

be the resulting continuous vector of cum-dividend prices. It is a finite executable menu of bonds, coupon accounts, collateral-conversion accounts, or other primitive traded assets. For $(h,e)\in \mathsf H_{n+1}$, write

All deterministic affine redundancies are removed. More precisely, for each $n$ fix a linear subspace $D_n\subseteq \R ^d$ such that every increment lies in $D_n$. Positions may be projected onto $D_n$, because the orthogonal component has identically zero gain.

This is the same normalized primitive price vector denoted by $X$ in Section 2; the letter $S$ is used in the Bellman proof to emphasize its role as the dynamically traded martingale.

The use of a common numeraire is load-bearing. Separate martingale measures for separate collateral conventions cannot be combined in one self-financing gain. In a term-structure setting, a known coupon is carried through its payment date by

and is then stopped. The corresponding stopped, common-numeraire price must be a component of $S$, or its carry must be an explicitly displayed predictable gain in components of $S$. Counting both the stopped cum-dividend price and a separate dividend would double count carry.

The relative ball is taken in the active space $D_n$, not in $\R ^d$. The assumption is stronger than mere one-step absence of arbitrage. Its uniform radius supplies attainment and a uniform bound on hedge ratios. The boundary example in Section 8 shows why it cannot simply be deleted from the exact-attainment statement.

5.2The full rectangular martingale class

We use three standard compact-state facts in their following precise forms. Berge’s theorem says that the maximum of a continuous function over a nonempty compact-valued continuous correspondence is continuous. The Arsenin–Kunugui Borel uniformization theorem says that a Borel subset of a standard-Borel–Polish product whose nonempty sections are sigma-compact admits a Borel selector; the compact sections below are sigma-compact. Finally, the finite-horizon Ionescu–Tulcea theorem says that an initial law and a finite sequence of Borel probability kernels determine a unique law on the product history space. These are the only measurable-selection and kernel-concatenation results used below; see Kurka (2016, Theorem 2.1) for Arsenin–Kunugui and Kallenberg (2021) for finite kernel concatenation. We use Aliprantis and Border (2006) separately for the compact-correspondence facts.

For $h\in \mathsf H_n$, define the local martingale-kernel set

By (38), this set is nonempty. It is compact, and its graph is closed in $\mathsf H_n\times \mathcal P(E_{n+1})$. Indeed, support is preserved under a joint limit by upper hemicontinuity of $\Gamma _n$, and the barycenter condition passes to the limit by continuity of $\Delta S_n$. For the second assertion, first extend $\Delta S_n$ continuously from the closed graph $\mathsf H_{n+1}\subseteq \mathsf H_n\times E_{n+1}$ to the ambient product by the Tietze theorem, exactly as in the proof of Theorem 6.6; the extension is defined on a common domain, is uniformly continuous on the compact product, and therefore lets one compare $\int \Delta S_n(h^j,\cdot )\dd p^j$ with $\int \Delta S_n(h,\cdot )\dd p$. The value of the integral is unaffected, because each $p^j$ is carried by $\Gamma _n(h^j)$, where the extension agrees with $\Delta S_n$.

Let

Equivalently, $Q\in \mathcal M_\Omega$ if and only if

Continuous tests suffice: bounded Borel tests follow by the monotone-class theorem, coordinate by coordinate.

General nondominated robust duality is developed, under substantially more flexible analytic hypotheses, by Bouchard and Nutz (2015). The efficient scenario qualification in pointwise arbitrage theory is emphasized by Burzoni et al. (2019). The compact finite-calendar theorem below uses the stronger uniform relative-interior hypothesis because it yields bounded attained slopes, continuous epsilon-selectors, and, by Lemma 5.3, a certificate on every declared path rather than only on a martingale-efficient subset.

The adjective full in (40) is essential. The class contains every martingale law supported by the declared prediction set, not merely a preferred model library, its convex hull, or a statistically selected subset. In particular, a compact model-class envelope $\mathcal M$ may replace the dual class in the dynamic theorem only under the saturation identity

For a particular continuous separator, it is instead enough to establish its strict lower bound over $\mathcal M_\Omega$, either by establishing it there directly or by a separate valid transfer from a smaller class (proved equality of the relevant lower values is one such transfer). Expectation separation over a strict subset alone need not have a tradeable pathwise interpretation.

5.3Admissible gains and carried static claims

A predictable strategy is a sequence $H=(H_n)_{n<N}$, where $H_n:\mathsf H_n\to D_n$ is Borel. Its normalized terminal gain is

All strategies constructed below are bounded. Since $S$ is bounded on the compact histories, their gains are integrable and

Write $\mathcal H^b$ for the bounded Borel-predictable class and $\mathcal H^c\subseteq \mathcal H^b$ for the subclass having continuous $H_n$.

Every quoted static object used in the Bellman and separation argument must have a continuous common-unit representative $g\in C(\Omega )$. This representative may be the actual terminal ledger when that ledger is continuous. If a physically delivered ledger is discontinuous because of its exercise indicator, use instead its continuous fixing-date mark $g^{\rm fix}$; the market bid and ask remain those of the actual contract. In either case the required carry interface is an exact pathwise identity