Martingale Coupling via Entropy Maximization

• The paper introduces a framework for constructing martingale transport plans that minimizes relative entropy relative to a reference measure.
• It integrates entropic regularization with martingale constraints, leading to dual formulations with utility maximization and Schrödinger-type bridges.
• Efficient numerical methods, including Sinkhorn-type algorithms and fixed-point iterations, provide tractable solutions for high-dimensional robust pricing problems.

Martingale coupling via entropy maximization addresses the problem of constructing martingale transport plans between prescribed marginals that are optimally “close” to a reference coupling, most commonly in the sense of minimal relative entropy. This framework, emerging from developments in optimal transport, stochastic control, and mathematical finance, rigorously integrates martingale constraints into entropic regularization. It leads to a new class of Schrödinger-type problems—martingale Schrödinger bridges—and natural duality with utility maximization. Its formulation unlocks both theoretical clarity and numerical tractability for a host of applications, notably in robust pricing and computation of multistage stochastic couplings.

1. Entropic Martingale Optimal Transport: Primal and Dual Problems

In the two-period setting, let $(X, Y)$ be a process under a “physical” probability law $P$ on $\mathbb{R}^2$, and let $\nu$ be the target risk-neutral law for $Y$, typically derived from observed terminal option prices. The set of admissible couplings is the family of equivalent martingale measures calibrated to $\nu$: $$M(\nu) := \{\, Q \in \mathcal{P}(\mathbb{R}^2): Q \sim P,\, Q^2 = \nu,\, \mathbb{E}^Q[Y\mid X] = X \,\}.$$ The entropic martingale coupling, or martingale Schrödinger bridge, is defined as the solution to: $$\inf_{Q\in M(\nu)} H(Q \mid P), \qquad H(Q \mid P) = \begin{cases} \mathbb{E}^Q \bigl[ \ln \frac{dQ}{dP} \bigr],& Q \ll P,\ +\infty, & \text{otherwise.} \end{cases}$$ With mild feasibility conditions, convexity, and lower semicontinuity, there exists a unique minimizer $Q_\ast$ (Nutz et al., 2022).

The dual problem, in exponential utility or semistatic portfolio terms, is

$$\frac{1}{\gamma} \inf_{Q \in M(\nu)} H(Q \mid P) = \sup_{V \in \mathsf{Adm}} u^{-1} \bigl( \mathbb{E}^P [u(V)] \bigr)$$

for risk aversion parameter $P$0 and $P$1, with $P$2 encompassing both dynamic and static hedging terms.

In the general discrete-time case, the entropy-martingale optimal transport (EMOT) problem takes the form (Doldi et al., 2020): $P$3 where $P$4 is the set of martingale laws, $P$5 is a cost, and $P$6 is a penalty term arising from a concave utility $P$7. The dual is a nonlinear subhedging or pricing–hedging problem, generally characterized by Fenchel conjugacy with nonlinear “optimized-certainty-equivalent” (OCE) functionals.

2. Structure and Potentials of the Entropic Martingale Coupling

In the classic one-period martingale Schrödinger bridge, given $P$8 with $P$9 (convex order), and under mild “irreducibility” and endpoint-atom conditions, there exists a unique minimizer $\mathbb{R}^2$0 of $\mathbb{R}^2$1 (Nutz et al., 2024). This optimizer is absolutely continuous with respect to $\mathbb{R}^2$2 and has log-density

$\mathbb{R}^2$3

where:

• $\mathbb{R}^2$4, $\mathbb{R}^2$5 enforce the marginal constraints,
• $\mathbb{R}^2$6 operates as a Lagrange multiplier field for the martingale constraint,
• The potentials $\mathbb{R}^2$7 are unique up to affine transformations.

This representation encodes both fixed-marginal and conditional mean constraints, closely linking optimality with the notions of Schrödinger potentials in entropy-regularized transport. The corresponding density is

$\mathbb{R}^2$8

3. Duality, Variational Characterizations, and Existence Results

Strong duality holds between the entropy-minimizing primal problem and a supremum over potentials $\mathbb{R}^2$9: $\nu$0 The maximizers correspond directly to the potentials defining the optimizer $\nu$1. Existence, uniqueness, and integrability of these potentials are established under convex order and further mild technical conditions (Nutz et al., 2024, Nutz et al., 2022).

The duality further generalizes to path-dependent and continuous-time settings, integrating constraints, penalty terms, and nonlinear utilities, yielding robust existence and uniqueness results even where standard semistatic strategies may not be closed (Doldi et al., 2020, Nutz et al., 2022).

4. Numerical Methods and Sinkhorn-type Algorithms

Martingale entropy-regularized problems admit efficient numerical schemes via generalizations of Sinkhorn-type and iterative proportional fitting (IPFP) algorithms. The martingale variants iteratively update potentials to enforce (i) marginals, (ii) martingale moment constraints, and (iii) normalization. For example, the updates alternate between

• Marginal-matching via coordinate updates of $\nu$2 and $\nu$3,
• Moment-matching of $\nu$4 for martingale conditions.

In continuous time, Poisson or trace-normalized approximations (SEMOT) yield discretizations that avoid high-dimensional grid-refinement, facilitating numerically tractable computations even in moderate to high dimensions. Fixed-point iterations, involving Hamilton–Jacobi–Bellman (HJB) and Fokker–Planck (FP) equations, are implemented with provable monotone convergence properties (Nutz et al., 2022, Buet-Golfouse et al., 21 May 2026, Nutz et al., 2024).

5. Applications in Mathematical Finance and Robust Pricing

Martingale entropic couplings and their duality theory underpin nonlinear pricing–hedging duality in mathematical finance. In particular, the minimal-entropy martingale measure $\nu$5 coincides with the solution to a utility-maximization problem over semistatic portfolios—portfolios involving both static positions in terminal options and dynamic trading. This duality is structurally robust, capturing both indifference pricing for exponential utility and general robust subhedging with convex penalty terms (Doldi et al., 2020, Nutz et al., 2022).

Dense subsets of calibrated martingale measures and explicit constructions overcome the non-closedness of semistatic strategies, enabling integrable and well-defined portfolio solutions (Nutz et al., 2022). The entropy-regularized framework also extends to nonlinear and Wasserstein-type penalty settings, interpolating between relaxed and strict martingale and marginal constraints (Doldi et al., 2020).

6. Extensions: Continuous Time and Specific Entropy

Classical relative entropy regularization is ill-suited for continuous-time martingale transport, as it enforces alignment of volatility characteristics. A recent approach introduces generalized specific entropy via Poisson-jump approximations, yielding explicit entropy functionals whose limits preserve the martingale property while incorporating local volatility structure (Buet-Golfouse et al., 21 May 2026). The resulting SEMOT (specific-entropy martingale optimal transport) problems feature compactness, strong duality, and are characterized by coupled HJB–FP systems, with practical numerical schemes feasible even in two spatial dimensions. This approach avoids the curse of dimensionality inherent in traditional multimarginal Sinkhorn algorithms.

7. Illustrative Examples and Special Cases

• Discrete two-point marginals: The explicit solution for the martingale Schrödinger bridge in this case is given by

$\nu$6

Potentials $\nu$7 admit closed forms in this scenario (Nutz et al., 2024).

• Gaussian marginals: The minimizer is again Gaussian, and potentials are quadratic in $\nu$8 and $\nu$9 with $Y$0 linear in $Y$1, matching the form predicted by the theory.
• General discrete/continuous-time settings: EMOT and SEMOT frameworks apply to robust finance, where the entropic martingale coupling provides a foundation for model uncertainty quantification and the design of computationally tractable nonlinear valuations (Doldi et al., 2020, Buet-Golfouse et al., 21 May 2026).

This synthesis of optimal transport, entropy minimization, and martingale constraints establishes the entropic martingale coupling as a canonical model in problems requiring both structure-preserving transport and computational feasibility, with far-reaching implications for finance, statistics, and stochastic analysis.