# Geometric Martingale Benamou-Brenier transport and geometric Bass
martingales

Julio Backhoff-Veraguas111University of Vienna Gregoire Loeper222BNP Paribas
Jan Obłój333University of Oxford

###### Abstract

We introduce and study _geometric_ Bass martingales. Bass martingales were
introduced in [6] and studied recently in a series of works, including [3, 4],
where they appear as solutions to the martingale version of the Benamou-
Brenier optimal transport formulation. These _arithmetic_ , as well as our
novel _geometric_ , Bass martingales are continuous martingale on $[0,1]$ with
prescribed initial and terminal distributions. An arithmetic Bass martingale
is the one closest to Brownian motion: its quadratic variation is as close as
possible to being linear in the averaged $L^{2}$ sense. Its geometric
counterpart we develop here, is the one closest to a geometric Brownian
motion: the quadratic variation of its logarithm is as close as possible to
being linear. By analogy between Bachelier and Black-Scholes models in
mathematical finance, the newly obtained _geometric Bass martingales_ have the
potential to be of more practical importance in a number of applications.

Our main contribution is to exhibit an explicit bijection between geometric
Bass martingales and their arithmetic counterparts. This allows us, in
particular, to translate fine properties of the latter into the new geometric
setting. We obtain an explicit representation for a geometric Bass martingale
for given initial and terminal marginals, we characterise it as a solution to
an SDE, and we show that geometric Brownian motion is the only process which
is both an arithmetic and a geometric Bass martingale. Finally, we deduce a
dual formulation for our geometric martingale Benamou-Brenier problem. Our
main proof is probabilistic in nature and uses a suitable change of measure,
but we also provide PDE arguments relying on the dual formulation of the
problem, which offer a rigorous proof under suitable regularity assumptions.

##  1 Introduction

Optimal transport refers to a wide range of problems all concerned with
constructing couplings of measures with optimal properties. It is an area of
mathematics with a long history, finding applications across a wide spectrum
of fields.

A particularly interesting way of constructing optimal couplings derives from
a fluid mechanics perspective, known as the Benamou-Brenier formulation of
optimal transport [10]: consider a fluid moving with time, driven by an
unknown velocity field, starting with a given mass distribution $\nu_{0}$. The
problem is to find the velocity having the smallest average kinetic energy (in
other words, least action) such that the final distribution of mass is
$\nu_{1}$. Mathematically this amounts to solving the variational problem

$\displaystyle\inf_{\begin{subarray}{c}X_{0}\sim\nu_{0},X_{1}\sim\nu_{1}\\\
X_{t}=X_{0}+\int_{0}^{t}V_{s}ds\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}|V_{t}|^{2}dt\right].$
(1.1)

Remarkably, [10] showed that this time-continuous problem is in fact
equivalent to a static one of minimising the average squared distance among
couplings of $\nu_{0}$ and $\nu_{1}$. In this way, the marginal distributions
$\\{\text{Law}(X_{t})\\}_{t\in[0,1]}$ of the continuously moving fluid trace
the geodesic connecting $\nu_{0}$ and $\nu_{1}$ in the space of measures,
endowed with the celebrated 2-Wasserstein distance.

More recently, [3] considered the martingale analogue of this problem, which
when specialized to the one-dimensional setting becomes:

$\displaystyle\inf_{\begin{subarray}{c}M_{0}\sim\nu_{0},M_{1}\sim\nu_{1}\\\
M_{t}=M_{0}+\int_{0}^{t}\Sigma_{s}dB_{s}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}(\Sigma_{t}-\bar{\Sigma})^{2}dt\right],$
(A-mBB)

where the optimisation is taken over filtered probability spaces with a
Brownian motion $(B_{t})_{t\geq 0}$, possibly starting from a non-trivial
position $B_{0}$. By the seminal results of [17, 16], continuous real-valued
martingales are all time-changes of a Brownian motion and are thus
fundamentally characterised by their quadratic variation. Here,
$\nu_{0},\nu_{1}$ are given, in convex order and with finite second moments.
Problem (A-mBB) looks for a particle evolving as a martingale which is the
closest to the constant volatility martingale: a constant $\bar{\Sigma}$
multiple of a Brownian motion. This problem mas been dubbed martingale Benamou
Brenier problem in [3]. To stress that the reference process is the
_arithmetic_ Brownian motion, we refer to it as the _arithmetic_ -mBB problem.
This problem was studied in detail in [3, 4], who show in particular that
(A-mBB) admits a unique (in distribution) optimiser, known as the _stretched
Brownian motion_ from $\mu_{0}$ to $\mu_{1}$. Under mild regularity
assumption, known as irreducibility and explained in Section 2 below, the
optimiser is further shown to be a _Bass martingale_. This means in particular
that $M_{t}=f(t,B_{t})$, for all $t$, where $f(t,\cdot)$ is non-decreasing;
see Definition 6.1 below. Further, they also show that this continuous time
problem is equivalent to a static _weak_ -OT problem in the sense of [19]. We
refer to [6] for a classical application of Bass martingales to the Skorokhod
embedding problem and to [28] for a much more recent application concerning
Kellerer’s theorem.

Martingales, and diffusion processes in particular, are a backbone of
mathematical finance: they describe the dynamics of risky assets under a
pricing measure. Selecting a model involves its calibration, a process
ensuring that model prices match the observed market prices. The canonical
example of a calibration problem is that of matching European call and put
prices. This, via the classical argument of [11], is equivalent to matching
marginal distributions at some fixed times, the maturities. Finding robust
bounds on prices of an exotic option then corresponds to minimising, or
maximising, the expectation of a certain path functional over all such
martingales. Starting with [23] this observation underpinned new interplay
between Skorokhod embeddings and robust finance, e.g., [12, 15], and
subsequently led to the introduction of _martingale optimal transport_ in [7,
18] and the ensuing rapid and rich growth of this field. More recently,
optimal transport techniques have also been used as means for non-parametric
calibration: OT is used as a means to project one’s favourite model onto the
set of calibrated martingales, i.e., martingales which satisfy a set of given
distributional constraints, see [22, 21]. In general, this OT-calibration
problem is solved via its dual, numerically optimizing over solutions to a
non-linear PDE, which can be challenging. The Bass martingale can be seen as a
particular case of the OT-calibration problem, but one which can be reduced to
a static problem, and is hence much easier to solve; see Section 8 below. The
main drawback of the resulting solution is that the _arithmetic_ Brownian
motion is not a desirable model for risky assets. Instead, the _geometric_
Brownian motion is, and in mathematical finance one quantifies the variability
of a model using the quadratic variation of the logarithm of the price
process.

Motivated by the above remarks, we consider a _geometric_ version of the
martingale Benamou-Brenier problem. We suppose $\mu_{0},\mu_{1}$ are supported
on $(0,+\infty)$ and study the problem

$\displaystyle\mathbf{GmBB}_{\mu_{0},\mu_{1}}$
$\displaystyle=\inf_{\textstyle\begin{subarray}{c}S_{0}\sim\mu_{0},S_{1}\sim\mu_{1}\\\
S_{t}=S_{0}+\int_{0}^{t}\sigma_{u}S_{u}dB_{u}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}(\sigma_{t}-\bar{\sigma})^{2}dt\right],$
(G-mBB)

where the optimisation is taken over filtered probability spaces
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ with a
Brownian motion $(B_{t})_{t\geq 0}$. We will show that a suitable change of
measure argument allows to build a 1-1 relationships between (G-mBB) for
$(\mu_{0},\mu_{1})$ and the (A-mBB) for a different set of marginals
$(\nu_{0},\nu_{1})$. This, in particular, yields uniqueness (in distribution)
of the solution to (G-mBB). We will present a probabilistic proof of the
equivalence between the two problems, as well as a PDE motivation behind it.
Furthermore, the 1-1 relationship between arithmetic and geometric problems
means that we can efficiently translate results from the arithmetic setting
into the geometric one. For instance [3, 4] explore in detail the fine
structure of the optimiser of Problem (A-mBB), [14, 2, 24] propose a numerical
method for it, and [8] establishes its stability w.r.t. perturbation of the
data of the problem. We will show in Section 6 what this implies for the
structure of our geometric optimiser and we will explain in Section 8 how a
numerical method for the geometric setting looks like.

We stress that the idea of transforming one martingale transport problem into
another via a change of measure, but in discrete time, was pioneered by Campi,
Laachir and Martini in [13]. This is a particular case of the well-known
change of numeraire argument. We will see how this same idea is fruitful in
our continuous time setting as well.

##  2 Preliminaries

We let ${\mathbb{R}}_{+}=(0,+\infty)$ and denote
$\mathcal{P}_{r,p}({\mathbb{R}}_{+})$ for $r<0<p$ the set of probability
measures on ${\mathbb{R}}_{+}$ for which
$\int_{0}^{\infty}|x|^{s}\mu(dx)<\infty$ for $r\leq s\leq p$. The push-forward
operator is denoted with a subscript $\\#$, i.e., for a function
$G:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ and probability measures $\mu$ and
$\nu$, we write

$G_{\\#}\mu=\nu\Leftrightarrow\nu=\mu\circ
G^{-1}\Leftrightarrow\forall\Gamma\in\mathcal{B}({\mathbb{R}}_{+}),\nu(\Gamma)=\mu(G^{-1}(\Gamma)).$

For a $\mu$-integrable $f:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$, we consider
the $f$-reflected measure

$\textstyle
f_{\dagger}\mu=\left(y\to\frac{1}{f(y)}\right)_{\\#}\left(\frac{f(y)}{\int
f(x)\mu(dx)}\mu(dy)\right),$

that is we define a new probability measure with density proportional to $f$
w.r.t. $\mu$ and then consider its pushforward using $1/f$. We will be mostly
interested in the case $f(y)=y$, in which case the measure resulting from the
above is denoted $Id_{\dagger}\mu$. Note that $Id_{\dagger}\cdot$ reflects
moments and is an involution: $Id_{\dagger}(Id_{\dagger}\mu)=\mu$ for
$\mu\in\mathcal{P}_{-1,2}({\mathbb{R}}_{+})$.

It is immediate that (G-mBB) is invariant under multiplicative scaling: the
value remains the same and the optimisers are constant multiples of each
other, see Remark 3.4 below. It is thus useful to normalise probability
measures and for a $\mu\in\mathcal{P}_{0,1}({\mathbb{R}}_{+})$ we let
$\tilde{\mu}:=(x\to x/m)_{\\#}\mu$, where $m=\int x\mu(dx)$. In this way,
$\int x\tilde{\mu}(dx)=1$. We let

$Id_{\ddagger}\mu:=Id_{\dagger}\tilde{\mu}=\widetilde{Id_{\dagger}\mu}.$

We say that $\eta,\rho\in\mathcal{P}_{0,1}({\mathbb{R}}_{+})$ are in convex
order, and write $\eta\preccurlyeq_{cx}\rho$, if $\int f(x)\eta(dx)\leq\int
f(x)\rho(dx)$ for all convex functions
$f:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$. In fact, working on ${\mathbb{R}}$,
it is enough to check the inequality for a representative family of convex
functions, which allows to reconstruct other convex functions as their
weighted averages or integrals. Specifically, $\eta\preccurlyeq_{cx}\rho$ if
and only if $U_{\eta}\leq U_{\rho}$, pointwise on ${\mathbb{R}}$, where
$U_{\rho}(z):=\int_{\mathbb{R}}|x-z|\rho(dx)$ is known as the potential of
$\rho$. As potential functions are continuous, the set
$\\{U_{\eta}<U_{\rho}\\}$ is open and hence equal to an at most countable
union of open maximal intervals. We write $\mathcal{I}_{[\eta,\rho]}$ for the
collection of these intervals and notice that each such intervals are
contained in ${\mathbb{R}}_{+}$. We say that $\eta,\rho$ are irreducible if
$\mathcal{I}_{[\eta,\rho]}$ contains a single interval $I$. In this case,
without any loss of generality, we may and will assume that
$\eta({\mathbb{R}}\setminus I)=0$. We refer to [27, 9] for further details on
potentials and their applications in Skorokhod embeddings and in martingale
optimal transport.

######  Lemma 2.1.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$ and
$\nu_{i}:=Id_{\dagger}\mu_{i}$, $i=0,1$. Then
$\nu_{0},\nu_{1}\in\mathcal{P}_{0,2}({\mathbb{R}}_{+})$ and

$\mu_{0}\preccurlyeq_{cx}\mu_{1}\iff\nu_{0}\preccurlyeq_{cx}\nu_{1}.$

In case $\mu_{0}\preccurlyeq_{cx}\mu_{1}$, we have
$I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}\iff\left\\{\frac{1}{x}:x\in
I\right\\}\in\mathcal{I}_{[\mu_{0},\mu_{1}]},$ and conversely
$J\in\mathcal{I}_{[\mu_{0},\mu_{1}]}\iff\left\\{\frac{1}{x}:x\in
J\right\\}\in\mathcal{I}_{[\nu_{0},\nu_{1}]}$.

###### Proof.

That $\nu_{0},\nu_{1}\in\mathcal{P}_{0,2}({\mathbb{R}}_{+})$ if
$\mu_{0},\mu_{1}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$, is immediate. Let
$m_{i}=\int x\mu_{i}(dx)$. Now, for $z\in\mathbb{R}_{+}$, we have

$\textstyle
U_{\nu_{i}}(z)=\int_{\mathbb{R}_{+}}|1/x-z|(x/m_{i})\mu_{i}(dx)=\frac{z}{m_{i}}\int_{\mathbb{R}_{+}}|1/z-x|\mu_{i}(dx),$

so $\frac{m_{i}}{z}U_{\nu_{i}}(z)=U_{\mu_{i}}(1/z)$. As all measures involved
are supported in $\mathbb{R}_{+}$, we also have $U_{\mu_{i}}(z)=m_{i}-z$ and
$U_{\nu_{i}}(z)=1/m_{i}-z$ for $z<0$. It follows that $U_{\mu_{0}}(z)\leq
U_{\mu_{1}}(z)$ for all $z\in{\mathbb{R}}$ if and only if $U_{\nu_{0}}(z)\leq
U_{\nu_{1}}(z)$ for all $z\in{\mathbb{R}}$, and in this case $m_{1}=m_{2}$.

All open intervals considered are likewise subsets of the positive reals. It
follows, for $z>0$, that $U_{\nu_{1}}(z)>U_{\nu_{0}}(z)$ if and only if
$U_{\mu_{1}}(1/z)>U_{\mu_{0}}(1/z)$. This exhibits the desired bijection
between $\mathcal{I}_{[\nu_{0},\nu_{1}]}$ and
$\mathcal{I}_{[\mu_{0},\mu_{1}]}$. ∎

We note that the same remains true if $\nu_{i}=Id_{\ddagger}\mu_{i}$ with the
only difference that

$\textstyle I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}\iff\left\\{\frac{m}{x}:x\in
I\right\\}\in\mathcal{I}_{[\mu_{0},\mu_{1}]},$ (2.1)

where $m=\int x\mu_{0}(dx)=\int x\mu_{1}(dx)$.

We denote by $\gamma_{s}$ the centred one-dimensional Gaussian distribution
with variance $s$. We use $\ast$ to denote the convolution between a function
and a measure and $v^{\ast}$ to denote the convex conjugate (Legendre
transform) of a convex function $v$. We write $MC(\eta,\rho)$ for the optimal
transport problem of maximal covariance between measures $\eta,\rho$, that is

$\textstyle MC(\eta,\rho)=\sup_{\pi\in\Pi(\eta,\rho)}\int xy\pi(dx,dy)$

where $\Pi(\eta,\rho)$ denotes measures on ${\mathbb{R}}^{2}$ with marginals
$\eta$ and $\rho$. This problem is naturally equivalent to the squared
Wasserstein distance since

$\textstyle 2MC(\eta,\rho)=\int
x^{2}(\mu(dx)+\nu(dx))-\inf_{\pi\in\Pi(\eta,\rho)}\int(x-y)^{2}\pi(dx,dy).$

##  3 Main results

First, we make a straightforward observation that allows us to rewrite (A-mBB)
and (G-mBB) as maximisation problems. Consider a martingale $M$ as in (A-mBB),
then

$\textstyle{\mathbb{E}}\left[\int_{0}^{1}\Sigma_{t}^{2}dt\right]={\mathbb{E}}[\langle
M\rangle_{1}]=\int x^{2}d\nu_{1}-\int x^{2}d\nu_{0}$

and hence if $\bar{\Sigma}>0$ the original problem is equivalent to

$\mathbf{AP}_{\nu_{0},\nu_{1}}=\sup_{\textstyle\begin{subarray}{c}M_{0}\sim\nu_{0},M_{1}\sim\nu_{1}\\\
M_{t}=M_{0}+\int_{0}^{t}\Sigma_{s}dB_{s}\\\ M\text{
martingale}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}\Sigma_{t}dt\right]=\sup_{\textstyle\begin{subarray}{c}M_{0}\sim\nu_{0},M_{1}\sim\nu_{1}\\\
M_{t}=M_{0}+\int_{0}^{t}\Sigma_{s}dB_{s}\\\ M\text{
martingale}\end{subarray}}{\mathbb{E}}\left[M_{1}B_{1}\right],$ (AP)

in the sense that the two problems share the optimisers and the value in
(A-mBB) is equal to $\bar{\Sigma}^{2}+\int x^{2}d\nu_{1}-\int
x^{2}d\nu_{0}-2{\bar{\Sigma}}\mathbf{AP}_{\nu_{0},\nu_{1}}$. Similarly, for a
martingale $S$ in (G-mBB) we have444Assuming, as we will, that
$\mu_{i}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$, it follows that
$\int|\log(x)|d\mu_{i}(x)<\infty$.

$\textstyle{\mathbb{E}}\left[\int_{0}^{1}\sigma_{t}^{2}dt\right]=2{\mathbb{E}}[\log(S_{0}/S_{1})]=2\int\log(x)d\mu_{0}-2\int\log(x)d\mu_{1}$

and hence (G-mBB) is equivalent to the following problem:

$\mathbf{GP}_{\mu_{0},\mu_{1}}=\sup_{\textstyle\begin{subarray}{c}S_{0}\sim\mu_{0},S_{1}\sim\mu_{1}\\\
S_{t}=S_{0}+\int_{0}^{t}\sigma_{u}S_{u}dB_{u}\\\ S\text{
martingale}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}\sigma_{t}dt\right],$
(GP)

where $(B_{t})_{t\geq 0}$ is a Brownian motion on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$.

We can state our first main result:

######  Theorem 3.1.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$ satisfy
$\mu_{0}\preccurlyeq_{cx}\mu_{1}$. Let $\nu_{i}=Id_{{\ddagger}}\mu_{i}$,
$i=0,1$. Then

$\mathbf{GP}_{\mu_{0},\mu_{1}}=\mathbf{AP}_{\nu_{0},\nu_{1}},$

and (GP) admits a unique optimiser in distribution.

In fact the proof of the above result establishes a 1-1 mapping between the
optimisers to $\mathbf{GP}_{\mu_{0},\mu_{1}}$ and to
$\mathbf{AP}_{\nu_{0},\nu_{1}}$. Using our understanding of the latter we can
deduce a detailed description of the former. In the case of multiple
irreducible components the full description is more involved and we present it
in Section 6. We state here the result for the important special case of
irreducible measures. The existence and structure of the optimiser in (AP)
used below follows from [3, Thm. 3.1] or [4, Thm. 1.3], see also Definition
6.1 below.

######  Theorem 3.2.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$ with
$\mu_{0}\preccurlyeq_{cx}\mu_{1}$ be irreducible. Set
$\nu_{i}=Id_{{\ddagger}}\mu_{i}$, $i=0,1$, and let $F(1,\cdot)$ be an
increasing function, $\alpha\in\mathcal{P}({\mathbb{R}})$, such that
$(F(t,B_{t}),t\in[0,1])$ is an optimiser for $\mathbf{AP}_{\nu_{0},\nu_{1}}$,
where $B$ is a Brownian motion with an initial distribution $B_{0}\sim\alpha$
and $F(t,\cdot)=F*\gamma_{1-t}(\cdot)$. Then the distribution of the optimiser
$S$ in $\mathbf{GP}_{\mu_{0},\mu_{1}}$ is characterised by

${\mathbb{E}}\Big{[}g\big{(}\\{S_{t}:t\in[0,1]\\}\big{)}\Big{]}={\mathbb{E}}\Big{[}g\big{(}\\{m/F(t,B_{t}):t\in[0,1]\\}\big{)}\cdot
F(1,B_{1})\Big{]},$ (3.1)

for any measurable functional $g:C([0,1];{\mathbb{R}})\to{\mathbb{R}}_{+}$,
where $m:=\int x\mu_{0}(dx)$.

In particular, using that $(F(t,B_{t}):t\in[0,1])$ is a martingale, for
$g:{\mathbb{R}}\to{\mathbb{R}}_{+}$ we have

$\textstyle{\mathbb{E}}[g(S_{t})]={\mathbb{E}}[g(m/F(t,B_{t}))F(t,B_{t})]=\int
g\left(\frac{m}{F\ast\gamma_{1-t}(y)}\right)F\ast\gamma_{1-t}(y)(\alpha\ast\gamma_{t})(dy).$

This gives us a quick access to computations involving the distribution of $S$
once we know $\alpha$ and $F$. These, in turn, can be computed efficiently
using the fixed point scheme of [14] or, equivalently, the measure preserving
martingale Sinkhorn algorithm in [24]; see [2] for the proof of convergence.
We note that Theorem 3.2 also allows for an autonomous description of the
optimiser $S$ in $\mathbf{GP}_{\mu_{0},\mu_{1}}$ as a solution to an SDE, see
Proposition 6.2.

Having the connection between geometric and arithmetic problems at hand, the
following result is immediate from [5, Thm. 1.5] and [4, Thm. 1.4].

######  Corollary 3.3.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{-1,1}({\mathbb{R}}_{+})$ such that
$\mu_{0}\preccurlyeq_{cx}\mu_{1}$. Then

$\displaystyle\mathbf{GP}_{\mu_{0},\mu_{1}}$
$\displaystyle=\textstyle\inf\left\\{\int\psi
d\nu_{1}-\int(\psi^{\ast}\ast\gamma_{1})^{\ast}d\nu_{0}:\,\psi\mbox{
convex}\right\\}$
$\displaystyle=\sup\left\\{MC(\nu_{1},\alpha\ast\gamma_{1})-MC(\nu_{0},\alpha):\,\alpha\in\mathcal{P}_{2}({\mathbb{R}})\right\\},$

where $\nu_{i}=Id_{{\ddagger}}\mu_{i}$, $i=0,1$.

######  Remark 3.4.

If $S$ is feasible for (GP) for $(\mu_{0},\mu_{1})$, and $m=\int
x\mu_{0}(dx)=\int x\mu_{1}(dx)$ then $\frac{1}{m}S$ is feasible for (GP) for
the measures $\tilde{\mu}_{i}=(x\to x/m)_{\\#}\mu_{i}$, which have mean $1$.
In particular, the value of the problem (GP) is the same for
$(\mu_{0},\mu_{1})$ and $(\tilde{\mu}_{0},\tilde{\mu}_{1})$ and the optimisers
are constant multiples of each other. This explains why we take
$\nu_{i}=Id_{\ddagger}\mu_{i}=Id_{\dagger}\tilde{\mu}_{i}$ in Theorem 3.1.

In Section 4 we provide an argument for Theorems 3.1-3.2 in the irreducible
case by means of duality and PDE techniques. In Section 5 we provide a
probabilistic proof of Theorem 3.1, and in Section 6 we prove Theorem 3.2 as a
particular case of a more general statement wherein the irreducibility
assumption is dropped.

##  4 Kantorovitch duality perspective on geometric Bass martingale

The geometric martingale Benamou-Brenier problem (G-mBB) problem falls into
the general class of optimal transportation under controlled stochastic
dynamics. The duality for such problems is well understood, see [29, 20], and
offers a rich source of insights into their structure. To wit, recently [24]
used this approach to present a PDE perspective on the Bass martingale and, in
particular, offered an alternative justification for the duality result in [4,
Thm. 1.4] that we used above to obtain Corollary 3.3. We apply now an
analogous approach to (G-mBB). As explained in Remark 3.4, without any loss of
generality, we can assume that $\int x\mu_{0}(dx)=\int x\mu_{1}(dx)=1$. The
dual to (G-mBB) is found by considering

$\displaystyle\mathbf{DualGmBB}_{\mu_{0},\mu_{1}}$
$\displaystyle=\textstyle\sup_{u}\left\\{\int u(1,s)d\mu_{1}(s)-\int
u(0,s)d\mu_{0}(s)\right\\}$ (G-Dual)

among the solutions $u$ of

$\displaystyle\textstyle\partial_{t}u+\frac{\bar{\sigma}^{2}}{2}\frac{s^{2}\partial_{ss}u}{1-s^{2}\partial_{ss}u}=0,$
(4.1)

which satisfy $s^{2}\partial_{ss}u<1$. We refer to [29, Thm. 4.2] for a
statement allowing to derive the above, but note that our arguments remain
formal. In particular, we assume the existence and uniqueness of the dual
optimiser $u$, which may be hard to establish independently but which will
follow, in the irreducible case, from our proofs in sections 5 and 6. Once, we
have the optimal $u$ (G-Dual), the optimal $\sigma$ in (G-mBB) is obtained
directly via

$\displaystyle\textstyle\sigma=\frac{\bar{\sigma}}{1-s^{2}\partial_{ss}u},$
(4.2)

hence under ${\mathbb{P}}$

$\displaystyle\textstyle\frac{dS_{t}}{S_{t}}=\frac{\bar{\sigma}}{1-s^{2}\partial_{ss}u}dW^{\mathbb{P}}_{t}=\sigma
dW^{\mathbb{P}}_{t}.$ (4.3)

A suboptimal but sufficient statement for our pedagogic purpose is the
following:

######  Proposition 4.1.

Let $\mu_{0}$ be a smooth probability measure, $u$ be a $C^{4}$ smooth
classical solution to (4.1). Consider then $\sigma$ as in (4.2), and $\mu_{1}$
the density at time 1 of the process $(S_{t})_{0\leq t\leq 1}$ having density
$\mu_{0}$ at time 0 and lognormal volatility $\sigma$. Then the duality

$\mathbf{DualGmBB}_{\mu_{0},\mu_{1}}=\mathbf{GmBB}_{\mu_{0},\mu_{1}}$

holds, the l.h.s. is attained by $u$, the r.h.s. is attained by $S$, and
$S_{t}=1/F(t,W^{\tilde{\mathbb{P}}})$, for some $F,W^{\tilde{\mathbb{P}}}$
obtained from $u$ as explained below.

At this point, it may seem that the optimal $\sigma$ depends on the reference
level $\bar{\sigma}$, where we know from the equivalence between (G-mBB) and
(GP) that this does not happen. To understand this, we continue analogously to
[26, sec. 5]. We let

$d{\widetilde{\mathbb{P}}}=S_{1}d{\mathbb{P}}\quad\text{on }\mathcal{F}_{1},$

and consider

$\textstyle
v(t,s)=\left(-u(t,s)-\ln(s){+\frac{\bar{\sigma}^{2}}{2}t}\right)/\bar{\sigma}.$

Direct verification shows that $v$ satisfies
$\partial_{t}v-\frac{1}{2s^{2}\partial_{ss}v}=0$ and $\sigma$ is derived from
$v$ as $\sigma=\frac{1}{s^{2}\partial_{ss}v}$, and in particular is
independent of $\bar{\sigma}$. Differentiating (4.1), Itô’s formula and Lévy’s
characterisation of Brownian motion yield the following observations.

######  Lemma 4.2.

We have

  1. (i)

$S_{t}\partial_{s}u(t,S_{t}),S_{t}\partial_{s}v(t,S_{t})$ are local
martingales under ${\mathbb{P}}$;

  2. (ii)

$\partial_{s}u(t,S_{t}),\partial_{s}v(t,S_{t})$ are local martingales under
${\widetilde{\mathbb{P}}}$;

  3. (iii)

$d\partial_{s}v(t,S_{t})=\frac{1}{S_{t}}dW^{\widetilde{\mathbb{P}}}_{t}$ for a
${\widetilde{\mathbb{P}}}$–Brownian motion $W^{\widetilde{\mathbb{P}}}$.

The condition $s^{2}\partial_{ss}u<1$ implies that $v$ is convex. Its Legendre
transform $v^{*}$, satisfies

$\displaystyle\textstyle\partial_{t}v^{*}+\frac{\partial_{yy}v^{*}}{2(\partial_{y}v^{*})^{2}}$
$\displaystyle=$ $\displaystyle 0,$

as well as $\partial_{y}v^{*}(t,\partial_{s}v(t,s))=s$, the former from the
PDE satisfied by $v$ and the latter by the usual properties of Legendre
transform. Then, letting $X_{t}=v^{*}(t,\partial_{s}v(t,S_{t}))$, Itô’s
formula gives

$\displaystyle\textstyle dX_{t}$ $\displaystyle=$
$\displaystyle\partial_{y}v^{*}(\partial_{s}v(t,S_{t}))\frac{1}{S_{t}}dW^{\widetilde{\mathbb{P}}}_{t}=dW^{\widetilde{\mathbb{P}}}_{t}.$

We now let $w=\left({v^{*}}\right)^{-1}.$ Then $w$ sends a
${\widetilde{\mathbb{P}}}$–Brownian motion $W^{\widetilde{\mathbb{P}}}$ onto
the ${\widetilde{\mathbb{P}}}$ martingale $\partial_{s}v(t,S_{t})$,

$\textstyle w(t,W^{\widetilde{\mathbb{P}}}_{t})=\partial_{s}v(t,S_{t}),$

therefore $w$ must satisfy the heat equation

$\displaystyle\textstyle\partial_{t}w+\frac{1}{2}\partial_{xx}w=0.$ (4.4)

We finally have the following relationships:

$S_{t}\qquad\longleftrightarrow\qquad
Y_{t}=\partial_{s}v(t,S_{t})\qquad\longleftrightarrow\qquad
X_{t}=\left(v^{*}\right)^{-1}(t,Y_{t})$

and moreover

$\textstyle
S_{t}=\partial_{y}v^{*}(t,Y_{t})=\frac{1}{\partial_{x}w(t,W^{\widetilde{\mathbb{P}}}_{t})},$

and we can now let $F=\partial_{x}w$. This gives a fast numerical recipe for
solving the HJB equation (4.1). Under ${\widetilde{\mathbb{P}}}$, the process
$(\frac{1}{S_{t}}:t\leq 1)$ is a Bass-martingale given by

$\displaystyle\textstyle
1/S_{t}=F\big{(}t,W^{\widetilde{\mathbb{P}}}_{t}\big{)}.$ (4.5)

This, in particular, allows to simulate $(S_{t}:t\leq 1)$ efficiently and to
price options, including path-dependent ones, efficiently via (3.1). We
explore this further in Section 8 and link to recent works [24, 2, 14] on
numerics for (AP). We close this section with a summary of the main
similarities and differences between the arithmetic and geometric Bass
martingales. The former was denoted $(M_{t})$ in (A-mBB) but we write
$(X_{t})$ below keeping with notation of $x$ and $s$ for the state variables.

The Arithmetic Bass Martingale $(X_{t}:t\leq 1)$

$\displaystyle X_{t}$ $\displaystyle\text{ Martingale under }{\mathbb{P}}$
$\displaystyle\textstyle\mathrm{d}X_{t}=\frac{1}{\partial_{xx}v}\mathrm{d}W^{\mathbb{P}}_{t}$
$\displaystyle Y_{t}=\partial_{x}v(t,X_{t})$ $\displaystyle\text{ BM under
}{\mathbb{P}}$ $\displaystyle\mathrm{d}Y_{t}=\mathrm{d}W^{\mathbb{P}}_{t}$
$\displaystyle Z_{t}=v^{*}(t,Y_{t})$ $\displaystyle\text{ Martingale under
}{\mathbb{P}}$
$\displaystyle\mathrm{d}Z_{t}=X_{t}\mathrm{d}W^{\mathbb{P}}_{t}$
$\displaystyle X_{t}$ $\displaystyle=$
$\displaystyle\partial_{y}v^{*}(t,Y_{t}),\;v^{*}\text{ is harmonic.}$

The Geometric Bass Martingale $(S_{t}:t\leq 1)$

$\displaystyle S_{t}$ $\displaystyle\text{ Martingale under }{\mathbb{P}}$
$\displaystyle\textstyle\mathrm{d}S_{t}=\frac{1}{S_{t}\partial_{ss}v}\mathrm{d}W^{\mathbb{P}}_{t}$
$\displaystyle Y_{t}=\partial_{s}v(t,S_{t})$ $\displaystyle\text{ Martingale
under }{\widetilde{\mathbb{P}}}$
$\displaystyle\textstyle\mathrm{d}Y_{t}=\frac{1}{S_{t}}\mathrm{d}W^{\widetilde{\mathbb{P}}}_{t}$
$\displaystyle X_{t}=v^{*}(t,Y_{t})$ $\displaystyle\text{ BM under
}{\widetilde{\mathbb{P}}}$
$\displaystyle\mathrm{d}X_{t}=\mathrm{d}W^{\widetilde{\mathbb{P}}}_{t}$
$\displaystyle\textstyle\frac{1}{S_{t}}$ $\displaystyle=$
$\displaystyle\partial_{x}w\big{(}t,X_{t}\big{)},\;w=[v^{*}]^{-1}\text{ is
harmonic.}$

##  5 Change of measure and probabilistic proof of Theorem 3.1

We provide the promised probabilistic proof of Theorem (3.1):

###### Proof of Theorem (3.1).

Consider a stochastic base and a martingale $S$ admissible for (GP). Denote
$m=\int x\mu_{0}(dx)=\int x\mu_{1}(dx)$ and define $\tilde{\mathbb{P}}$, a
probability measure on $\mathcal{F}_{1}$, via
$d\tilde{\mathbb{P}}:=\frac{S_{1}}{m}d\mathbb{P}$. We will use the notation
$\tilde{\mathbb{E}}$ for expectation under this measure. As $S$ is a non-
negative martingale, by Girsanov’s theorem
$\tilde{B_{t}}:=B_{t}-\int_{0}^{t}\sigma_{s}ds$ is a
$\tilde{\mathbb{P}}$-Brownian motion. Note that $S_{1}>0$ ${\mathbb{P}}$-a.s.,
and hence ${\mathbb{P}}$ and $\tilde{\mathbb{P}}$ are equivalent on
$\mathcal{F}_{1}$ and

$\textstyle
R_{1}:=\frac{d{\mathbb{P}}}{d\tilde{\mathbb{P}}}\big{|}_{\mathcal{F}_{1}}=\frac{m}{S_{1}}\quad\text{and}\quad
R_{t}:=\tilde{{\mathbb{E}}}[R_{1}|\mathcal{F}_{t}]=\frac{\mathbb{E}[\frac{S_{1}}{m}R_{1}|\mathcal{F}_{t}]}{{\mathbb{E}}[\frac{S_{1}}{m}|\mathcal{F}_{t}]}=\frac{1}{\frac{S_{t}}{m}}=\frac{m}{S_{t}},$
(5.1)

for $t\in[0,1]$. Hence

$\textstyle\mathbb{E}\left[\int_{0}^{1}\sigma_{t}dt\right]=\tilde{\mathbb{E}}\left[R_{1}\int_{0}^{1}\sigma_{t}dt\right]=\tilde{\mathbb{E}}\left[\int_{0}^{1}R_{t}\sigma_{t}dt\right]=\tilde{\mathbb{E}}\left[\int_{0}^{1}\Sigma_{t}dt\right],$
(5.2)

where we defined $\Sigma_{t}:=R_{t}\sigma_{t}$. An application of the Itô
formula gives

$\displaystyle dR_{t}$ $\displaystyle=\textstyle
m\left[\frac{-1}{S_{t}^{2}}S_{t}\sigma_{t}dB_{t}+\frac{S_{t}^{2}\sigma_{t}^{2}}{S_{t}^{3}}dt\right]$
$\displaystyle=-R_{t}\sigma_{t}dB_{t}+R_{t}\sigma^{2}_{t}dt$
$\displaystyle=-R_{t}\sigma_{t}[dB_{t}-\sigma_{t}dt]=\Sigma_{t}dW_{t},$

where $W_{t}:=-\tilde{B_{t}}$ is likewise a $\tilde{\mathbb{P}}$-Brownian
motion. Furthermore we observe that for any bounded, smooth test function $g$
we have

$\textstyle\tilde{\mathbb{E}}[g(R_{1})]=\mathbb{E}\left[\frac{g(R_{1})}{R_{1}}\right]=\mathbb{E}\left[g\left(\frac{m}{S_{1}}\right)\frac{S_{1}}{m}\right]=\int
g\left(\frac{m}{y}\right)\frac{y}{m}\mu_{1}(dy)=\int gd\nu_{1}.$

Similarly, using the martingale property of $S$ under $\mathbb{P}$, we have

$\textstyle\tilde{\mathbb{E}}[g(R_{0})]=\mathbb{E}\left[\frac{g(R_{0})}{R_{1}}\right]=\mathbb{E}\left[g\left(\frac{m}{S_{0}}\right)\frac{S_{1}}{m}\right]=\mathbb{E}\left[g\left(\frac{m}{S_{0}}\right)\frac{S_{0}}{m}\right]=\int
gd\nu_{0}.$

We conclude that the initial and terminal marginals of $R$ do not depend on
the choice of $S$ as long as $S$ is feasible. We thus see that $R$
defined555How we extend dynamics for $t>1$ is irrelevant, we take
$\sigma_{t}=0$ for $t>1$ for concreteness. on
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\tilde{\mathbb{P}})$ is
admissible for (AP). Combined with (5.2), this shows that
$\mathbf{GP}_{\mu_{0},\mu_{1}}\leq\mathbf{AP}_{\nu_{0},\nu_{1}}$.

The reverse inequality is obtained in the same fashion. Consider a martingale
$R$ defined on a stochastic basis
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\tilde{\mathbb{P}})$ with a
Brownian motion $W$, such that $dR_{t}=\Sigma_{t}dW_{t}$ and
$R_{0}\sim\nu_{0}$, $R_{1}\sim\nu_{1}$. Observe that $R_{t}>0$ a.s.,
$t\in[0,1]$ and let $\sigma_{t}:=\frac{\Sigma_{t}}{R_{t}}$. Define a new
probability measure ${\mathbb{P}}$ via
$\frac{d{\mathbb{P}}}{d\tilde{\mathbb{P}}}\big{|}_{\mathcal{F}_{1}}=R_{1}$,
then $B_{t}=-W_{t}+\int_{0}^{t}\sigma_{s}ds$ is a ${\mathbb{P}}$-Brownian
motion. We let $S_{t}=\frac{m}{R_{t}}$ and by a direct computation, analogous
to (5.1), we see that $S$ is a ${\mathbb{P}}$-martingale and Itô’s formula
gives $dS_{t}=S_{t}\sigma_{t}dB_{t}$. Finally, for a test function $g$ we have

$\textstyle{\mathbb{E}}[g(S_{1})]=\tilde{\mathbb{E}}\left[g\left(\frac{m}{R_{1}}\right)R_{1}\right]=\int
g\left(\frac{m}{y}\right)y\nu_{1}(dy)=\int g(y)\mu_{1}(dy)$

using $\nu_{1}=Id_{\ddagger}\mu_{1}$, and likewise $S_{0}\sim\mu_{0}$. The
equality of values (5.2) still holds and we conclude that
$\mathbf{GP}_{\mu_{0},\mu_{1}}\geq\mathbf{AP}_{\nu_{0},\nu_{1}}$, as required.

Finally, we note that if (GP) had two maximisers $S^{1}$ and $S^{2}$ with
different distributions (as processes), then the above construction would give
us two distinct (in distribution) optimisers $R^{1}$ and $R^{2}$ for (AP)
which would contradict the uniqueness (in distribution) established in [3]. ∎

######  Remark 5.1.

From the proof of Theorem (3.1) it is clear that several immediate
generalisations are possible, linking arithmetic and geometric problems.
Specifically, in analogy to (5.2), we can write

$\displaystyle\textstyle\mathbb{E}\left[\int_{0}^{1}c(t,S_{t},\sigma_{t}^{2})dt\right]=\tilde{\mathbb{E}}\left[\int_{0}^{1}R_{t}c(t,S_{t},\sigma_{t}^{2})dt\right]=\tilde{\mathbb{E}}\left[\int_{0}^{1}R_{t}c\left(t,\frac{m}{R_{t}},\frac{\Sigma_{t}^{2}}{R_{t}^{2}}\right)dt\right].$

Hence to
$c:[0,1]\times{\mathbb{R}}_{+}\times\overline{{\mathbb{R}}_{+}}\to{\mathbb{R}}$
we can associate $\tilde{c}(t,x,z):=xc(t,m/x,z/x^{2})$, thus obtaining the
equivalence of the geometric and arithmetic problem, respectively

$\displaystyle\textstyle\inf_{\begin{subarray}{c}S_{0}\sim\mu_{0},S_{1}\sim\mu_{1}\\\
S_{t}=S_{0}+\int_{0}^{t}\sigma_{s}S_{s}dB_{s}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}c(t,S_{t},\sigma_{t}^{2})dt\right]$
$\displaystyle\textstyle\,\,\text{ and
}\,\,\inf_{\begin{subarray}{c}M_{0}\sim\nu_{0},M_{1}\sim\nu_{1}\\\
M_{t}=M_{0}+\int_{0}^{t}\Sigma_{s}dB_{s}\end{subarray}}{\mathbb{E}}\left[\int_{0}^{1}\tilde{c}(t,M_{t},\Sigma_{t}^{2})dt\right],$

with $\nu_{i}=Id_{{\ddagger}}\mu_{i}$, $i=0,1$. In this article we took
$c(t,x,\sigma^{2})=-\sigma$ as this is essentially the only case where the
arithmetic version of the problem has an explicit solution.

##  6 Structure of the Geometric Bass martingale

We now turn to a detailed study and characterisation of the optimiser in (GP).
Our proof in section 5 shows that the optimiser $S$ to (GP) is obtained by a
change of measure procedure starting with the optimiser $R$ to (AP), which was
dubbed _stretched Brownian motion_ and shown to be unique in [3]. We recall a
crucial concept towards understanding the structure of $R$.

######  Definition 6.1.

A real-valued martingale $M$ is a Bass martingale, if there is an increasing
function $F$ and a Brownian motion $B$ with a possibly non-trivial initial
distribution s.t.

$M_{t}=F(t,B_{t}),$

with $F(t,\cdot)=F*\gamma_{1-t}(\cdot)$. We refer to $F$ as the generating
function and to $\alpha:=\text{Law}(B_{0})$ as the Bass measure.

Following [3, Thm. 3.1] or [4, Thm. 1.3], we know that the unique optimizer of
(AP) has a very specific structure, namely: conditionally on it starting in
$I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}$, it is a Bass martingale. In other
words, for any $I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}$, there exists
$\alpha_{I}$ a probability measure and $F_{I}:\mathbb{R}\to\overline{I}$
increasing, such that, conditionally on $R_{0}\in I$, we have
$R_{t}=F_{I}(t,W^{I}_{t})$, with $F_{I}(t,x)=F_{I}\ast\gamma_{1-t}(x)$ and
$W^{I}$ being a Brownian motion started according to $\alpha_{I}$.

To understand better the dynamics of the optimiser $S$ in (GP), it is
convenient to consider a representation for the optimiser $R$ in (AP) where
the Brownian motions $W^{I}$ are coupled together. Let $(I_{i})_{i\geq 1}$ be
a numbering of intervals in $\mathcal{I}_{[\nu_{0},\nu_{1}]}$ and
$I_{0}:=\\{z:U_{\nu_{0}}(z)=U_{\nu_{1}}(z)\\}$, and write $F_{i}=F_{I_{i}}$,
$\alpha_{i}=\alpha_{I_{i}}$. We consider a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\tilde{\mathbb{P}})$ with a
standard Brownian motion $\tilde{B}$ and $\mathcal{F}_{0}$ rich enough so
support an integer-valued random variable $\xi$ with
$\tilde{\mathbb{P}}(\xi=i)=\nu_{0}(I_{i})$, $i\geq 0$, and
$\zeta_{0}\sim{\nu_{0}}_{|I_{0}}/\nu_{0}(I_{0})$, $\zeta_{i}\sim\alpha_{i}$,
$i\geq 1$, all of these being $\mathcal{F}_{0}$-measurable, independent of
each other, and independent of the Brownian motion $\tilde{B}$. Let
$W_{0}=\sum_{i\geq 1}\zeta_{i}\mathbf{1}_{\xi=i}$ and
$W_{t}=W_{0}+\tilde{B}_{t}$. Let
$R_{1}=\zeta_{0}\mathbf{1}_{\xi=0}+\sum_{i\geq
1}F_{i}(W_{1})\mathbf{1}_{\xi=i}$ and
$R_{t}=\tilde{\mathbb{E}}[R_{1}|\mathcal{F}_{t}]$. It follows that
$R_{t}=\zeta_{0}\mathbf{1}_{\xi=0}+\sum_{i\geq
1}F_{i}(t,W_{t})\mathbf{1}_{\xi=i}$. We therefore have

$\textstyle dR_{t}=R_{t}\left(\sum_{i\geq
1}\frac{\partial_{x}F_{i}(t,W_{t})}{R_{t}}\mathbf{1}_{\xi=i}\right)dW_{t}=R_{t}\sigma_{t}dW_{t},$

where $\sigma_{t}:=\sum_{i\geq
1}\partial_{x}\log(F_{i}(t,W_{t}))\mathbf{1}_{\xi=i}$. As above, with
$\frac{d{\mathbb{P}}}{d\tilde{\mathbb{P}}}\big{|}_{\mathcal{F}_{1}}=R_{1}$, we
have $B_{t}=-W_{t}+\int_{0}^{t}\sigma_{s}ds$ is a ${\mathbb{P}}$-Brownian
motion and $S_{t}=\frac{m}{R_{t}}$ is a ${\mathbb{P}}$-martingale with
$dS_{t}=S_{t}\sigma_{t}dB_{t}$.

For any measurable test functional
$g:C([0,1];{\mathbb{R}})\to{\mathbb{R}}_{+}$ we have

${\mathbb{E}}\Big{[}g\big{(}\\{S_{t}:t\in[0,1]\\}\big{)}\Big{]}=\tilde{\mathbb{E}}\Big{[}g\big{(}\\{m/R_{t}:t\in[0,1]\\}\big{)}\cdot
R_{1}\Big{]},$

which thus fully characterises the distribution of $S$. It reduces to (3.1) in
the irreducible case and thus establishes Theorem 3.2. Note that the marginals
of $S$ are recovered from the marginals of $R$ via
${\mathbb{E}}[g(S_{t})]=\tilde{\mathbb{E}}[g(m/R_{t})R_{t}]$, i.e.,

$\textstyle S_{t}\sim\left(y\to\frac{y}{m}\right)_{\dagger}\nu_{t},$

where $\nu_{t}\sim R_{t}$. In the general, not necessarily irreducible
setting, we note that for $J(I)\in\mathcal{I}_{[\mu_{0},\mu_{1}]}$ associated
to $I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}$ according to the bijection from Lemma
2.1, we have $\\{R_{0}\in I\\}=\\{S_{0}\in J(I)\\}$. Importantly,
$F_{i}(t,\cdot)$ is a smooth and strictly increasing function which admits an
inverse, for $t<1$. On the set $\\{S_{0}\in J(I_{i})\\}$, we have
$W_{t}=F_{i}^{-1}(t,m/S_{t})$ and

$\displaystyle S_{t}$ $\displaystyle=\textstyle
S_{0}\exp\left\\{-\int_{0}^{t}\partial_{x}\log
F_{i}(s,W_{s})dW_{s}+\frac{1}{2}\int_{0}^{t}(\partial_{x}\log
F_{i}(s,W_{s}))^{2}ds\right\\}$ $\displaystyle=\textstyle
S_{0}\exp\left\\{\int_{0}^{t}\partial_{x}\log
F_{i}(s,W_{s})dB_{s}-\frac{1}{2}\int_{0}^{t}(\partial_{x}\log
F_{i}(s,W_{s}))^{2}ds\right\\}$ $\displaystyle=\textstyle
S_{0}\exp\left\\{\int_{0}^{t}\frac{S_{s}}{m\partial_{x}F_{i}^{-1}(s,\frac{m}{S_{s}})}dB_{s}-\frac{1}{2}\int_{0}^{t}\left(\frac{S_{s}}{m\partial_{x}F_{i}^{-1}(s,\frac{m}{S_{s}})}\right)^{2}ds\right\\}.$

We note that the last representation offers an intrinsic characterisation of
the dynamics of $S$ under ${\mathbb{P}}$ without the need to consider any
dynamics under $\tilde{\mathbb{P}}$. This is summarised in the following
proposition.

######  Proposition 6.2.

Let $S$ be the unique optimizer of Problem (GP). Then, conditioned on
$\\{S_{0}\in J\\}$, with $J\in\mathcal{I}_{[\mu_{0},\mu_{1}]}$ corresponding
to $I\in\mathcal{I}_{[\nu_{0},\nu_{1}]}$, $S$ solves

$\textstyle
dS_{u}=S_{u}\frac{S_{u}/m}{\partial_{x}F_{I}^{-1}(u,\frac{m}{S_{u}})}dB_{u},\quad
0<u<1,$

for a Brownian motion $B$.

##  7 Relations between classes of Bass martingales

As observed in [3, Remark 1.9], the solution to (AP) for lognormal marginals
is given by the usual geometric Brownian motion which, trivially, is also the
solution to (GP) for lognormal marginals. So for log-normal marginals
Arithmetic-Bass and Geometric-Bass martingales coincide and are equal to
geometric Brownian motion. This is clear from the following simple
computation: let $\mu_{0}=\delta_{1}$ and $\mu_{1}$ be the distribution of
$S_{1}$ for a geometric Brownian motion $S$ solving
$dS_{t}=\bar{\sigma}S_{t}dB_{t}$. Then, for a test function $g$ and
$\nu_{1}=Id_{\ddagger}\mu_{1}$ we have

$\displaystyle\textstyle\int g(y)\nu_{1}(dy)$ $\displaystyle=\textstyle\int
g\left(\frac{1}{y}\right)y\mu_{1}(dy)=\int
g(e^{-z})e^{z}e^{\frac{-(z+\bar{\sigma}^{2}/2)^{2}}{2\bar{\sigma}^{2}}}\frac{dz}{\bar{\sigma}\sqrt{2\pi}}$
$\displaystyle=\textstyle\int
g(e^{-z})e^{\frac{-(z-\bar{\sigma}^{2}/2)^{2}}{2\bar{\sigma}^{2}}}\frac{dz}{\bar{\sigma}\sqrt{2\pi}}=\int
g(e^{z})e^{\frac{-(z+\bar{\sigma}^{2}/2)^{2}}{2\bar{\sigma}^{2}}}\frac{dz}{\bar{\sigma}\sqrt{2\pi}}$
$\displaystyle=\textstyle\int g(y)\mu_{1}(dy),$

so $\mu_{1}=Id_{{\ddagger}}\mu_{1}$ is a fixed point for the ${\ddagger}$
operator. We now show that in fact the geometric Brownian motion, with an
arbitrary starting distribution, is the only process which is both an
arithmetic- and a geometric- Bass martingale.

######  Proposition 7.1.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{2}({\mathbb{R}}_{+})$ in convex order
$\mu_{0}\preccurlyeq_{cx}\mu_{1}$ and irreducible. Then the optimisers in (AP)
and (GP) coincide if and only if

$\log_{\\#}{\mu_{1}}=\left(\log_{\\#}{\mu_{0}}\right)\ast\mathcal{N}(-\sigma^{2}/2,\sigma^{2}),$

for some $\sigma^{2}>0$, where $\mathcal{N}(-\sigma^{2}/2,\sigma^{2})$ is the
Gaussian with mean $-\sigma^{2}/2$ and variance $\sigma^{2}$.

###### Proof.

For sufficiency, let $\alpha=(y\to\frac{1}{\sigma}\log(y))_{\\#}\mu_{0}$ and

$S_{t}=\exp\left\\{\sigma
B_{0}+\sigma(B_{t}-B_{0})-\sigma^{2}t/2\right\\},\quad t\in[0,1].$

Then $S$ is a Bass martingale with $S_{0}\sim\mu_{0}$ and $S_{1}\sim\mu_{1}$,
hence it attains $\mathbf{AP}_{\mu_{0},\mu_{1}}$. Equally, since (G-mBB) and
(GP) are equivalent, the optimiser is the same for any choice of
$\bar{\sigma}$ in (G-mBB), but for $\bar{\sigma}=\sigma$, the process $S$
attains value zero, which is clearly the lower bound, and hence is the
optimiser.

For the converse implication, suppose $S$ is the optimiser in (GP) and a Bass
martingale. We thus have $S_{t}=H(t,B_{t})$ for some function $H$ and a
Brownian motion $B$. At the same time, by the proof of Theorem 3.1,
$S_{t}=\frac{m}{R_{t}}=\frac{m}{F(t,W_{t})}$, for the
$\tilde{\mathbb{P}}$-Brownian motion $W$, since $R$ is a Bass martingale under
$\tilde{\mathbb{P}}$. It follows that, noting that both $H$ and $F$ are smooth
for $0<t<1$ and strictly increasing in the spatial argument,

$\textstyle
H^{-1}\left(t,\frac{m}{F(t,W_{t})}\right)=B_{t}=-W_{t}+\int_{0}^{t}\partial_{x}\log
F(s,W_{s})ds$

and hence, comparing the dynamics and equating the $dW_{t}$ terms,

$\textstyle\partial_{x}H^{-1}\left(t,\frac{m}{F(t,W_{t})}\right)=-1\
d\tilde{\mathbb{P}}-a.s.,\quad\Longrightarrow\frac{m}{F(t,x)}=H(t,-x+at),$

for some constant $a$. Recall however that both $H$ and $F$ solve the heat
equation, so

$\displaystyle 0$
$\displaystyle=\textstyle\partial_{t}F+\frac{1}{2}\partial_{xx}F=-\frac{m}{H^{2}}\left(\partial_{t}H+a\partial_{x}H\right)+\frac{m}{2}\frac{-H^{2}\partial_{xx}H+2H(\partial_{x}H)^{2}}{H^{4}}$
$\displaystyle\textstyle=\frac{m}{H^{2}}\left(-\partial_{t}H-a\partial_{x}H-\frac{1}{2}\partial_{xx}H+\frac{(\partial_{x}H)^{2}}{H}\right)=\frac{m}{H^{2}}\left(-a\partial_{x}H+\frac{(\partial_{x}H)^{2}}{H}\right)$

from which we deduce that $\partial_{x}H=aH$ and hence $H(t,x)=q(t)e^{ax}$.
Plugging into the heat equation we obtain $H(t,x)=e^{ax-a^{2}/2t}$ as
required. ∎

To end this section, we discuss the relation between the martingale Benamou-
Brenier problems and projections using the adapted Wasserstein distance. The
latter, also known as the (bi)causal Wasserstein distance has been shown to be
the natural analogue of the classical Wasserstein distance when measuring the
distance between the distributions of two stochastic processes, see [25, 1].
We focus here on the distance $\mathcal{AW}_{2}$ between matingale laws
arising from measuring distances between paths $\omega,\omega^{\prime}$ via
$(w_{0}-w^{\prime}_{0})^{2}+\langle\omega-\omega^{\prime}\rangle_{1}$.
Repeating the arguments used to show the equivalence between (A-mBB) and (AP),
and establishing suitable representation properties for bicausal couplings so
that we can identify martingales with their distributions, [3, Sec. 6] show
that (A-mBB) is equivalent to projecting the Wiener measure on the set
$\mathcal{M}(\nu_{0},\nu_{1})$ of the distributions of continuous martingales
with marginals $\nu_{i}$ and times $i=0,1$, where the projection is in the
sense of squared adapted Wasserstein distance $\mathcal{AW}_{2}^{2}$. At
first, one might expect that the geometric Bass martingale in (G-mBB) arises
as the $\mathcal{AW}_{2}^{2}$-projection of the geometric Brownian motion.
This is not true. In fact, the resulting process is a continuous extension of
the _q_ -Bass martingale recently introduced by [30].

To see this, we consider a slightly more general setup. Let $M$ be an
arithmetic Bass martingale with $M_{1}=F(B_{1})$, where $(B_{t})$ is a
standard Brownian motion on some stochastic basis. For simplicity, we assume
$B_{0}$ is a constant and hence, also $M_{0}=F\ast\gamma_{1}(B_{0})$ is a
constant. Then, the problem of projecting $M$ onto
$\mathcal{M}(\mu_{0},\mu_{1})$ in the $\mathcal{AW}_{2}^{2}$-sense is
equivalent to the problem

$\sup_{S\in\mathcal{M}(\mu_{0},\mu_{1})}{\mathbb{E}}[S_{1}F(B_{1})],$

where we continue to use the martingale and its distribution interchangeably
for simplicity (and $B$ is supposed to be a Brownian motion in the same
filtration where $S$ is a martingale). This amounts to a static weak optimal
transport problem in the sense of [19] which we can write as

$\sup_{\pi}\int MC(\pi^{x},q)d\mu_{0}(x),$

where now we optimize over 1-period martingale couplings with the given
marginals $\mu_{0},\mu_{1}$ and where $q:=Law(M_{1})$. To be precise, this
static problem gives an upper bound to the above continuous problem. Its
solution was recently studied in [30], going under the name of _q_ -Bass
martingales. Assuming $q$ does not charge points, this problem admits a unique
optimiser. While its full characterisation is, to the best of our knowledge,
an open problem, in some situations [30] shows that the optimiser may be
written as $S_{1}=G(\xi+F(B_{1})),$ for an $\mathcal{F}_{0}$-measurable random
variable $\xi$ independent of $B$ and $G$ an increasing function. It then
follows that $(G\ast q)_{\\#}Law(\xi)=\mu_{0}$ and in our setting this means
we can extend this one-period solution to a continuous martingale setting via

$S_{t}={\mathbb{E}}[S_{1}|\mathcal{F}_{t}]=\int
G(\xi+F(B_{t}+z))d\gamma_{1-t}(z):=G_{t}(\xi,B_{t}).$

Note that $S$ has an absolutely continuous quadratic variation, and the above
construction provides a bicausal coupling between $S$ and $M$ (recall that
$M_{t}=F\ast\gamma_{1-t}(B_{t})$), so the martingale $S$ saturates the upper
bound, and is the optimizer to our projection problem. In the particular case
when $M=B$, we recover that $S$ is the _arthimetic_ Bass martingale. However,
when $M$ is the geometric Brownian motion, or some other Bass martingale, the
resulting projection appears to be a (continuous) _q_ -Bass martingale. We
believe this provides a natural motivation to study these processes further
and to understand better the $\mathcal{AW}_{2}^{2}$-projection relations
between different types of Bass martingales. We leave this topic for future
research.

##  8  Martingale Sinkhorn systems

We work in the setting of Theorem 3.1 and further (without loss of generality)
assume

$\displaystyle\textstyle 1=\int sd\mu_{0}(s)=\int sd\mu_{1}(s).$ (8.1)

For simplicity, we suppose $\mu_{0},\mu_{1}$ are irreducible (otherwise our
analysis has to be repeated for each irreducible component), and admit
densities. For simplicity of notation, we identify the measure with its
density.

The unique Bass martingale which solves (A-mBB) is characterised by
$\alpha_{0},F_{1}$ s.t.
$\displaystyle\nu_{0}$ $\displaystyle=$
$\displaystyle(\gamma_{1}*F_{1})_{\\#}\alpha_{0},$ (8.2)
$\displaystyle\nu_{1}$ $\displaystyle=$ $\displaystyle
F_{1\\#}(\gamma_{1}*\alpha_{0}),$ (8.3) $\displaystyle\nu_{t}$
$\displaystyle=$
$\displaystyle(\gamma_{1-t}*F_{1})_{\\#}(\gamma_{t}*\alpha_{0}).$ (8.4)
The above system has been introduced in [24] and referred to as a martingale
Sinkhorn algorithm thanks to structural parallels with the classical Sinkhorn
algorithm. It is equivalent to the fixed point characterisation in [14].
Either formulation readily yields an algorithm to compute $\alpha,F$, see [2,
24] for a proof of its convergence.

In order to solve (GP), we may compute $\nu_{i}=Id_{\dagger}\mu_{i}$, i=0,1,
and solve the system (8.2-8.4) for $\nu_{0},\nu_{1}$. This will yield
$F,\alpha$, and for any time $t$ we have

$\textstyle s\mu_{t}(s)=\left(\frac{1}{F(t,\cdot)}\right)_{\\#}\alpha_{t}.$

Alternatively, we may write a system solved by the $F,\alpha$ directly in
terms of $\mu_{0},\mu_{1}$:
$\displaystyle s\mu_{0}(s)$ $\displaystyle=$
$\displaystyle([\gamma_{1}*F_{1}]^{-1})_{\\#}\alpha_{0},$ (8.5) $\displaystyle
s\mu_{1}(s)$ $\displaystyle=$
$\displaystyle([F_{1}]^{-1})_{\\#}(\gamma_{1}*\alpha_{0}),$ (8.6)
$\displaystyle s\mu_{t}(s)$ $\displaystyle=$
$\displaystyle([\gamma_{1-t}*F_{1}]^{-1})_{\\#}(\gamma_{t}*\alpha_{0}),,$
(8.7)

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