Bi-Martingale Framework in Transport

• Bi-martingale framework is a transport formulation using two coupled martingale constraints via an intermediate barycentric variable, relaxing strict convex order conditions.
• It extends classical martingale optimal transport by encoding dual constraints in a convex variational form, offering stable approximations even when convex order fails.
• The framework connects concepts like Zolotarev distance, quadratic cost structures, and backward Monge transports, enhancing robustness in high-dimensional settings.

Searching arXiv for the cited papers to ground the article in current records. Bi-martingale framework denotes a class of constructions in which a probabilistic coupling, transport plan, or conditional-expectation dynamics is governed by two martingale-type directions rather than a single forward martingale condition. In the most explicit recent formulation, bi-martingale optimal transport introduces an intermediate barycentric variable between two probability measures with common barycentre and requires both induced two-point couplings to be martingale plans, thereby linking optimal transport, convex order, Zolotarev distances, convex dominance, and robust approximation of classical martingale optimal transport (MOT) (Bołbotowski, 31 Oct 2025). Related literature develops complementary versions of the same two-sided idea: backward Monge martingale transports impose determinism over the second marginal rather than the first (Nutz et al., 2022), and martingale representation theory under non-monotone information decomposes dynamics into infinitesimal forward and backward martingale components, with the backward term quantifying information loss (Christiansen, 2018).

1. Foundational definition and basic geometry

The classical MOT problem augments Monge–Kantorovich transport by the martingale constraint

$\mathbb E[Y\mid X]=X,$

or equivalently, for a coupling $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$,

$$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$

By Strassen’s theorem, this is feasible if and only if $\mu\preceq_c \nu$ in convex order. The bi-martingale framework replaces this one-sided condition by two martingale-type constraints against an intermediate map $\zeta(x,y)$, thereby removing convex order as the basic feasibility condition and replacing it with the weaker requirement that the marginals share a common barycentre (Bołbotowski, 31 Oct 2025).

Formally, given $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$ with common barycentre $b$, a coupling $\gamma\in\Gamma(\mu,\nu)$ is a bi-martingale plan with respect to a Borel map $\zeta:X\times Y\to\mathbb R^d$ if both pushforwards $(\pi_1,\zeta)_\#\gamma$ and $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$0 are martingale plans. In disintegrated form, this is equivalent to

$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$1

These are the two martingale conditions that give the framework its name. The common barycentre condition is necessary and sufficient for the existence of such pairs $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$2; a simple witness is the product coupling $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$3 together with $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$4 (Bołbotowski, 31 Oct 2025).

A recurrent misconception is to treat the framework as merely “two martingale constraints at once.” In the transport formulation, the structure is more rigid: the two constraints are coupled through a single intermediate barycentric variable. This makes the framework symmetric in the marginals and, in the quadratic case, turns the intermediate variable into the main geometric carrier of convex-order information.

2. Convex variational formulation in optimal transport

A central reformulation introduces vector-valued couplings

$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$5

and the admissible set

$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$6

For $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$7, the Radon–Nikodym derivative $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$8 recovers the barycentric coupling map. The defining constraints of $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$9 are equivalent to weak moment–martingale conditions against scalar test functions $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$0 and vector fields $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$1, so the state space $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$2 encodes the bi-martingale condition in a convex-analytic form (Bołbotowski, 31 Oct 2025).

With a three-point cost $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$3 satisfying joint lower semicontinuity, boundedness from below, convexity in $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$4, and a superlinear coercive lower bound in $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$5, the bi-martingale optimal transport problem is

$$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$6

The functional is jointly convex in $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$7, weakly lower semicontinuous, and admits minimizers whenever the infimum is finite. An equivalent formulation uses three-point plans $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$8 whose $$\int y\,\gamma_x(dy)=x \quad\text{for }\mu\text{-a.e. }x.$$9 and $\mu\preceq_c \nu$0 marginals are martingale plans; strict convexity in $\mu\preceq_c \nu$1 forces concentration on the graph of a single map $\mu\preceq_c \nu$2 (Bołbotowski, 31 Oct 2025).

Three special cost classes organize the theory. If

$\mu\preceq_c \nu$3

finiteness forces $\mu\preceq_c \nu$4, the second martingale condition becomes trivial, and the problem reduces to classical MOT. If $\mu\preceq_c \nu$5 with $\mu\preceq_c \nu$6 convex and superlinear, the framework becomes a convex-dominance problem over common convex dominants $\mu\preceq_c \nu$7. If

$\mu\preceq_c \nu$8

the resulting problem yields an optimal transport representation of the second Zolotarev distance $\mu\preceq_c \nu$9 (Bołbotowski, 31 Oct 2025).

3. Zolotarev distance, convex dominance, and convex-order geometry

On $\zeta(x,y)$0, the Zolotarev-2 distance is defined by

$\zeta(x,y)$1

The admissible class can be characterized by

$\zeta(x,y)$2

and a three-point inequality due to Le Gruyer gives the exact variational structure behind the optimal barycentric map (Bołbotowski, 31 Oct 2025).

A key identity from Bołbotowski–Bouchitté, re-derived and extended in the bi-martingale framework, is

$\zeta(x,y)$3

where

$\zeta(x,y)$4

Equivalently,

$\zeta(x,y)$5

and therefore

$\zeta(x,y)$6

Thus the quadratic bi-martingale problem is an exact transport representation of $\zeta(x,y)$7 (Bołbotowski, 31 Oct 2025).

For general convex lower semicontinuous superlinear $\zeta(x,y)$8, bi-martingale transport with cost $\zeta(x,y)$9 is equivalent to

$\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$0

If $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$1 is strictly convex, optimal intermediate measures $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$2 correspond to optimal pairs $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$3 via $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$4. In dimension $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$5, convex order is a join-semilattice, and the least upper bound $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$6 is the unique minimizer for any convex $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$7 whenever the infimum is finite (Bołbotowski, 31 Oct 2025).

These results show that bi-martingale transport does not merely approximate convex-order problems; it re-encodes them as transport problems with an endogenous barycentric variable.

4. Quadratic structure, rigidity, and symmetric projections

In the quadratic setting, the optimal barycentric map is determined by any maximizer $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$8 of the Zolotarev dual problem: $\mu,\nu\in\mathcal P_1^b(\mathbb R^d)$9 For each $b$0, this $b$1 is the unique minimizer in $b$2 of the associated three-point quadratic expression. The corresponding admissible class

$b$3

is always nonempty, and the set of minimizers of the quadratic bi-martingale problems is exactly

$b$4

Hence the barycentric component is rigid—completely determined by the Zolotarev potential—while the coupling $b$5 may remain nonunique (Bołbotowski, 31 Oct 2025).

Convex order reappears as a special degenerate case. The following are equivalent: $b$6; the potential $b$7 is optimal; $b$8; and $b$9 coincides with the classical martingale coupling set $\gamma\in\Gamma(\mu,\nu)$0. In that case $\gamma\in\Gamma(\mu,\nu)$1, the first bi-martingale condition reduces to the usual martingale condition, and the second becomes automatic. This clarifies why quadratic MOT under convex order is trivial at the level of total cost: all martingale couplings have the same quadratic cost (Bołbotowski, 31 Oct 2025).

The same quadratic structure yields a symmetric projection theory. The forward Zolotarev projection of $\gamma\in\Gamma(\mu,\nu)$2 onto the cone $\gamma\in\Gamma(\mu,\nu)$3 coincides with the forward Zolotarev projection of $\gamma\in\Gamma(\mu,\nu)$4 onto $\gamma\in\Gamma(\mu,\nu)$5, and both equal the set $\gamma\in\Gamma(\mu,\nu)$6 of common convex dominants of minimal second moment. In dimension $\gamma\in\Gamma(\mu,\nu)$7 this projection is the unique least upper bound $\gamma\in\Gamma(\mu,\nu)$8; in higher dimension it is a nonempty convex set and may be non-singleton (Bołbotowski, 31 Oct 2025). This symmetry is specific to the Zolotarev metric and contrasts with asymmetric Wasserstein projection phenomena.

5. Robust approximation of martingale optimal transport

One of the principal applications of the framework is a $\gamma\in\Gamma(\mu,\nu)$9-convergent approximation scheme for MOT. Let $\zeta:X\times Y\to\mathbb R^d$0 be continuous, bounded from below, with $\zeta:X\times Y\to\mathbb R^d$1-growth for some $\zeta:X\times Y\to\mathbb R^d$2, and let $\zeta:X\times Y\to\mathbb R^d$3 converge to $\zeta:X\times Y\to\mathbb R^d$4 in $\zeta:X\times Y\to\mathbb R^d$5, equivalently in $\zeta:X\times Y\to\mathbb R^d$6. The discrete approximants are not required to preserve convex order. With parameters $\zeta:X\times Y\to\mathbb R^d$7 satisfying

$\zeta:X\times Y\to\mathbb R^d$8

the penalized problems are

$\zeta:X\times Y\to\mathbb R^d$9

The corresponding functionals $(\pi_1,\zeta)_\#\gamma$0 $(\pi_1,\zeta)_\#\gamma$1-converge in the weak topology to

$(\pi_1,\zeta)_\#\gamma$2

Minimizers exist, are relatively compact, and every limit point solves the symmetric limit problem

$(\pi_1,\zeta)_\#\gamma$3

If, in addition, $(\pi_1,\zeta)_\#\gamma$4, then $(\pi_1,\zeta)_\#\gamma$5, so the limit recovers classical MOT exactly (Bołbotowski, 31 Oct 2025).

This construction addresses two structural deficiencies of MOT emphasized in the motivation: instability in dimension $(\pi_1,\zeta)_\#\gamma$6 under small perturbations of the marginals, and ill-posedness when convex order fails. The bi-martingale penalty tolerates loss of convex order at the approximation level and still converges to the correct MOT minimizers when convex order holds at the limit. In the Brückerhoff–Juillet instability example in $(\pi_1,\zeta)_\#\gamma$7, the paper reports numerically that the scheme produces non-trivial bi-martingale plans converging to the correct non-trivial MOT minimizer, whereas the naive discrete MOT sequence does not (Bołbotowski, 31 Oct 2025).

A plausible implication is methodological: stability theory for martingale representations under convergence of martingales and filtrations, developed in a separate one-martingale/one-filtration framework, provides exactly the kind of architecture one would want to adapt to bi-martingale or multi-filtration settings, especially when the limiting martingale is quasi-left-continuous and has the predictable representation property (Papapantoleon et al., 2018).

6. Directional martingale constructions on the line

A related strand of literature studies one-dimensional martingale transports from a reversed structural perspective. For $(\pi_1,\zeta)_\#\gamma$8, a forward Monge martingale coupling $(\pi_1,\zeta)_\#\gamma$9 is impossible unless $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$00, because the martingale property forces $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$01 almost surely. By contrast, backward Monge martingale transports seek couplings of the form

$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$02

so that $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$03 and $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$04. Under atomless $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$05, such couplings exist in great generality and are weakly dense in the full martingale coupling set $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$06 (Nutz et al., 2022).

The constructive object is the barcode transport, built by decomposing $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$07 into countably many “bars,” transporting each bar through a left-curtain shadow coupling that is backward Monge on that piece, and aggregating the result. When $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$08 is atomless, the aggregate coupling is a global backward Monge martingale transport. The density theorem then implies a Monge–Kantorovich equivalence for MOT in the backward direction: for continuous costs with an integrable linear bound,

$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$09

The same paper constructs backward deterministic martingales, in which $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$10 is $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$11-measurable for every $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$12, so the current state encodes the entire history (Nutz et al., 2022).

This is not the same formalism as bi-martingale optimal transport, but it illuminates a closely related principle: nontrivial martingale structure can emerge from reversing the deterministic direction. The paper explicitly notes that this backward Monge structure is exactly the kind of dual temporal direction one might want in a bi-directional or bi-martingale framework.

7. Non-monotone information and bidirectional martingale representation

A different, explicitly two-sided use of the term arises when information is not monotone in time. Instead of a filtration $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$13, one considers sub-$\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$14-algebras $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$15 with no monotonicity assumption, so information may be gained and later deleted. In a marked point process model with deletion times, classical martingale notions are replaced by infinitesimal forward martingales (IF-martingales) and infinitesimal backward martingales (IB-martingales), defined through partition limits of conditional increments given $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$16 or $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$17, respectively (Christiansen, 2018).

The corresponding representation theory extends classical jump-martingale representations by adding a symmetric counterpart that quantifies the effect of information loss. At the level of conditional expectations $\gamma(dx\,dy)=\mu(dx)\otimes\gamma_x(dy)$18, the dynamics split into a forward innovation term driven by forward compensators and a backward term driven by backward compensators. In the filtration case, the backward term vanishes and the theory collapses to the classical single-martingale representation. In non-monotone settings such as life insurance with data deletion or credit-risk models based on state compression, the backward term measures reserve changes or hedging errors caused specifically by information loss (Christiansen, 2018).

This representation-theoretic version clarifies that a bi-martingale framework need not mean two separate stochastic processes. It may also mean one conditional-expectation process decomposed into dual infinitesimal martingale components, one associated with information gain and the other with information deletion. Across transport, one-dimensional MOT, and non-monotone-information theory, the common principle is the same: a symmetric or bidirectional martingale structure captures phenomena that a one-sided martingale formulation either excludes or renders unstable.