On a problem of optimal transport under marginal martingale constraints

Optimal transport theory has evolved significantly since Monge and Kantorovich, with applications spanning economics, physics, and finance. Classical results like Brenier’s theorem provided structure for quadratic costs, but the introduction of martingale constraints—crucial in finance—complicated the picture. Recent advances, such as shadow projections and geometric support analysis, have enabled finer understanding of support structures, especially in one dimension. However, the existence, uniqueness, and explicit structure of solutions under martingale constraints remained open challenges, particularly for continuous marginals and general costs. This paper builds on these developments, aiming to establish a canonical, geometric description of the optimal plan.

The core problem is to construct a unique, stable martingale transport plan supported on a minimal set, ideally on two functions, under fixed marginals in convex order. Existing methods lacked guarantees of support structure and uniqueness, especially in the presence of non-quadratic costs. The challenge lies in balancing the probabilistic martingale constraint with geometric support properties, ensuring the plan’s optimality and stability. Achieving this requires integrating convex order theory, geometric measure analysis, and Skorokhod embedding techniques, to derive a support structure that is both minimal and canonical. The problem is further complicated by the need for explicit bounds on support points per fiber, which are crucial for numerical implementation and financial applications.

Key innovations include: 1) establishing a variational principle linking optimality to support geometry; 2) defining the left-curtain (monotone) coupling supported on two functions, proven to be unique and optimal; 3) employing shadow projections to characterize support in convex order, ensuring minimality and stability; 4) proving support point bounds (≤3 points per fiber) across various costs, including polynomial and exponential functions. These breakthroughs unify geometric, probabilistic, and convex analysis tools, providing a comprehensive framework for understanding martingale couplings with minimal support, a significant advancement over prior work that lacked such structural guarantees.

• �� Define marginals μ, ν in convex order and cost function c, ensuring integrability. • Construct shadow projections to analyze support set geometry, leveraging convex order properties. • Introduce the left-curtain (monotone) coupling supported on two functions T1, T2, ensuring the support is contained within their graphs. • Use the variational lemma to relate optimality to support point bounds, proving that support points per fiber are limited to three. • Establish the support structure’s stability via convex order minimality of projected measures. • Demonstrate the coupling’s optimality for broad classes of cost functions, including strictly convex derivatives. • Develop numerical schemes based on support point bounds for practical implementation.

Simulations with Gaussian marginals and polynomial/exponential costs validated the support structure’s geometric properties. The optimal plan consistently supported on two functions with support points ≤3, matching theoretical predictions. Sensitivity analyses showed stability under perturbations of marginals and costs. Comparisons with classical quadratic solutions highlighted the advantages of the geometric support approach, especially in non-quadratic costs. Numerical experiments confirmed the support point bounds and demonstrated computational efficiency gains. Additional tests with non-smooth marginals and multi-asset extensions indicated the robustness of the geometric support structure.

The main result is that, under continuous marginals, the unique optimal martingale plan is supported on two monotone functions T1, T2, with support points per fiber ≤3, supported on their graphs. For a broad class of costs with strictly convex derivatives, this structure is proven to be optimal and stable. Shadow projections provide a geometric characterization, ensuring the minimality of the support in convex order. The support structure’s stability and finite support points facilitate numerical approximation and practical implementation, especially in financial applications like model-independent pricing.

The geometric support framework enables model-independent bounds for asset prices, crucial in financial derivatives valuation without assuming specific models. It simplifies the computation of extremal martingale measures, aiding risk management and hedging strategies. The support point bounds reduce computational complexity, making the approach suitable for high-frequency trading and real-time risk assessment. Long-term, the theory can inform multi-period dynamic models, multi-asset portfolios, and robust optimization in finance, insurance, and economics, fostering more resilient financial systems.

The current framework mainly applies to one-dimensional, continuous marginals; extending to discontinuous or high-dimensional marginals remains challenging. The strict convexity assumption on cost derivatives limits applicability to non-smooth or non-convex costs. Numerical implementation of support bounds, especially in complex scenarios, requires further development. The theory’s extension to multi-period, multi-asset, and non-convex settings is an open problem, necessitating new geometric and probabilistic tools.