Geometric Bass Martingales

• Geometric Bass martingales are continuous-time positive martingales on [0,1] with prescribed log-normal marginals, serving as the multiplicative analogue of arithmetic Bass martingales.
• They are derived as the unique solution to a martingale optimal transport problem, optimizing an averaged L2 deviation from a geometric Brownian motion.
• Their explicit SDE formulation and bijective correspondence with arithmetic counterparts offer actionable insights for stochastic analysis and mathematical finance.

A geometric Bass martingale is a continuous-time positive martingale on $[0,1]$ with prescribed initial and terminal marginals on $(0,\infty)$, characterized as the unique solution to a martingale optimal transport problem that is the multiplicative analogue of the arithmetic Bass martingale. The geometric Bass martingale is defined to be as close as possible (in an averaged $L^2$ sense) to a geometric Brownian motion, mirroring in the multiplicative setting the role of Brownian motion in the additive (arithmetic) case. The theory establishes a precise correspondence between geometric Bass martingales and their arithmetic counterparts, enabling the transfer of structural and variational properties across both settings. These constructions provide fundamental martingale interpolants with applications in stochastic analysis, mathematical finance, and optimal transport.

1. Martingale Benamou–Brenier Problem and Bass Martingales

The martingale Benamou–Brenier (mBB) framework generalizes classical optimal transport to martingales with prescribed marginals. Two main cases are distinguished:

• Arithmetic (additive) mBB: Given laws $\nu_0, \nu_1$ on $\mathbb{R}$ in convex order with finite second moments and reference volatility $\bar\Sigma > 0$, one considers continuous martingales of the form $M_t = M_0 + \int_0^t \Sigma_s\,dB_s$, with $M_0 \sim \nu_0$, $M_1 \sim \nu_1$, minimizing the cost functional

$\int_0^1 (\Sigma_t - \bar\Sigma)^2\,dt.$

The optimal process, called an arithmetic Bass martingale (or stretched Brownian motion), is the martingale which, among all those with the given marginals, has quadratic variation as close as possible to a deterministic linear path in $(0,\infty)$0 sense (Backhoff et al., 2024).

• Geometric mBB: For strictly positive laws $(0,\infty)$1 in convex order and reference volatility $(0,\infty)$2, one considers positive continuous martingales $(0,\infty)$3 with $(0,\infty)$4, minimizing

$(0,\infty)$5

Here the interpolant is a geometric Bass martingale: the unique martingale whose log-quadratic variation is as close as possible to linear in $(0,\infty)$6 and which matches the prescribed laws at times $(0,\infty)$7 and $(0,\infty)$8 (Backhoff et al., 2024, Backhoff-Veraguas et al., 2017).

An arithmetic Bass martingale is of the form $(0,\infty)$9 for an increasing function $L^2$0 and Brownian motion $L^2$1, with $L^2$2. Analogously, a geometric Bass martingale satisfies $L^2$3 for an increasing $L^2$4 and Brownian motion $L^2$5, again with $L^2$6.

2. Bijection Between Arithmetic and Geometric Bass Martingales

A central result is the explicit and invertible correspondence between arithmetic and geometric Bass martingales (Backhoff et al., 2024). Given a maximizer $L^2$7 for the geometric problem, define its mean $L^2$8 and set the reflected marginals $L^2$9. Under the measure $\nu_0, \nu_1$0, the process $\nu_0, \nu_1$1 is a martingale associated to the arithmetic problem with marginals $\nu_0, \nu_1$2, and its volatility process $\nu_0, \nu_1$3 [(Backhoff et al., 2024), Theorem 3.1]. This provides a bijection: each geometric Bass martingale corresponds to an arithmetic Bass martingale (and vice versa) via this change of measure and transformation.

Uniqueness in distribution for geometric Bass martingales directly follows from the uniqueness of the arithmetic Bass martingale optimizer under irreducibility.

3. Explicit Representations and Stochastic Differential Equations

In the irreducible case (i.e., initial and final marginals cannot be decomposed further in convex order), a full explicit description is available [(Backhoff et al., 2024), Theorem 3.2]:

• Let $\nu_0, \nu_1$4 and $\nu_0, \nu_1$5 denote the generating function and Bass measure of the arithmetic optimizer; then the associated geometric Bass martingale $\nu_0, \nu_1$6 is characterized, for any bounded measurable $\nu_0, \nu_1$7, by

$\nu_0, \nu_1$8

thus determining all finite-dimensional distributions.

The SDE satisfied by $\nu_0, \nu_1$9 (on an irreducible component $\mathbb{R}$0) is

$\mathbb{R}$1

where $\mathbb{R}$2 is the restriction of $\mathbb{R}$3 to $\mathbb{R}$4 [(Backhoff et al., 2024), Proposition 5.3].

The only process which is both an arithmetic and a geometric Bass martingale is geometric Brownian motion with log-normal marginals. Specifically, if both $\mathbb{R}$5 and $\mathbb{R}$6 and $\mathbb{R}$7 in law, then $\mathbb{R}$8 for some $\mathbb{R}$9, leading to $\bar\Sigma > 0$0 [(Backhoff et al., 2024), Proposition 5.4].

4. Duality and PDE Formulation

The geometric martingale Benamou–Brenier (G-mBB) problem admits a convex dual representation directly analogous to Kantorovich duality [(Backhoff et al., 2024), Corollary 3.3]. Writing $\bar\Sigma > 0$1,

$\bar\Sigma > 0$2

where $\bar\Sigma > 0$3 is the maximal covariance.

A dynamic programming/Hamilton–Jacobi–Bellman PDE for the dual potential $\bar\Sigma > 0$4 is

$\bar\Sigma > 0$5

with prescribed Cauchy data. The optimal $\bar\Sigma > 0$6 leads to the equivalent fully nonlinear equation

$\bar\Sigma > 0$7

and the optimal volatility is $\bar\Sigma > 0$8 [(Backhoff et al., 2024), Section 4]. The optimal process is then recovered from the dual via inversion and Legendre transformation.

5. Geometric Bass Martingales in the Context of Martingale Optimal Transport

Geometric Bass martingales naturally realize the multiplicative structure in martingale optimal transport. They provide the martingale which is most similar to geometric Brownian motion among those matching prescribed initial and terminal marginals (in the sense of quadratic logarithmic variation). This mirrors the arithmetic Bass martingale's role as the closest martingale to Brownian motion in the additive structure (Backhoff-Veraguas et al., 2017).

These martingales exhibit time-consistent interpolation (martingale displacement interpolation), covariance maximization with respect to geometric Brownian motion, and explicit Markovian dynamics governed by a unique SDE with a time-dependent, law-determined volatility coefficient. Their construction extends the Benamou–Brenier transport theory into the multiplicative/martingale setting, offering a convex-analytic and PDE-based toolkit for problems in stochastic analysis and mathematical finance.

6. Structural and Uniqueness Results

A key structural property is the bijective correspondence between arithmetic and geometric Bass martingales, allowing explicit inheritance of regularity, uniqueness, and optimality properties from the arithmetic setting. In particular:

• Geometric Brownian motion is the unique process that is simultaneously arithmetic and geometric Bass martingale, and it is characterized by log-normal marginal laws (Backhoff et al., 2024).
• For irreducible marginals, the geometric Bass martingale is unique in law. For general marginals, a decomposition into irreducible components is possible, with each component supporting its own (local) Bass martingale (Schachermayer et al., 2024).
• The Bass martingale structure is preserved under both convexity and variational minimization frameworks developed in martingale optimal transport (Backhoff-Veraguas et al., 2023, Schachermayer et al., 2024).

7. Related Developments and Open Questions

Discrete-time analogues, such as $\bar\Sigma > 0$9-Bass martingales, further generalize the construction by replacing the Gaussian kernel (Brownian motion) with arbitrary reference measures $M_t = M_0 + \int_0^t \Sigma_s\,dB_s$0. In the context of a geometric law $M_t = M_0 + \int_0^t \Sigma_s\,dB_s$1, the so-called "geometric Bass martingale" associated with geometric random walks has been defined and partially characterized, though in these settings, only two-point (time-0 and time-1) bridges are straightforwardly described (Tschiderer, 2024). The extension to fully continuous-time geometric Bass martingales for general $M_t = M_0 + \int_0^t \Sigma_s\,dB_s$2 remains an open problem.

Research continues on structural, variational, and practical aspects of geometric Bass martingales, with ongoing connections to convex analysis, stochastic control, parabolic PDEs, and mathematical finance.

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Key References:

• "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales" (Backhoff et al., 2024)
• "Martingale Benamou–Brenier: a probabilistic perspective" (Backhoff-Veraguas et al., 2017)
• "The Bass functional of martingale transport" (Backhoff-Veraguas et al., 2023)
• "The decomposition of stretched Brownian motion into Bass martingales" (Schachermayer et al., 2024)
• "$M_t = M_0 + \int_0^t \Sigma_s\,dB_s$3-Bass martingales" (Tschiderer, 2024)