---
title: Geometric Bass Martingales
url: https://www.emergentmind.com/topics/geometric-bass-martingales
type: topic
---

# Geometric Bass Martingales

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 $L^2$ sense [2406.04016].

- **Geometric mBB**: For strictly positive laws $\mu_0, \mu_1$ in convex order and reference volatility $\bar\sigma>0$, one considers positive continuous martingales $S_t = S_0 + \int_0^t \sigma_s S_s\,dB_s$ with $S_0 \sim \mu_0, S_1 \sim \mu_1$, minimizing  
  $$
  \int_0^1 (\sigma_t - \bar\sigma)^2\,dt.
  $$
  Here the interpolant is a geometric Bass martingale: the unique martingale whose log-quadratic variation is as close as possible to linear in $L^2$ and which matches the prescribed laws at times $0$ and $1$ [2406.04016, 1708.04869].

An arithmetic Bass martingale is of the form $M_t = F(t, B_t)$ for an increasing function $F$ and Brownian motion $B$, with $F(t,\cdot) = F*\gamma_{1-t}$. Analogously, a geometric Bass martingale satisfies $\log S_t = G(t, W_t)$ for an increasing $G$ and Brownian motion $W$, again with $G(t,\cdot) = G*\gamma_{1-t}$.

## 2. Bijection Between Arithmetic and Geometric Bass Martingales

A central result is the explicit and invertible correspondence between arithmetic and geometric Bass martingales [2406.04016]. Given a maximizer $S$ for the geometric problem, define its mean $m = \int x\,\mu_0(dx) = \int x\,\mu_1(dx)$ and set the reflected marginals $\nu_i = Id_\ddag \mu_i$. Under the measure $\widetilde{\mathbb P} = (S_1/m)\,\mathbb P$, the process $R_t = m/S_t$ is a martingale associated to the arithmetic problem with marginals $\nu_0, \nu_1$, and its volatility process $\Sigma_t = R_t \sigma_t$ [2406.04016, 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 [2406.04016, Theorem 3.2]:

- Let $F$ and $\alpha$ denote the generating function and Bass measure of the arithmetic optimizer; then the associated geometric Bass martingale $S$ is characterized, for any bounded measurable $g\colon C([0,1];\mathbb R)\rightarrow\mathbb R_+$, by
  $$
  \mathbb E\left[g(S)\right] = \mathbb E\left[g\left(m / F(t, B_t)\right)\, F(1, B_1)\right],
  $$
  thus determining all finite-dimensional distributions.

The SDE satisfied by $S$ (on an irreducible component $J \subset (0,\infty)$) is
$$
dS_t = S_t\, \frac{S_t / m}{\partial_x[F_J^{-1}](t, m / S_t)}\, dB_t, \quad S_0 \in J,
$$
where $F_J$ is the restriction of $F$ to $I=1/J$ [2406.04016, 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 $M_t=F(t,B_t)$ and $S_t=G(t,W_t)$ and $M \equiv S$ in law, then $F(t,x) = \exp(a x - \tfrac{a^2}{2} t)$ for some $a\in \mathbb R$, leading to $d S_t = a S_t dB_t$ [2406.04016, 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 [2406.04016, Corollary 3.3]. Writing $\nu_i = Id_\ddag \mu_i$,
$$
\mathbf{GP}_{\mu_0,\mu_1} = \inf_{\psi\ \text{convex}}\Bigl[\int \psi\,d\nu_1 - \int (\psi^*\ast\gamma_1)^*\,d\nu_0\Bigr],
$$
where $MC(\cdot,\cdot)$ is the maximal covariance.

A dynamic programming/Hamilton–Jacobi–Bellman PDE for the dual potential $u(t,s)$ is
$$
\sup_{\sigma\ge0}\left\{\sigma s u_s + \tfrac12\sigma^2 s^2 u_{ss} - (\sigma-\bar\sigma)^2\right\} + u_t = 0,
$$
with prescribed Cauchy data. The optimal $\sigma$ leads to the equivalent fully nonlinear equation
$$
u_t + \frac{\bar\sigma^2}{2} \frac{s^2 u_{ss}}{1-s^2 u_{ss}} = 0, \qquad 1 - s^2 u_{ss} > 0,
$$
and the optimal volatility is $\sigma(t,s) = \bar\sigma / (1-s^2 u_{ss})$ [2406.04016, 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 [1708.04869].

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 [2406.04016].
- 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 [2406.10656].
- The Bass martingale structure is preserved under both convexity and variational minimization frameworks developed in martingale optimal transport [2309.11181, 2406.10656].

## 7. Related Developments and Open Questions

Discrete-time analogues, such as $q$-Bass martingales, further generalize the construction by replacing the Gaussian kernel (Brownian motion) with arbitrary reference measures $q$. In the context of a geometric law $q$, 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 [2402.05669]. The extension to fully continuous-time geometric Bass martingales for general $q$ 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" [2406.04016]  
- "Martingale Benamou–Brenier: a probabilistic perspective" [1708.04869]  
- "The Bass functional of martingale transport" [2309.11181]  
- "The decomposition of stretched Brownian motion into Bass martingales" [2406.10656]  
- "$q$-Bass martingales" [2402.05669]

Source: https://www.emergentmind.com/topics/geometric-bass-martingales