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Bi-martingale optimal transport and its applications

Karol Bołbotowski

TL;DR

Bi-martingale optimal transport (M^2OT) introduces a nonlinear OT framework for probability measures with a common barycenter by imposing two martingale-type constraints through a coupling map. It unifies OT, MOT, convex-order optimization, Zolotarev distances, and convex-dominance projections, and establishes a second-order Kantorovich–Rubinstein duality within this bi-martingale setting. The work proves Γ-convergence of a bi-martingale approximation (M^2OT_n) to MOT, provides a robust computational path via 3-plan reformulations and conic optimizations, and offers detailed analysis and illustrative examples including non-uniqueness of Zolotarev projections and stabilization effects in higher dimensions. Overall, the framework yields stable, calculable projections onto convex-dominance cones and reliable MOT approximations under marginal perturbations, with potential impact on numerical MOT and risk-averse decision-making under model uncertainty.

Abstract

We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on $\mathrm{P}_2(\mathbb{R}^d)$. For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a $Γ$-convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions.

Bi-martingale optimal transport and its applications

TL;DR

Bi-martingale optimal transport (M^2OT) introduces a nonlinear OT framework for probability measures with a common barycenter by imposing two martingale-type constraints through a coupling map. It unifies OT, MOT, convex-order optimization, Zolotarev distances, and convex-dominance projections, and establishes a second-order Kantorovich–Rubinstein duality within this bi-martingale setting. The work proves Γ-convergence of a bi-martingale approximation (M^2OT_n) to MOT, provides a robust computational path via 3-plan reformulations and conic optimizations, and offers detailed analysis and illustrative examples including non-uniqueness of Zolotarev projections and stabilization effects in higher dimensions. Overall, the framework yields stable, calculable projections onto convex-dominance cones and reliable MOT approximations under marginal perturbations, with potential impact on numerical MOT and risk-averse decision-making under model uncertainty.

Abstract

We introduce a new non-linear optimal transport formulation for a pair of probability measures on sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on . For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a -convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions.

Paper Structure

This paper contains 26 sections, 24 theorems, 133 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Selected topics on the space of probabilities $\mathcal{P}({\mathbb{R}^d})$
  3. Bi-martingale optimal transport as the unifying framework
  4. Second-order Kantorovich--Rubinstein duality via $(\mathrm{M^2OT})$
  5. A robust approximation of the classical martingale transport
  6. Other results and organization of the paper
  7. Additional remarks on the similarities with the weak optimal transport
  8. Notation and basic facts
  9. Bi-martingale formulation for a general cost
  10. Existence of optimal bi-martingale plans
  11. Reformulation via 3-plans
  12. Application to convex dominance with minimum cost
  13. Optimal convex dominance on the real line
  14. Link with the bi-martingale optimal transport
  15. The quadratic case and the Zolotarev-2 distance
  16. ...and 11 more sections

Key Result

Theorem 1.1

The following identities hold true for $\mu,\nu \in \mathcal{P}_2({\mathbb{R}^d})$ that share the barycentre $[\mu] = [\nu]$, and both problems share a non-empty set of solutions. Moreover, the following conditions are equivalent for any pair $(\gamma,q) = (\gamma,\zeta \gamma) \in \Gamma Q(\mu,\nu)$:

Figures (5)

  • Figure 1: Example of a bi-martingale plan $\gamma$ (green dashed lines) with respect to a coupling map $\zeta$ for two-point data $\mu$ (gray) and $\nu$ (black). The martingale plans $\gamma_1 = (\pi_1,\zeta) \# \gamma$ (blue) and $\gamma_2 = (\pi_2,\zeta) \# \gamma$ (red) share the second marginal $\rho = \zeta \# \gamma$ (magenta) that dominates both $\mu$ and $\nu$ for convex order.
  • Figure 2: Data $\mu$ (gray), $\nu$ (black), the optimal convex dominant $\rho$ (magenta), and the associated martingale plans $\gamma_1 \in \Gamma_{\mathrm{M}}(\mu,\rho)$ (blue), $\gamma_2 \in \Gamma_{\mathrm{M}}(\nu,\rho)$ (red). Subsequent figures display different solutions.
  • Figure 3: Data $\mu$ (discretized, gray), $\nu$ (black), the optimal convex dominant $\overline\rho$ (magenta) for the cost $f = {\left \lvert {\,\cdot\,} \right \rvert}^p$. For visibility, a different and finer scale was applied to the measure $\nu$ in comparison to $\mu,\rho$.
  • Figure 4: Left column: optimal martingale transport (blue) between $\mu$ (gray) and $\nu_n$ (black). Right column: bi-martingale transport (blue and red) solving the approximation $(\mathrm{M^2OT}_n)$ for the same data. In the right column the probability $\rho_n = \zeta_n \# \gamma_n$ (magenta) is also displayed.
  • Figure 5: The sequence of solutions $\gamma_n = \sum_{i,j} \gamma^n_{ij} \, \delta_{(x_i,y_j)}$ to the bi-martingale approximations $(\mathrm{M^2OT_n})$: (a) the transports emanating from the mass at $x_1$; (b) the total cost $\iint {\left \lvert x-y \right \rvert} \,\gamma_n(dxdy)$ (without the penalizing term).

Theorems & Definitions (50)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 40 more