Martingale KR Duality in Finance

• Martingale KR Duality is a framework that extends classical optimal transport by incorporating martingale constraints to enforce no-arbitrage conditions.
• It features varied formulations—including discrete-time, continuous-time, and path-space approaches—that support robust superhedging and model-free pricing.
• Strong duality and optimizer attainment under mild conditions are key strengths, though challenges remain in multidimensional and irregular settings.

Martingale Kantorovich–Rubinstein (KR) duality is a fundamental theoretical framework extending classical optimal transport duality to problems constrained by the martingale property. It arises in mathematical finance, stochastic analysis, and probability theory, encapsulating robust superhedging, model-free pricing, and pathwise transport across prescribed marginals. The duality ensures strong equality between the optimal transport cost (or price) under martingale constraints and the value of associated dual optimization problems often involving both static and dynamic hedging. The martingale KR duality has several formulations—including process-level, PDE, and path-space approaches—and underpins modern theoretical advances in robust finance and stochastic optimal control.

1. Classical and Martingale Optimal Transport: Problem Statement

The classical Kantorovich–Rubinstein optimal transport framework considers two probability measures $\mu, \nu$ on $\mathbb R^d$ and seeks a coupling $\pi$ minimizing the expected cost: $$\inf_{\pi \in \Pi(\mu, \nu)} \int c(x, y) \, \pi(dx, dy),$$ where $\Pi(\mu, \nu)$ denotes couplings with given marginals. The dual is

$$\sup_{\varphi, \psi} \left\{\int \varphi \, d\mu + \int \psi \, d\nu: \varphi(x) + \psi(y) \leq c(x, y) \right\},$$

with equality under mild conditions (Beiglböck et al., 2015).

Martingale optimal transport (MOT) augments this by enforcing the martingale constraint on the coupling: $$M(\mu, \nu) = \left\{P \in \Pi(\mu, \nu): \mathbb E^P[Y \mid X] = X \;\; P\text{-a.s.} \right\}.$$ The primal problem is then

$\sup_{P \in M(\mu, \nu)} \mathbb E^P[c(X, Y)],$

and the dual must encode this additional linear (martingale) constraint (Beiglböck et al., 2015, Herrmann et al., 2017, Zitridis, 12 Oct 2025).

2. Martingale KR Duality: General Formulations

Two-Marginal (Discrete-Time) Duality

For general cost $c: \mathbb R \times \mathbb R \to \mathbb R$ and marginals in convex order, the dual reads: $$\inf_{(\varphi, \psi, H)} \left\{ \int \varphi \, d\mu + \int \psi \, d\nu : \varphi(x) + \psi(y) + H(x)(y - x) \geq c(x, y) \;\; \text{for all } x, y \right\}$$ where $H$ enforces the martingale constraint via a dynamic hedging term (Beiglböck et al., 2015, Zitridis, 12 Oct 2025).

Continuous-Time Benamou–Brenier Formulation

In continuous-time, with margins $\mu, \nu \in \mathcal P(\mathbb R^d)$, the Benamou–Brenier process-level problem is: $$BB(\mu, \nu) := \inf_{X} \int_0^1 \mathbb E\big[c(t, X_t, \dot{X}_t)\big] dt,$$ where $X$ is a continuous martingale with prescribed endpoints and $\dot{X}_t$ is its (absolutely continuous) quadratic variation density (Huesmann et al., 2017). The PDE/Fokker–Planck version is: $$FPE(\mu, \nu) := \inf_{(\varrho_t, a_t)} \int_0^1 \int_{\mathbb R^d} c(t, x, a_t(x)) \, d\varrho_t(x) dt,$$ subject to the Fokker–Planck equation

$$\partial_t \varrho_t = \frac{1}{2} \nabla^2[a_t \varrho_t], \quad \varrho_0 = \mu, \; \varrho_1 = \nu.$$

Dual variables arise from the backward Hamilton–Jacobi–Bellman (HJB) PDE and Legendre transforms (Huesmann et al., 2017).

Path-Space and Multi-Marginal Extensions

For $N$ marginals $\{\mu_i\}_{i=0}^N$, semi-static and fully dynamic duals introduce static options at each marginal and predictable processes for dynamic trading: $$W(G) = \inf \left\{ \sum_{i=1}^N \int \phi_i \, d\mu_i : \sum_{i=1}^N \phi_i(X_{T_i}) + \int H \cdot dX \geq G(X), \forall X \right\}$$ This duality is pathwise and encompasses multi-step martingale constraints (Dolinsky et al., 2014, Sester, 2023).

3. Dual Problem Structure and PDE Characterization

The martingale dual involves variable triplets (or higher tuples for vectorial/multi-marginal problems):

• Static potentials: $\varphi, \psi$ (and potentially many $\phi_i$)
• Dynamic hedge: $H(x)$ or a process $H_t(X)$
• For continuous models, HJB PDE super-solutions: $$\partial_t \phi(t, x) + c^*\left(t, x, \frac{1}{2} \nabla^2 \phi(t, x)\right) \leq 0$$ with the partial Legendre transform $c^*$ as in (Huesmann et al., 2017).

Dual attainment may depend on convexity and regularity in the cost (e.g., pointwise dual maximizers are guaranteed when $c(x, \cdot)$ is $C^{0,2}$ in $y$ and marginals are compactly supported (Beiglboeck et al., 2017)), but can fail for lower regularity or in higher dimensions (March, 2018).

4. Strong Duality, No Duality Gap, and Attainment

The key feature of Martingale KR duality is strong duality (no gap) and optimizer attainment under mild, explicit conditions:

• Convex order and finite (moment) bounds on marginals
• Strict convexity, lower semicontinuity, or $p$-growth in cost functions
• Regularity and boundedness of dual potentials or associated PDE super-solutions

For the Benamou–Brenier formulation, Theorem 4.3 in (Huesmann et al., 2017) asserts: $$BB(\mu, \nu) = FPE(\mu, \nu) = \mathcal D(\mu, \nu) = \sup_{\phi} \left\{ \int \phi(1) \, d\nu - \int \phi(0) \, d\mu \right\},$$ with supremum over bounded $C^{1,2}$ super-solutions of HJB, and both infimum and supremum attained.

Similarly, in the discrete-time KR duality, no gap and optimizer existence follow from Komlós-type compactness and Choquet capacitability arguments (Beiglböck et al., 2015, Dolinsky et al., 2014). The quasi-sure dual formulation is essential to this generality, as classical pointwise duality can fail (Beiglböck et al., 2015, March, 2018).

5. Extensions: Vectorial, Multi-marginal, and Path-space Duality

The framework generalizes to:

• Multi-period: Recursive biconjugate/C-convex transforms for dual potentials (Sester, 2023)
• Vectorial marginals (joint martingale for several assets), with dual potentials indexed by asset/vector coordinates (Lim, 2016)
• Path-dependency: Martingale KR duality for payoffs $G(S)$ or functionals on càdlàg space, with semi-static and dynamic hedging (Dolinsky et al., 2012, Dolinsky et al., 2014, Guo et al., 2015)

Tables organizing dual variable structure:

Dual Problem Type	Static Potential(s)	Dynamic Component
Two-marginal discrete	$\varphi, \psi$	$H(x)$
Continuous Benamou–Brenier	$\phi(t, x)$ HJB	Induced via PDE (Legendre)
Multi-marginal	$\{\phi_i\}$ for each $T_i$	$H_t$ or $\{\Delta_i\}$
Vectorial MOT (VMOT)	$\{\varphi_i,\psi_i,h_i\}$	--

The dynamic component encodes the martingale constraint, which in the dual appears as either a hedging process or HJB-type PDE condition.

6. Relation to Classical KR Duality and Financial Interpretation

Martingale KR duality recovers classical Kantorovich–Rubinstein optimal transport in the absence of the martingale constraint (i.e., setting $H \equiv 0$). In the martingale setting, the dynamic adjustments enforce no arbitrage and yield minimal robust super-hedging cost for path-dependent contingent claims (Dolinsky et al., 2012, Huesmann et al., 2017).

Financially, the duality provides:

• Model-independent bounds for exotic derivative pricing under marginal constraints and no-arbitrage
• Construction of robust semi-static superhedges attaining the minimal replication cost (Dolinsky et al., 2012, Herrmann et al., 2017)
• Calibration and pricing under uncertainty in stochastic volatility models via martingale Schrödinger bridges (Zitridis, 12 Oct 2025)

7. Analytical and Probabilistic Challenges; Open Problems

Martingale KR duality entails technical challenges for existence, regularity, and quasi-sure formulation:

• Pointwise dual attainment requires semiconvexity in cost and may fail below $C^2$ regularity (Beiglboeck et al., 2017, March, 2018)
• Quasi-sure (measure-theoretic) formulation overcomes failure of classical (everywhere) duality (Beiglböck et al., 2015)
• Multi-dimensional and multi-marginal decomposition is required to address "irreducible components," monotonicity sets, and the geometry of optimal transport plans (March, 2018, Lim, 2016)
• Extensions to path-dependent and entropy-penalized transport have been developed (e.g., entropic martingale OT duality) (Doldi et al., 2020)

Further research directions include sharp regularity thresholds, geometric structure of primal optimizers under dual attainment, higher-dimensional duality and capacity theory, and martingale KR duality in robust model-free financial frameworks.