Martingale Benamou–Brenier Formula

• The Martingale Benamou–Brenier formula generalizes classical optimal transport by controlling volatility under martingale constraints, yielding the Bass martingale as the unique solution.
• It leverages convex analytic duality and a static Brenier-type structure to connect weak transport theory, PDE duality, and stochastic control in a unified framework.
• The formulation covers both arithmetic and geometric cases and enables algorithmic approaches like measure-preserving martingale Sinkhorn procedures for optimal dynamic interpolation.

The Martingale Benamou–Brenier formula generalizes the classical Benamou–Brenier dynamic formulation for optimal transport to the martingale-constrained setting. It characterizes the optimal dynamic interpolation between probability measures in convex order by continuous martingale diffusions that are as close as possible to Brownian motion (or geometric Brownian motion in the multiplicative case) subject to prescribed initial and terminal laws. The optimal process is shown to be a Bass martingale—a specific transformation of Brownian motion via the gradient of a convex potential—arising as the unique solution to this variational problem. The theory uses convex analytic duality and a static Brenier-type theorem for martingale weak optimal transport, revealing deep structural connections to weak transport theory, PDE duality, and stochastic control.

1. Primal Dynamic Formulation and Covariance Characterization

Given $\mu,\nu\in\mathcal{P}_2(\mathbb{R}^d)$ satisfying $\mu \preceq \nu$ in convex order, the continuous-time martingale Benamou–Brenier (MBB) problem is to find a law for a martingale $M_t=M_0+\int_0^t \sigma_s\,dB_s$ such that $M_0\sim\mu$, $M_1\sim\nu$ and the following quadratic penalty is minimized: $$MT(\mu,\nu) = \inf \mathbb{E}\left[\int_0^1 \|\sigma_t - I_d\|_{HS}^2\,dt\right].$$ Alternatively, the problem can be formulated as maximizing the averaged instantaneous covariance: $$P(\mu,\nu) = \sup \mathbb{E}\left[\int_0^1 \mathrm{tr}\,\sigma_t\,dt\right].$$ Both formulations have the same unique optimizer, realized by the Bass martingale (Backhoff-Veraguas et al., 2023, Backhoff-Veraguas et al., 2017). This establishes a martingale analogue of the classical energy minimization in dynamic optimal transport, but with volatility (not drift) controlling the “action.”

2. Dual Problem and Dual Attainment

The dual formulation is built via the weak martingale transport (WMT) problem: $$\widetilde{P}(\mu,\nu) = \sup_{\pi\in MT(\mu,\nu)} \int MCov(\pi_x, \gamma)\, \mu(dx),$$ where $MCov(p,\gamma)$ is the maximal covariance over couplings $(p,\gamma)$, with $\gamma$ the standard Gaussian (Backhoff-Veraguas et al., 2023, Beiglböck et al., 27 Jan 2025).

Dual variables are convex functions $\psi:\mathbb{R}^d\to(-\infty,+\infty]$, and the dual value is given by: $$\widetilde{D}(\mu,\nu) = \inf_{\psi\in\mathcal{C}} \left\{ \int\psi\,d\nu - \int\varphi^\psi(x)\,\mu(dx) \right\},$$ with

$$\varphi^\psi(x) = \inf_{p \in P_2^x(\mathbb{R}^d)} \left\{\int\psi\,dp - MCov(p,\gamma)\right\}.$$

No duality gap exists provided $\mu(\mathrm{ri}\,\mathrm{dom}\,\psi)=1$, and dual attainment requires irreducibility of $(\mu,\nu)$. The optimizer can be taken to be l.s.c. and convex (Backhoff-Veraguas et al., 2023).

3. Brenier-Type Structure and Static Solution

The static Brenier-type theorem for weak martingale transport shows that, for convex $\psi$, the maximizer of $MCov(p,\gamma)-\int\psi\,dp$ is $p=\nabla v_{\#}\gamma$ with $v=\psi^*$ convex. More generally,

$$\sup_{p\in P_2(\mathbb{R}^d)} \left[MCov(p,\gamma)-\int\psi\,dp\right] = \int \psi^*\,d\gamma.$$

The optimizer in $P_2^x$ arises as the image of $\gamma$ under the gradient map associated with a tilted convex conjugate (Backhoff-Veraguas et al., 2023). The proof involves sequential dualizations—a Kantorovich duality for covariance, followed by Fenchel–Moreau in $x$—plus convolution identities ensuring uniqueness.

4. Bass Martingale: Construction and Characterization

Bass martingales are Markovian processes constructed from Brownian motion and convex potentials: $$M_t = \mathbb{E}\left[\nabla v(B_1) | B_t\right], \quad 0\le t\le 1,$$ where $B_0\sim\alpha$ and $v$ depends on a dual optimizer $\psi$ via $v=\psi^*$. The optimal martingale $M^*$ that solves (MBB) is uniquely a Bass martingale, with

$$M_t = \nabla(v*\gamma^{1-t})(B_t), \quad B_0 \sim (\nabla(v*\gamma))^{-1}_{\#}\mu,$$

and the transport plan disintegrates as

$$\pi^*_x = \nabla v\left(\zeta(x)+\,\cdot\,\right)_{\#}\gamma, \quad \zeta(x) = (\nabla v*\gamma)^{-1}(x).$$

These martingales interpolate between prescribed laws in convex order and recover classical models (Brownian, geometric Brownian, Bass embedding) as extremal cases (Backhoff-Veraguas et al., 2023, Backhoff-Veraguas et al., 2017).

5. Regularity, Irreducibility, and Dual Attainment

Dual attainment for the MBB problem is delicate: the dual optimizer $\psi$ may not be integrable under $\nu$. Extended convexity hypotheses and compactness arguments guarantee existence only if $(\mu,\nu)$ is irreducible (i.e., its De March–Touzi cell is the relative interior of the convex support of $\nu$). The necessity relates to the connectivity by Bass martingales, while the sufficiency uses measurable selection and Komlós-type limiting procedures (Backhoff-Veraguas et al., 2023).

6. Comparative and Geometric Extensions

Both the arithmetic (Brownian reference) and geometric (Black–Scholes reference) versions fit into the MBB framework. In the geometric case, one seeks martingales $S_t$ satisfying

$dS_t = \sigma_t S_t\,dB_t$

with volatility penalized from a constant, or equivalently, maximizes average log-volatility. There exists a bijection between geometric and arithmetic Bass martingales via reciprocal change of measure, preserving optimizer uniqueness and allowing explicit SDE representations (Backhoff et al., 6 Jun 2024). Duality involves a nonlinear Hamilton–Jacobi PDE in log-coordinates, connecting the process dynamics to feedback control derived from its solution.

Classical Benamou–Brenier minimizes drift-based kinetic energy, while MBB controls volatility or log-volatility under martingale constraints, with static duality given by maximal covariance or its transformation through reciprocal mappings (Backhoff et al., 6 Jun 2024, Backhoff-Veraguas et al., 2017).

7. Technical Challenges and Algorithmic Developments

Key technical challenges include failure of dual attainment in general, resolution by convexification and compactness techniques, and detailed structural finds for static/dynamic optimizers. The time-consistency property (constant-speed martingale geodesics) mirrors the Wasserstein geodesics of classical theory, and interpolations extend to local volatility limits and extensions of Kellerer’s theorem.

Algorithmically, the dual structure enables measure-preserving martingale Sinkhorn procedures converging monotonically to the optimal dual, analogous to the entropic Sinkhorn algorithm in Schrödinger bridge problems. The iterative renormalization alternates between forward/backward heat equations and push-forward normalizations (Joseph et al., 2023).

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The Martingale Benamou–Brenier formula thus provides a rigorous dynamical foundation for martingale optimal transport, extends weak optimal transport through convex duality and static structure theorems, and interconnects variational principles, stochastic processes, and PDE duality. Its canonical optimal process is the Bass martingale, which interpolates laws in convex order by minimal deviation from Brownian reference dynamics. The theory is robust to extensions—including geometric analogues and relaxed barycentric transport—providing a unified paradigm for continuous-time stochastic transport under martingale constraints (Backhoff-Veraguas et al., 2023, Beiglböck et al., 27 Jan 2025, Guo et al., 26 Nov 2025, Huesmann et al., 2017, Backhoff-Veraguas et al., 2017, Backhoff et al., 6 Jun 2024, Joseph et al., 2023).