On the Martingale Schrödinger Bridge between Two Distributions

The evolution of optimal transport theory has seen significant progress with the introduction of entropy regularization, notably through Cuturi's Sinkhorn algorithm. Schrödinger bridges, originating from Schrödinger's 1931 question, have been developed as a stochastic counterpart, providing a probabilistic framework for path reconstruction with applications in physics, statistics, and finance. Classical Schrödinger bridges rely on potential functions (f, g) to describe optimal couplings, with well-understood existence and regularity results. However, extending these results to incorporate martingale constraints, especially with multiple marginals, introduces substantial complexity. Recent advances in martingale optimal transport (MOT) by Beiglböck, Henry-Labordère, and others have addressed the existence and structure of martingale couplings, but the construction of potentials analogous to Schrödinger's remains unresolved. This paper builds on these foundations, aiming to develop a comprehensive potential theory for the martingale Schrödinger bridge in multi-marginal settings.

The core challenge lies in constructing potentials (f, g, h) that characterize the optimal martingale coupling with minimal relative entropy. Unlike classical cases, the martingale constraint introduces a term h(x)(y−x) that complicates the potential's structure, affecting integrability and uniqueness. Existing methods fail to guarantee the existence of such potentials under broad conditions, especially when the marginals have complex support or lack regularity. The problem becomes more acute in multi-marginal scenarios, where the interaction between marginals and the martingale condition creates a highly non-convex landscape. Overcoming these difficulties requires innovative analytical tools to establish the existence, regularity, and explicit form of the potentials, ensuring they satisfy both marginal and martingale constraints simultaneously.

This paper's key innovation is the introduction of an auxiliary relaxed problem cM(μ, ν), replacing the strict martingale constraint with a weaker inequality, facilitating potential construction. The authors leverage duality theory, tightness, and convex analysis to prove the existence of a unique potential triplet (f, g, h) that describes the optimal coupling density as an exponential function. The potential h(x) acts as a Lagrange multiplier for the martingale constraint, enabling a unified potential framework that extends classical Schrödinger theory to the martingale setting. This approach overcomes previous limitations related to integrability and regularity, providing a rigorous foundation for the existence and uniqueness of potentials in multi-marginal martingale transport.

• �� Define a relaxed auxiliary problem cM(μ, ν) with entropy minimization over a broader set of couplings satisfying a weaker inequality.
• �� Use Sion’s minimax theorem to relate the primal problem to a dual maximization over potentials h, f, g.
• �� Prove the existence of a saddle point (h_m, π_m) in the space of Lipschitz functions and probability measures.
• �� Show that the optimal measure π_m satisfies the martingale constraint by limit and regularity arguments.
• �� Derive the explicit exponential form of the density using duality and tightness, ensuring the potentials meet the integrability and support assumptions.
• �� Verify the potentials' uniqueness up to affine shifts and establish the strong duality between primal and dual problems.

Simulations involve constructing multiple marginals with known distributions (e.g., Gaussian), verifying the existence of potentials (f, g, h) numerically. Using iterative algorithms inspired by Sinkhorn, the stability and convergence of potentials are tested under various support and regularity conditions. The experiments compare the entropy values, potential functions, and the satisfaction of the martingale constraint, demonstrating the robustness of the theoretical results. Sensitivity analyses explore the impact of support assumptions and regularity conditions, confirming the theoretical predictions about existence and uniqueness.

Numerical results confirm that the constructed potentials (f, g, h) satisfy the exponential density form, with the dual maximization attaining a unique solution. The potentials exhibit stability under perturbations, and the entropy values closely match theoretical bounds. The experiments validate the theoretical assumptions, particularly the support conditions, and demonstrate the potentials' effectiveness in modeling complex multi-marginal martingale processes. These findings support the broader applicability of the framework in financial modeling and stochastic process calibration.

The framework can be directly applied to multi-period risk-neutral pricing, calibration of financial models with multiple assets, and complex derivative valuation. It enables practitioners to construct optimal martingale couplings that respect market constraints while minimizing information loss. Additionally, the potentials facilitate efficient numerical algorithms, making the approach suitable for high-dimensional problems in finance, statistics, and machine learning. The theory also opens pathways for data-driven calibration and robust risk assessment in uncertain environments.

The current results rely on support regularity and boundary conditions (assumption 2.3), limiting applicability to distributions with irregular support or heavy tails. Extending the theory to higher dimensions presents significant challenges due to increased complexity and computational costs. Numerical algorithms may face stability issues in high-dimensional settings, and the assumptions may not hold in all practical scenarios. Future work should aim to relax these restrictions, improve computational scalability, and develop adaptive algorithms for broader classes of distributions.