q-Bass Martingales: Theory and Applications

• q-Bass martingales are canonical objects in martingale optimal transport that generalize the classical Bass martingale by using arbitrary reference measures.
• They are constructed via convex duality and optimal transport couplings that maximize cross-covariance with a chosen measure q under martingale constraints.
• They provide a unifying interpolation between deterministic, entropic, and martingale transports, with applications in stochastic control and financial modelling.

A $q$-Bass martingale is a canonical object in martingale optimal transport, generalizing the classical Bass martingale by replacing the standard Gaussian reference kernel with an arbitrary measure $q$. This construction yields a martingale (in discrete or continuous time) that "mimics" the behavior of a reference law $q$ as closely as possible, given prescribed initial and terminal marginals, and subject to the martingale constraint. The $q$-Bass martingale framework includes, as special cases, both the classical Bass martingale (when $q$ is Gaussian) and processes arising from alternative reference laws such as $q$-Gaussians, $q$-Hermite, or more general non-Gaussian distributions. This notion provides a central unifying structure in martingale transport, geometric and arithmetic settings, and represents a fundamental interpolation between classical deterministic transport, stochastic (Schrödinger bridge) transport, and martingale transport.

1. Variational Definition and Martingale Optimal Transport

Let $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$ be Borel probability measures with finite second moments, and suppose $\mu\preceq_c\nu$ (convex order). The $q$-martingale optimal transport problem seeks a martingale coupling $q$0 between $q$1 and $q$2 which at each initial point maximizes the maximal cross-covariance with a reference measure $q$3: $q$4 where $q$5 and $q$6 denotes the set of couplings with $q$7, $q$8 such that the disintegration $q$9 satisfies $q$0 for $q$1-almost every $q$2 (Tschiderer, 2024).

The $q$3-Bass martingale is the canonical stochastic process that, starting at $q$4 and ending at $q$5, realizes this optimal transport in the sense that its transitions are as close as possible to the reference law $q$6 at each step, subject to the martingale condition.

2. Dual Formulation and Characterization via Convex Potentials

The dual of the $q$7-martingale transport problem takes the form: $q$8 where $q$9 is the convex conjugate, $q$0, and

$q$1

The double conjugate $q$2 is then taken back in the $q$3 variable. There is no duality gap: $q$4 (Tschiderer, 2024).

At the optimizer, one finds that the subgradients of $q$5 or its analogues yield the transport maps for the $q$6-Bass martingale. This duality is the convex-analytic core of the $q$7-Bass construction.

3. Explicit Construction of Discrete-Time $q$8-Bass Martingales

A pair $q$9 with $q$0 convex and $q$1 is a $q$2-Bass martingale if: $q$3 where $q$4 and $q$5 denotes the convolution of measures (Tschiderer, 2024).

Operationally, this structure allows for the "lifting" of arbitrary reference kernels into the martingale setting: sample $q$6, $q$7, set $q$8, and choose $q$9 so that the marginal law at time $q$0 (as the law of $q$1) matches $q$2, and at time $q$3 matches $q$4. This process thus possesses the property that its transitions are as $q$5-like as allowed by the martingale and marginal constraints.

4. Existence and Sufficient Conditions

Existence of $q$6-Bass martingales is governed by the notion of a $q$7-Bass-generating function [Definition 2.4 in (Tschiderer, 2024)]: a convex function $q$8 is $q$9-Bass-generating for $q$0 if

• $q$1,
• $q$2 is finite, strictly convex, and $q$3 everywhere,
• the terminal law $q$4 has finite second moment.

Under these conditions, Theorem 1.5 in (Tschiderer, 2024) guarantees that the pair $q$5 (with $q$6) defines a legitimate $q$7-Bass martingale, and furthermore that $q$8 realizes the optimizer of the $q$9-martingale transport problem.

5. Special Cases: Gaussian, $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$0-Hermite, and Geometric Bass Martingales

• For $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$1 equal to the standard Gaussian, the $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$2-Bass martingale reduces to the classical Bass construction. The time-continuous process $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$3 gives the unique Markov optimizer of the Benamou–Brenier martingale transport problem (Backhoff-Veraguas et al., 2017, Backhoff-Veraguas et al., 2023).
• For reference measures $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$4 corresponding to $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$5-Hermite or $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$6-Gaussian distributions, the $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$7-Bass martingale connects to quadratic harnesses and $\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$8-deformed Wiener/Bass martingales (Szabłowski, 2012). In the "pure-$\mu,\nu,q\in\mathcal{P}_2(\mathbb{R}^d)$9" case (parameters $\mu\preceq_c\nu$0), the relevant three-term OMP recurrence coefficients encode the $\mu\preceq_c\nu$1-Hermite structure, generating martingales with compactly supported $\mu\preceq_c\nu$2-Gaussian marginals.
• In the geometric setting, the "geometric Bass martingale" (sometimes denoted as "$\mu\preceq_c\nu$3-Bass" in financial mathematics, but not to be confused with the $\mu\preceq_c\nu$4-deformation above) is defined as the process that is as close as possible to a geometric Brownian motion in the sense of minimizing the $\mu\preceq_c\nu$5-distance between the instantaneous volatility of the log-process and a constant benchmark (Backhoff et al., 2024). There exists a precise bijection between the arithmetic and geometric Bass problems via reciprocal-reflected marginal modifications, and the unique process that is both an arithmetic and a geometric Bass martingale is geometric Brownian motion.

6. Relation to Schrödinger–Bass Bridges and Interpolating Flows

The $\mu\preceq_c\nu$6-Bass martingale emerges naturally in the stochastic control problem that interpolates between the Schrödinger bridge (entropic optimal transport with fixed drift) and the Bass martingale (martingale transport with volatility penalty). The cost functional

$\mu\preceq_c\nu$7

with $\mu\preceq_c\nu$8 yields a continuum between the Bass (martingale) and Schrödinger (entropic) regimes as $\mu\preceq_c\nu$9 and $q$0 respectively (Henry-Labordere et al., 29 Mar 2026, Alouadi et al., 25 Jan 2026). The solution structure involves time-dependent transport via the gradient of a $q$1-convex potential, and the optimal process can be represented as a "stretched Schrödinger bridge"—the image of the base Schrödinger bridge process under a transport map derived from $q$2-Bass duality.

The time-marginals and path laws satisfy coupled forward-backward PDEs (heat equations and Hamilton–Jacobi–Bellman equations), and the entire system interpolates between classical, entropic, and martingale optimal transport.

7. Applications and Significance

$q$3-Bass martingales enrich the theory of martingale optimal transport by providing canonical Markovian representatives whose transitions are as close as possible (in a maximal cross-covariance sense) to a chosen reference law $q$4. They subsume the classical Bass martingales as well as variants connected to $q$5-Hermite and $q$6-Gaussian frameworks, and play a central role in the decomposition of stretched Brownian motion into irreducible Bass components (Schachermayer et al., 2024). The geometric and $q$7-deformed extensions further enhance their scope, with direct connections to local volatility approximations, causal transport, and applications in mathematical finance.

Through their characterization via convex-analytic duality, explicit construction via convolution and gradient flows, and interpolation role between fundamental types of stochastic transport, $q$8-Bass martingales serve both as crucial theoretical archetypes and as practical tools for the analysis of optimal interpolations between probability measures under martingale constraints (Tschiderer, 2024, Backhoff-Veraguas et al., 2017, Henry-Labordere et al., 29 Mar 2026, Backhoff et al., 2024, Szabłowski, 2012).