disposition section subsection

#  $q$-Bass martingales††thanks:  I am grateful to Julio Backhoff-Veraguas,
Mathias Beiglböck and Walter Schachermayer for their useful comments. This
research was funded by the Austrian Science Fund (FWF) [Grant DOI:
10.55776/P35519]. For open access purposes, the author has applied a CC BY
public copyright license to any author accepted manuscript version arising
from this submission.

Bertram Tschiderer  Faculty of Mathematics, University of Vienna (email:
bertram.tschiderer@univie.ac.at).

###### Abstract

Abstract. An intriguing question in martingale optimal transport is to
characterize the martingale with prescribed initial and terminal marginals
whose transition kernel is as Gaussian as possible. In this work we address an
extension of this question, in which the role of the Gaussian distribution is
replaced by an arbitrary reference measure $q$. Our first main result is a
dual formulation of the corresponding martingale optimization problem in terms
of convex functions.

In the well-studied case when $q$ is Gaussian, the careful analysis of the
solution to the above-mentioned optimization problem is a crucial building
block in the construction of Bass martingales, i.e., Brownian martingales
induced by gradients of convex functions with possibly non-degenerate starting
laws. In our second main result we extend this concept beyond the Gaussian
case by introducing the notion of $q$-Bass martingales in discrete time, and
give sufficient conditions for their existence.

MSC 2020 subject classifications: Primary 60G42, 60G44; secondary 91G20.

Keywords and phrases: optimal transport, Brenier’s theorem, Benamou–Brenier,
stretched Brownian motion, Bass martingale

##  1 Introduction

###  1.1 Martingale optimization problem

Let $\mu,\nu$ be elements of $\mathscr{P}_{2}(\mathds{R}^{d})$, the collection
of probability measures on $\mathds{R}^{d}$ with finite second moments. Assume
that $\mu$ is dominated by $\nu$ in convex order, denoted by
$\mu\preceq_{\textnormal{c}}\nu$, and meaning that $\int f\,d\mu\leqslant\int
f\,d\nu$ holds for all convex functions
$f\colon\mathds{R}^{d}\rightarrow\mathds{R}$. In the recent articles [BVBHK20,
CHL21, BVBST23, BVST23, AMP23], the martingale optimization problem

$P^{\gamma}(\mu,\nu)\coloneqq\sup_{\pi\in\mathsf{MT}(\mu,\nu)}\int\textnormal{MCov}(\pi_{x},\gamma)\,\mu(dx)$
(1.1)

received particular interest. The standard Gaussian measure $\gamma$ on
$\mathds{R}^{d}$ is used as reference measure in (1.1) and the cost function
is the maximal covariance defined as

$\textnormal{MCov}(\rho,\varrho)\coloneqq\sup_{\tilde{\pi}\in\mathsf{Cpl}(\rho,\varrho)}\int\langle
y,z\rangle\,\tilde{\pi}(dy,dz),$ (1.2)

for $\rho,\varrho\in\mathscr{P}_{2}(\mathds{R}^{d})$. The maximization in
(1.2) takes place over the set $\mathsf{Cpl}(\rho,\varrho)$ of all couplings
$\tilde{\pi}$ between $\rho$ and $\varrho$, i.e., probability measures
$\tilde{\pi}$ on $\mathds{R}^{d}\times\mathds{R}^{d}$ with first marginal
$\rho$ and second marginal $\varrho$. In the martingale optimization problem
(1.1), the supremum is taken over the collection $\mathsf{MT}(\mu,\nu)$ of
martingale transports, i.e., couplings $\pi\in\mathsf{Cpl}(\mu,\nu)$
satisfying $\operatorname{bary}(\pi_{x})\coloneqq\int y\,\pi_{x}(dy)=x$, for
$\mu$-a.e. $x\in\mathds{R}^{d}$. Here, the family of probability measures
$\\{\pi_{x}\\}_{x\in\mathds{R}^{d}}$ on $\mathds{R}^{d}$ is obtained by
disintegrating the coupling $\pi$ with respect to its first marginal $\mu$,
i.e., $\pi(dx,dy)=\pi_{x}(dy)\,\mu(dx)$.

The optimization problem (1.1) is a weak optimal transport problem in the
sense of [GRST17], since the function
$\pi\mapsto\int\textnormal{MCov}(\pi_{x},\gamma)\,\mu(dx)$ is non-linear, as
opposed to classical optimal transport, where linear problems of the form
$\pi\mapsto\int c(x,y)\,\pi(dx,dy)$ are studied. The problem (1.1) can also be
viewed as a discrete-time version of a continuous-time martingale optimization
problem, see [BVBHK20, BVBST23, BVST23]. Related problems also appear in
[Loe18, Loe23] and [HT19].

It is natural to consider more general reference measures than the standard
Gaussian $\gamma$ in (1.1). We thus fix a measure
$q\in\mathscr{P}_{2}(\mathds{R}^{d})$ which does not give mass to small sets,
i.e., measurable sets with Hausdorff dimension at most $d-1$ in
$\mathds{R}^{d}$. This leads to the following extension of (1.1):

$P^{q}(\mu,\nu)\coloneqq\sup_{\pi\in\mathsf{MT}(\mu,\nu)}\int\textnormal{MCov}(\pi_{x},q)\,\mu(dx).$
(1.3)

###  1.2 Dual formulation

Our first main result is a dual formulation of the martingale optimization
problem (1.3) in terms of convex functions.

######  Theorem 1.1.

Let $\mu,\nu,q\in\mathscr{P}_{2}(\mathds{R}^{d})$. Assume that
$\mu\preceq_{\textnormal{c}}\nu$ and that $q$ does not give mass to small
sets. The primal problem (1.3) is uniquely attained. Its value
$P^{q}(\mu,\nu)$ is finite and equal to

$D^{q}(\mu,\nu)\coloneqq\inf_{\begin{subarray}{c}\psi\in L^{1}(\nu),\\\
\textnormal{$\psi$
convex}\end{subarray}}\Big{(}\int\psi\,d\nu-\int(\psi^{\ast}\star
q)^{\ast}\,d\mu\Big{)}.$ (1.4)

A few comments on the notation used in the dual problem (1.4) are in order.
For a function $\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$, its
convex conjugate is denoted by $\psi^{\ast}$ and given by

$\psi^{\ast}(x)\coloneqq\sup_{y\in\mathds{R}^{d}}\big{(}\langle
x,y\rangle-\psi(y)\big{)}.$ (1.5)

For a probability measure $\varrho$ on $\mathds{R}^{d}$, we denote by
$\psi\star\varrho$ the function

$(\psi\star\varrho)(y)\coloneqq\int\psi(y+z)\,\varrho(dz)$ (1.6)

provided $\psi(y+\,\cdot\,)\in L^{1}(\varrho)$. If $\varrho$ is symmetric
(like, e.g., in the Gaussian case $\varrho=\gamma$), the function
$\psi\star\varrho$ coincides with the convolution of $f$ and $\varrho$ given
by

$(\psi\ast\varrho)(y)=\int\psi(y-z)\,\rho(dz).$

In the Gaussian case $q=\gamma$, the duality result of Theorem 1.1 is
contained in the statement of Theorem 1.4 in [BVBST23]. It turns out that for
the proof of this duality result it is not essential that the reference
measure $q$ is Gaussian. In fact, it is sufficient that $q$ is a probability
measure on $\mathds{R}^{d}$ with finite second moments (so that the maximal
covariance with respect to $q$ is well-defined) and which does not give mass
to small sets (recall that such sets are measurable subsets of
$\mathds{R}^{d}$ with Hausdorff dimension at most $d-1$). The latter condition
enables us to apply Brenier’s theorem (see, e.g., [Vil03, Theorem 2.12]).

###  1.3 $q$-Bass martingales

We recall the following special construction of Brownian martingales (see
[KS98, p. 182] for the classical definition) induced by gradients of convex
functions with possibly non-degenerate starting laws.

######  Definition 1.2.

Let $(B_{t})_{0\leqslant t\leqslant 1}$ be $d$-dimensional Brownian motion
with initial distribution $B_{0}\sim\hat{\alpha}$, where $\hat{\alpha}$ is an
arbitrary probability measure on $\mathds{R}^{d}$. Let
$\hat{v}\colon\mathds{R}^{d}\rightarrow\mathds{R}$ be a convex function such
that $\nabla\hat{v}(B_{1})$ is square-integrable. The martingale

$\hat{M}_{t}\coloneqq\mathds{E}[\nabla\hat{v}(B_{1})\,|\,\sigma(B_{s}\colon
s\leqslant t)]=\mathds{E}[\nabla\hat{v}(B_{1})\,|\,B_{t}],\qquad 0\leqslant
t\leqslant 1$ (1.7)

is called Bass martingale with Bass measure $\hat{\alpha}$, and with initial
marginal $\mu\coloneqq\operatorname{Law}(\hat{M}_{0})$ and terminal marginal
$\nu\coloneqq\operatorname{Law}(\hat{M}_{1})$.

Martingales of this form where introduced by Bass [Bas83] (in dimension $d=1$
and with $\hat{\alpha}$ a Dirac measure) in order to derive a solution of the
Skorokhod embedding problem. Definition 1.2 generalizes this concept to
multiple dimensions and non-degenerate starting laws; see [BVBHK20, BVBST23].

We write $\gamma^{t}$ for the $d$-dimensional centered Gaussian distribution
with covariance matrix $tI_{d}$ and set
$\hat{v}_{t}\coloneqq\hat{v}\ast\gamma^{1-t}\colon\mathds{R}^{d}\rightarrow\mathds{R}$,
for $0\leqslant t\leqslant 1$. In these terms, (1.7) amounts to

$\hat{M}_{t}=\nabla\hat{v}_{t}(B_{t}),\qquad 0\leqslant t\leqslant 1.$ (1.8)

Furthermore, denoting by $\ast$ the convolution operator (either between a
function and a measure or between two measures), we observe that the convex
function $\hat{v}$ and the Bass measure $\hat{\alpha}$ from Definition 1.2
satisfy the identities

$(\nabla\hat{v}\ast\gamma)(\hat{\alpha})=\mu\qquad\textnormal{ and
}\qquad\nabla\hat{v}(\hat{\alpha}\ast\gamma)=\nu.$ (1.9)

Motivated by (1.9), it is natural to extend the definition of Bass martingales
as follows.

######  Definition 1.3.

Let $\mu,\nu,q\in\mathscr{P}_{2}(\mathds{R}^{d})$ and assume that $q$ does not
give mass to small sets. A pair $(\hat{v},\hat{\alpha})$ consisting of a
convex function $\hat{v}\colon\mathds{R}^{d}\rightarrow\mathds{R}$ and a
probability measure $\hat{\alpha}$ on $\mathds{R}^{d}$ is called a $q$-Bass
martingale with initial marginal $\mu$ and terminal marginal $\nu$ if

$(\nabla\hat{v}\star q)(\hat{\alpha})=\mu\qquad\textnormal{ and
}\qquad\nabla\hat{v}(\hat{\alpha}\ast q)=\nu.$ (1.10)

Recall that the symbol $\star$ is defined as in (1.6). We summarize the
fundamental relations (1.10) in the following graphic.

${\hat{\alpha}\ast
q}$${\nu}$${\hat{\alpha}}$${\mu}$$\scriptstyle{\nabla\hat{v}}$$\scriptstyle{\ast}$$\scriptstyle{\nabla\hat{v}\,\star\,q}$

We note that $q$-Bass martingales in the sense of Definition 1.3 are discrete-
time stochastic processes. In the Gaussian case $q=\gamma$, the corresponding
continuous-time martingales can be represented as in (1.7), (1.8). We leave
the definition of general $q$-Bass martingales in continuous time as an open
question.

Next, we define an appropriate class of convex functions
$\hat{v}\colon\mathds{R}^{d}\rightarrow\mathds{R}$ which generate $q$-Bass
martingales in a natural way.

######  Definition 1.4.

Let $\mu,q\in\mathscr{P}_{2}(\mathds{R}^{d})$ and assume that $q$ does not
give mass to small sets. Let
$\hat{v}\colon\mathds{R}^{d}\rightarrow\mathds{R}$ be a convex function and
denote by $\hat{\psi}\coloneqq\hat{v}^{\ast}$ its convex conjugate. We say
that $\hat{v}$ is $q$-Bass martingale generating with initial marginal $\mu$
if

  1. (i)

$\mu(\operatorname{int}(\operatorname{dom}\hat{\psi}))=1$,

  2. (ii)

the function $\hat{v}\star q$ is finite-valued, strictly convex, and
$\nabla(\hat{v}\star q)=(\nabla\hat{v})\star q$,

  3. (iii)

the measure $\nu^{\hat{v}}\coloneqq\nabla\hat{v}(\hat{\alpha}^{\hat{v}}\ast
q)$ has finite second moment, where
$\hat{\alpha}^{\hat{v}}\coloneqq\nabla(\hat{v}\star q)^{\ast}(\mu)$.

In our second main result we show that $q$-Bass martingale generating
functions indeed induce $q$-Bass martingales in the natural way. Beyond that,
we establish the connection to the primal problem (1.3) as well as the dual
problem (1.4).

######  Theorem 1.5.

Let $\mu,q\in\mathscr{P}_{2}(\mathds{R}^{d})$ and assume that $q$ does not
give mass to small sets. Let $\hat{v}$ be $q$-Bass martingale generating with
initial marginal $\mu$ and let $\nu^{\hat{v}},\hat{\alpha}^{\hat{v}}$ be as in
Definition 1.4.

  1. (1)

The pair $(\hat{v},\hat{\alpha}^{\hat{v}})$ is a $q$-Bass martingale with
initial marginal $\mu$ and with terminal marginal $\nu^{\hat{v}}$.

  2. (2)

The optimizer $\hat{\pi}(dx,dy)=\hat{\pi}_{x}(dy)\,\mu(dx)$ of the primal
problem (1.3) between $\mu$ and $\nu=\nu^{\hat{v}}$ is given by

$\hat{\pi}_{x}=\nabla\hat{v}\Big{(}\nabla(\hat{v}\star
q)^{\ast}(x)+\,\cdot\,\Big{)}(q),$

for $\mu$-a.e. $x\in\mathds{R}^{d}$.

  3. (3)

The function $\hat{\psi}=\hat{v}^{\ast}$ is an optimizer of the dual problem
(1.4) between $\mu$ and $\nu=\nu^{\hat{v}}$.

The proof of Theorem 1.5 is presented in Subsection 3.1 of Section 3.

######  Remark 1.6.

  1. (i)

The conditions (i) – (iii) of Definition 1.4 do not necessarily imply that the
function $\hat{\psi}=\hat{v}^{\ast}$ is integrable with respect to the
probability measure $\nu^{\hat{v}}$. Therefore attainment of the dual problem
(1.4) as claimed in part (3) of Theorem 1.5 has to be understood in a
“relaxed” sense frequently encountered in martingale transport problems [BJ16,
BNT17, BNS22]; see Definition 3.4 below.

  2. (ii)

Already in the Gaussian case $q=\gamma$, the conditions (i) – (iii) of
Definition 1.4 are not only sufficient but also necessary for a result in the
form of Theorem 1.5 to be true. More precisely, if $q=\gamma$, it follows from
the results of [BVBST23] that a function $\hat{v}$ satisfying the properties
(1) – (3) of Theorem 1.5 has to be $\gamma$-Bass martingale generating in the
sense of Definition 1.4. $\diamond$

Finally, we raise the question under which assumptions a $q$-Bass martingale
with given marginals $\mu$ and $\nu$ exists. For $\gamma$-Bass martingales
this question was answered in [BVBST23, Theorem 1.3]: a $\gamma$-Bass
martingale with marginals $\mu$ and $\nu$ exists if and only if the pair
$(\mu,\nu)$ is irreducible in the following sense.

######  Definition 1.7.

For probability measures $\mu,\nu$ we say that the pair $(\mu,\nu)$ is
irreducible if for all measurable sets $A,B\subseteq\mathds{R}^{d}$ with
$\mu(A)>0$ and $\nu(B)>0$ there is a martingale $(X_{t})_{0\leqslant
t\leqslant 1}$ with $X_{0}\sim\mu$, $X_{1}\sim\nu$ such that
$\mathds{P}(X_{0}\in A,X_{1}\in B)>0$.

Given an irreducible pair $(\mu,\nu)$, we conjecture that a $q$-Bass
martingale with marginals $\mu$ and $\nu$ exists, provided that $q$ is
absolutely continuous and satisfies suitable integrability conditions.

###  1.4 Related literature

The formulation of optimal transport as a field in mathematics goes back to
Monge [Mon81] and Kantorovich [Kan42]. The modern theory is based on the
seminal contributions of Benamou, Brenier, and McCann [Bre87, Bre91, McC94,
McC95, BB99]. We refer to the books [Vil03, Vil09, AG13, San15] for
introductions to the field of optimal transport as well as a variety of
applications in different areas.

In martingale optimal transport (see e.g. [HN12, BHLP13, TT13, DS14, GHLT14,
CLM17] among many others), the transport plan satisfies an additional
martingale constraint. This additional requirement leads to notable
consequences in mathematical finance [BHLP13], the study of martingale
inequalities [BN15, OST15, HLOST16], and the Skorokhod embedding problem
[BCH17, KTT17, BNS22]. For structural descriptions of optimal martingale
transport plans we refer to [BJ16, OS17, DMT19, GKL19].

Interesting problems in the field of martingale optimal transport are
formulated in continuous time, see e.g. [DS14, BHLT17, COT19, GKP19, GLW19,
GKL20, CKPS21, GL21]. Most notably, the martingale version of the
Benamou–Brenier problem was introduced in [BVBHK20] in probabilistic language
and in [HT19] in PDE language. It was then analyzed from the point of view of
duality theory in [BVBST23] and complemented by a variational perspective in
[BVST23]. In the context of market impact in finance, the same kind of problem
appeared independently in [Loe18, Loe23]. For the interpretation of the
martingale Benamou–Brenier problem as a local volatility model we refer to the
recent articles [BVBHK20, CHL21, AMP23].

##  2 Proof of the duality result

Throughout the rest of this work we fix
$\mu,\nu\in\mathscr{P}_{2}(\mathds{R}^{d})$ with
$\mu\preceq_{\textnormal{c}}\nu$. We also fix a reference measure
$q\in\mathscr{P}_{2}(\mathds{R}^{d})$ which does not give mass to small sets.
The goal of this chapter is to provide the proof of the duality result as
stated in Theorem 1.1. The proof is based on [BVBST23, Theorem 1.4], which is
formulated in the Gaussian setting $q=\gamma$.

###  2.1 A first dual formulation

To formulate a first version of a dual problem to (1.3), we consider the set
of continuous functions with quadratic growth

$C_{2}(\mathds{R}^{d})\coloneqq\big{\\{}\psi\colon\mathds{R}^{d}\rightarrow\mathds{R}\textnormal{
continuous s.t.\ }\exists\,a,k,\ell\in\mathds{R}\textnormal{ with
}\ell+\tfrac{|\,\cdot\,|^{2}}{2}\leqslant\psi(\,\cdot\,)\leqslant
a+k|\cdot|^{2}\big{\\}}.$

We define the dual problem

$\tilde{D}^{q}(\mu,\nu)\coloneqq\inf_{\psi\in
C_{2}(\mathds{R}^{d})}\Big{(}\int\psi\,d\nu-\int\varphi^{\psi}\,d\mu\Big{)},$
(2.1)

where the function $\mathds{R}^{d}\ni x\longmapsto\varphi^{\psi}(x)$ is given
by

$\varphi^{\psi}(x)\coloneqq\inf_{p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\int\psi\,dp-\textnormal{MCov}(p,q)\Big{)}.$
(2.2)

Here, the set $\mathscr{P}_{2}^{x}(\mathds{R}^{d})$ denotes the elements $p$
of $\mathscr{P}_{2}(\mathds{R}^{d})$ with barycenter

$\operatorname{bary}(p)=\int y\,p(dy)=x\in\mathds{R}^{d}.$

######  Proposition 2.1.

There is no duality gap between the primal problem (1.3) and the dual problem
(2.1), i.e., $P^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu)$. Moreover, the primal
problem is uniquely attained and has a finite value, i.e., there exists a
unique $\hat{\pi}\in\mathsf{MT}(\mu,\nu)$ such that

$P^{q}(\mu,\nu)=\int\textnormal{MCov}(\hat{\pi}_{x},q)\,\mu(dx)<+\infty.$
(2.3)

###### Proof.

For $x\in\mathds{R}^{d}$ and $p\in\mathscr{P}_{2}(\mathds{R}^{d})$ we define
the cost function

$C(x,p)\coloneqq\begin{cases}-\textnormal{MCov}(p,q)+\tfrac{1}{2}\int|y|^{2}\,dp,&\textnormal{
if }\operatorname{bary}(p)=x,\\\ +\infty,&\textnormal{ if
}\operatorname{bary}(p)\neq x,\end{cases}$

which is bounded from below and convex in the second argument. If we equip
$\mathscr{P}_{2}(\mathds{R}^{d})$ with the topology induced by the quadratic
Wasserstein distance, one verifies that the function $(x,p)\mapsto C(x,p)$ is
jointly lower semicontinuous with respect to the product topology on
$\mathds{R}^{d}\times\mathscr{P}_{2}(\mathds{R}^{d})$. Hence we can apply
[BVBP19, Theorem 1.2], which guarantees the existence of an optimizer
$\hat{\pi}\in\mathsf{MT}(\mu,\nu)$ of the primal problem (1.3). This optimizer
is unique, as the value $P^{q}(\mu,\nu)$ is finite and $(x,p)\mapsto C(x,p)$
is strictly convex in the second argument. The former follows from the fact
that $\mu$ and $\nu$ have finite second moments, the latter is a consequence
of Brenier’s theorem.

In order to show that there is no duality gap, we introduce the space of
continuous functions which are bounded from below and have at most quadratic
growth

$C_{\textnormal{b},2}(\mathds{R}^{d})\coloneqq\big{\\{}\tilde{\psi}\colon\mathds{R}^{d}\rightarrow\mathds{R}\textnormal{
continuous s.t.\ }\exists\,a,k,\ell\in\mathds{R}\textnormal{ with
}\ell\leqslant\tilde{\psi}(\,\cdot\,)\leqslant a+k|\cdot|^{2}\big{\\}}.$

Then by [BVBP19, Theorem 1.3] the value $P^{q}(\mu,\nu)$ of the primal problem
is equal to

$\tilde{D}_{\textnormal{b},2}^{q}(\mu,\nu)\coloneqq\inf_{\tilde{\psi}\in
C_{\textnormal{b},2}(\mathds{R}^{d})}\Big{(}\int\big{(}\tilde{\psi}(\,\cdot\,)+\tfrac{|\,\cdot\,|^{2}}{2}\big{)}\,d\nu-\int\tilde{\varphi}^{\tilde{\psi}}\,d\mu\Big{)},$

where

$\tilde{\varphi}^{\tilde{\psi}}(x)\coloneqq\inf_{p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\int\big{(}\tilde{\psi}(\,\cdot\,)+\tfrac{|\,\cdot\,|^{2}}{2}\big{)}\,dp-\textnormal{MCov}(p,q)\Big{)}.$

Finally, passing from the functions $\tilde{\psi}\in
C_{\textnormal{b},2}(\mathds{R}^{d})$ to
$\psi(\,\cdot\,)\coloneqq\tilde{\psi}(\,\cdot\,)+\frac{|\,\cdot\,|^{2}}{2}\in
C_{2}(\mathds{R}^{d})$, we see that
$\tilde{D}_{\textnormal{b},2}^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu)$. ∎

We close this section by providing a heuristic argument why (2.1) is indeed a
plausible dual formulation of (1.3). Considering the “indicator function”

$\chi(\pi)\coloneqq\inf_{\psi\in
C_{2}(\mathds{R}^{d})}\Big{(}\int\psi\,d\nu-\int\psi\,d\pi_{x}\,d\mu(x)\Big{)}=\begin{cases}0,&\mbox{if
}\int\pi_{x}\,d\mu(x)=\nu,\\\ -\infty,&\mbox{else},\end{cases}$

we formally obtain the desired duality relation by interchanging $\inf$ and
$\sup$, to wit

$P^{q}(\mu,\nu)=\sup_{\begin{subarray}{c}\pi(dx,dy)=\pi_{x}(dy)\,\mu(dx),\\\
\pi_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})\end{subarray}}\,\Big{(}\int\textnormal{MCov}(\pi_{x},q)\,d\mu(x)+\chi(\pi)\Big{)}\\\
=\inf_{\psi\in
C_{2}(\mathds{R}^{d})}\bigg{(}\int\sup_{\pi_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(\pi_{x},q)+\int\psi\,d(\nu-\pi_{x})\Big{)}\,d\mu(x)\bigg{)}=\tilde{D}^{q}(\mu,\nu).$

###  2.2 Convexity of the dual variables

In this section we make the crucial observation that in the dual problem (2.1)
it suffices to optimize over the class of functions $\psi\in
C_{2}(\mathds{R}^{d})$ which are convex. We then show that it is also
equivalent to optimize over all convex functions
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ which are only
$\mu$-a.s. finite, but not necessarily of quadratic growth.

Given some $\psi\in C_{2}(\mathds{R}^{d})$, we denote by
$\operatorname{conv}\psi$ the convex hull of $\psi$, i.e., the greatest convex
function smaller or equal to $\psi$. It will be convenient to have an explicit
representation for the convex hull of a function, as in (2.4) below. This
identity is usually stated in the more specific form

$(\operatorname{conv}\psi)(y)=\inf\bigg{\\{}\sum_{i=1}^{d+1}\lambda_{i}\psi(y_{i})\colon\sum_{i=1}^{d+1}\lambda_{i}y_{i}=y\bigg{\\}},$

where the infimum is taken over all expressions of $y$ as a convex combination
of $d+1$ points, see [Roc70, Corollary 17.1.5].

######  Lemma 2.2.

For a function $\psi\in C_{2}(\mathds{R}^{d})$ its convex hull
$\operatorname{conv}\psi$ is given by

$\mathds{R}^{d}\ni
y\longmapsto(\operatorname{conv}\psi)(y)=\inf_{p\in\mathscr{P}_{2}^{y}(\mathds{R}^{d})}\int\psi\,dp$
(2.4)

and again $\operatorname{conv}\psi\in C_{2}(\mathds{R}^{d})$.

Recalling the dual problem (2.1), we define the dual function

$C_{2}(\mathds{R}^{d})\ni\psi\longmapsto\tilde{\mathcal{D}}(\psi)\coloneqq\int\psi\,d\nu-\int\varphi^{\psi}\,d\mu.$
(2.5)

Now we prove that it suffices to optimize the dual function over the class of
functions $\psi\in C_{2}(\mathds{R}^{d})$ which are convex.

######  Proposition 2.3.

For all $\psi\in C_{2}(\mathds{R}^{d})$ we have
$\tilde{\mathcal{D}}(\operatorname{conv}\psi)\leqslant\tilde{\mathcal{D}}(\psi)$
and consequently

$\tilde{D}^{q}(\mu,\nu)=\inf_{\psi\in
C_{2}(\mathds{R}^{d})}\tilde{\mathcal{D}}(\psi)=\inf_{\begin{subarray}{c}\psi\in
C_{2}(\mathds{R}^{d}),\\\ \textnormal{$\psi$
convex}\end{subarray}}\tilde{\mathcal{D}}(\psi).$ (2.6)

###### Proof.

Let $\varepsilon>0$, $\psi\in C_{2}(\mathds{R}^{d})$ and
$\\{p_{x}\\}_{x\in\mathds{R}^{d}}\subseteq\mathscr{P}_{2}(\mathds{R}^{d})$ be
a measurable collection of probability measures with
$\operatorname{bary}(p_{x})=x$. To show the claim, it is sufficient to
construct a measurable family
$\\{\bar{p}_{x}\\}_{x\in\mathds{R}^{d}}\subseteq\mathscr{P}_{2}(\mathds{R}^{d})$
with $\operatorname{bary}(\bar{p}_{x})=x$ such that

$\textnormal{MCov}(p_{x},q)+\int\operatorname{conv}\psi\,d(\nu-
p_{x})\leqslant\textnormal{MCov}(\bar{p}_{x},q)+\int\psi\,d(\nu-\bar{p}_{x})+\varepsilon.$
(2.7)

By Lemma 2.2 and a measurable selection argument we can choose a measurable
collection of probability measures
$\\{\tilde{p}_{y}\\}_{y\in\mathds{R}^{d}}\subseteq\mathscr{P}_{2}(\mathds{R}^{d})$
with $\operatorname{bary}(\tilde{p}_{y})=y$ such that

$\int\psi\,d\tilde{p}_{y}\leqslant(\operatorname{conv}\psi)(y)+\varepsilon.$
(2.8)

Then we define $\bar{p}_{x}(dz)\coloneqq\int_{y}\tilde{p}_{y}(dz)\,p_{x}(dy)$,
so that $\operatorname{bary}(\bar{p}_{x})=x$. Integrating (2.8) with respect
to $p_{x}(dy)$ yields

$\int\psi\,d\bar{p}_{x}\leqslant\int\operatorname{conv}\psi\,dp_{x}+\varepsilon.$
(2.9)

Since $\psi,\operatorname{conv}\psi\in C_{2}(\mathds{R}^{d})$ and
$p_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$ we conclude

$\ell+\tfrac{1}{2}\int|y|^{2}\,\bar{p}_{x}(dy)\leqslant
a+k\int|y|^{2}\,p_{x}(dy)+\varepsilon<+\infty,$

so that $\bar{p}_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$.

In order to show the inequality (2.7), we first observe that
$p_{x}\preceq_{\textnormal{c}}\bar{p}_{x}$ by Jensen’s inequality. Together
with the Kantorovich duality (see, e.g., [Vil09, Theorem 5.10]), we conclude
that111In fact, $p_{x}\preceq_{\textnormal{c}}\bar{p}_{x}$ is equivalent to
the inequality
$\textnormal{MCov}(p_{x},q)\leqslant\textnormal{MCov}(\bar{p}_{x},q)$ being
valid for all probability measures $q\in\mathscr{P}_{2}(\mathds{R}^{d})$;
see., e.g., [AP22, Theorem 1] or [WZ23, Corollary 1.2].

$\displaystyle\textnormal{MCov}(p_{x},q)$
$\displaystyle=\inf_{\begin{subarray}{c}f\colon\mathds{R}^{d}\rightarrow\mathds{R}\\\
\textnormal{convex}\end{subarray}}\Big{(}\int f\,dp_{x}+\int
f^{\ast}\,dq\Big{)}$ (2.10)
$\displaystyle\leqslant\inf_{\begin{subarray}{c}f\colon\mathds{R}^{d}\rightarrow\mathds{R}\\\
\textnormal{convex}\end{subarray}}\Big{(}\int f\,d\bar{p}_{x}+\int
f^{\ast}\,dq\Big{)}=\textnormal{MCov}(\bar{p}_{x},q).$

On the other hand, from $\operatorname{conv}\psi\leqslant\psi$ and (2.9) we
have the inequality

$\int\operatorname{conv}\psi\,d(\nu-
p_{x})\leqslant\int\psi\,d(\nu-\bar{p}_{x})+\varepsilon.$ (2.11)

Finally, summing (2.10) and (2.11), we obtain the inequality (2.7). ∎

Next, we extend the definition (2.5) of the dual function
$\tilde{\mathcal{D}}(\,\cdot\,)$ to the class of convex functions
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ which are not confined
to be in $C_{2}(\mathds{R}^{d})$, but which are only required to satisfy
$\mu(\operatorname{dom}\psi)=1$, where

$\operatorname{dom}\psi\coloneqq\\{y\in\mathds{R}^{d}\colon\psi(y)<+\infty\\}$

denotes the domain of $\psi$. In order for the difference of the integrals in
(2.5) still to be well-defined, the dual function has to be understood in a
“relaxed” sense, as frequently encountered in martingale transport problems;
see [BJ16, BNT17, BNS22]. More precisely, we define the “relaxed”
representation of (2.5) by

$\tilde{\mathcal{E}}(\psi)\coloneqq\int\Big{(}\int\psi(y)\,\pi_{x}(dy)-\varphi^{\psi}(x)\Big{)}\,\mu(dx),$
(2.12)

where $\pi(dx,dy)=\pi_{x}(dy)\,\mu(dx)$ is an arbitrary fixed element of
$\mathsf{MT}(\mu,\nu)$. Then $\tilde{\mathcal{E}}(\psi)$ is well-defined for
every convex function $\psi$ satisfying $\mu(\operatorname{dom}\psi)=1$.
Taking the infimum of (2.12) over all such $\psi$ again leads to the same
value as in (2.6). This is summarized in the following lemma.

######  Lemma 2.4.

Fix $\pi\in\mathsf{MT}(\mu,\nu)$ and let
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a convex function
with $\mu(\operatorname{dom}\psi)=1$. Then
$\tilde{\mathcal{E}}(\psi)\in[0,+\infty]$ and we have the inequality

$\tilde{D}^{q}(\mu,\nu)\leqslant\tilde{\mathcal{E}}(\psi).$ (2.13)

In particular, recalling (2.6), we have

$\tilde{D}^{q}(\mu,\nu)=\inf_{\begin{subarray}{c}\psi\in
C_{2}(\mathds{R}^{d}),\\\ \textnormal{$\psi$
convex}\end{subarray}}\tilde{\mathcal{D}}(\psi)=\inf_{\begin{subarray}{c}\mu(\operatorname{dom}\psi)=1,\\\
\textnormal{$\psi$ convex}\end{subarray}}\tilde{\mathcal{E}}(\psi).$ (2.14)

###### Proof.

Fix $\pi\in\mathsf{MT}(\mu,\nu)$ and let
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a convex function
with $\mu(\operatorname{dom}\psi)=1$. First, note that by Jensen’s inequality
we have

$\int\psi(y)\,\pi_{x}(dy)\geqslant\psi\Big{(}\int
y\,\pi_{x}(dy)\Big{)}=\psi(x),$

and by taking $p=\delta_{x}$ in (2.2) we obtain
$\varphi^{\psi}(x)\leqslant\psi(x)$. Hence

$\int\psi(y)\,\pi_{x}(dy)-\varphi^{\psi}(x)\geqslant 0,$ (2.15)

for $\mu$-a.e. $x\in\mathds{R}^{d}$, so that
$\tilde{\mathcal{E}}(\psi)\in[0,+\infty]$. In order to prove the inequality
(2.13), we distinguish two cases. In the case $\int\psi\,d\pi_{x}=+\infty$,
for $x\in\mathds{R}^{d}$ in a set of positive $\mu$-measure, we conclude from
(2.15) that $\tilde{\mathcal{E}}(\psi)=+\infty$, so that (2.13) is trivially
satisfied. Now suppose that $\int\psi\,d\pi_{x}<+\infty$, for $\mu$-a.e.
$x\in\mathds{R}^{d}$. Recall from Proposition 2.1 that there exists an
optimizer $\hat{\pi}\in\mathsf{MT}(\mu,\nu)$ of the primal problem (1.3).
Taking $p=\hat{\pi}_{x}$ in (2.2) we get

$\varphi^{\psi}(x)\leqslant\int\psi\,d\hat{\pi}_{x}-\textnormal{MCov}(\hat{\pi}_{x},q)$

and therefore

$\tilde{\mathcal{E}}(\psi)\geqslant\int\Big{(}\int\psi\,d\pi_{x}+\textnormal{MCov}(\hat{\pi}_{x},q)-\int\psi\,d\hat{\pi}_{x}\Big{)}\,\mu(dx)=P^{q}(\mu,\nu),$

where we also used that both $\hat{\pi}$ and $\pi$ have $\nu$ as second
marginal, and that $\hat{\pi}$ is a primal optimizer, i.e., satisfying (2.3).
Since $P^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu)$ by Proposition 2.1, we again see
that (2.13) does hold.

Finally, from (2.13) and recalling (2.6), we conclude

$\tilde{D}^{q}(\mu,\nu)\leqslant\inf_{\begin{subarray}{c}\mu(\operatorname{dom}\psi)=1,\\\
\textnormal{$\psi$
convex}\end{subarray}}\tilde{\mathcal{E}}(\psi)\leqslant\inf_{\begin{subarray}{c}\psi\in
C_{2}(\mathds{R}^{d}),\\\ \textnormal{$\psi$
convex}\end{subarray}}\tilde{\mathcal{D}}(\psi)=\tilde{D}^{q}(\mu,\nu),$

which shows (2.14). ∎

###  2.3 Proof of Theorem 1.1

According to Proposition 2.1, there is no duality gap between the primal
problem (1.3) and the dual problem (2.1), i.e.,
$P^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu)$. Hence for the proof of Theorem 1.1 we
have to show that

$\tilde{D}^{q}(\mu,\nu)=\inf_{\begin{subarray}{c}\psi\in L^{1}(\nu),\\\
\textnormal{$\psi$
convex}\end{subarray}}\Big{(}\int\psi\,d\nu-\int(\psi^{\ast}\star
q)^{\ast}\,d\mu\Big{)}.$ (2.16)

Recalling the definitions (2.1), (2.2), and (2.5), it should come as no
surprise that the main idea behind the proof of the identity (2.16) is to
apply Proposition 2.3 and to show that $\varphi^{\psi}=(\psi^{\ast}\star
q)^{\ast}$ holds for every convex function $\psi\in C_{2}(\mathds{R}^{d})$.
This motivates our next goal, namely to solve the minimization problem (2.2),
which we rewrite as a maximization problem, to wit

$-\varphi^{\psi}(x)=\sup_{p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(p,q)-\int\psi\,dp\Big{)}.$
(2.17)

As a preliminary step, we consider the simpler problem

$\phi^{\psi}\coloneqq\sup_{p\in\mathscr{P}_{2}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(p,q)-\int\psi\,dp\Big{)},$
(2.18)

where we do not prescribe the barycenter $x\in\mathds{R}^{d}$ of
$p\in\mathscr{P}_{2}(\mathds{R}^{d})$. In Lemma 2.6 below we will show for an
arbitrary proper convex function
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$, that the value
$\phi^{\psi}$ of (2.18) is equal to $\int\psi^{\ast}\,dq$. Recall that $\psi$
is called a proper convex function if $\psi$ is convex and
$\operatorname{dom}\psi\neq\varnothing$.

Solving the maximization problem (2.18) leads to an interesting connection
with Brenier maps, see Lemma 2.5 below. By Brenier’s theorem (see, e.g.,
[Vil03, Theorem 2.12]), the optimal transport for quadratic cost between
$q\in\mathscr{P}_{2}(\mathds{R}^{d})$ and
$p\in\mathscr{P}_{2}(\mathds{R}^{d})$ is induced by the $q$-a.e. defined
gradient $\nabla v$ (the “Brenier map”) of some convex function
$v\colon\mathds{R}^{d}\rightarrow\mathds{R}$ via $(\nabla v)(q)=p$.

######  Lemma 2.5.

Let $v\colon\mathds{R}^{d}\rightarrow\mathds{R}$ be a finite-valued convex
function and $\psi\coloneqq v^{\ast}$ its convex conjugate. Assume that the
probability measure $\hat{p}\coloneqq(\nabla v)(q)$ has finite second moment.
Then $\hat{p}$ is the unique maximizer of the optimization problem (2.18) and

$\phi^{\psi}=\int v\,dq=\int\psi^{\ast}\,dq<+\infty.$

###### Proof.

For $p\in\mathscr{P}_{2}(\mathds{R}^{d})$, we denote by $T_{q}^{p}$ the
Brenier map from $q$ to $p\in\mathscr{P}_{2}(\mathds{R}^{d})$ and note that
$T_{q}^{\hat{p}}=\nabla v$ by Brenier’s theorem. Furthermore, we have that

$v(z)=\sup_{y\in\mathds{R}^{d}}\big{(}\langle
y,z\rangle-v^{\ast}(y)\big{)}=\big{\langle}\nabla
v(z),z\big{\rangle}-v^{\ast}\big{(}\nabla v(z)\big{)},$

for $q$-a.e. $z\in\mathds{R}^{d}$. Using these observations, for
$p\in\mathscr{P}_{2}(\mathds{R}^{d})$ we get

$\displaystyle\textnormal{MCov}(p,q)-\int\psi\,dp$
$\displaystyle=\int\Big{(}\big{\langle}T_{q}^{p}(z),z\big{\rangle}-v^{\ast}\big{(}T_{q}^{p}(z)\big{)}\Big{)}\,q(dz)$
$\displaystyle\leqslant\int\sup_{y\in\mathds{R}^{d}}\big{(}\langle
y,z\rangle-v^{\ast}(y)\big{)}\,q(dz)=\int v(z)\,q(dz)$
$\displaystyle=\int\Big{(}\big{\langle}\nabla
v(z),z\big{\rangle}-v^{\ast}\big{(}\nabla v(z)\big{)}\Big{)}\,q(dz)$
$\displaystyle=\textnormal{MCov}(\hat{p},q)-\int\psi\,d\hat{p},$

with equality if and only if $T_{q}^{p}(z)=T_{q}^{\hat{p}}(z)$, for $q$-a.e.
$z\in\mathds{R}^{d}$. This in turn is the case if and only if $p=\hat{p}$.
Finally, from the convexity of $v$ and the Cauchy–Schwarz inequality we obtain

$\int|v|\,dq\leqslant|v(0)|+\sqrt{\int|\nabla v|^{2}\,dq}\
\sqrt{\int|z|^{2}\,dq(z)}<+\infty,$

which proves that $\phi^{\psi}<+\infty$. ∎

Next, we relax the rather strong assumptions of Lemma 2.5. If we are given
just a proper convex function
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$, we can still compute
the value of the supremum in (2.18), without explicitly constructing a
maximizer of this optimization problem.

######  Lemma 2.6.

Let $\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a proper convex
function. Then

$\phi^{\psi}=\int\psi^{\ast}\,dq.$ (2.19)

###### Proof.

Using probabilistic notation, we rewrite the supremum in (2.18) as

$\phi^{\psi}=\sup\mathds{E}\big{[}\langle Y,Z\rangle-\psi(Y)\big{]},$ (2.20)

where the supremum in (2.20) is taken over all probability spaces such that
$Z\sim q$ and $Y$ is an $\mathds{R}^{d}$-valued random variable with finite
second moment. Replacing $Y$ by $\mathds{E}[Y\,|\,Z]$, we observe that the
maximization in (2.20) can be restricted to random variables $Y$ which are
measurable functions of $Z$, and we obtain

$\phi^{\psi}=\sup_{Y\in
L^{2}(q;\mathds{R}^{d})}\int\Big{(}\big{\langle}Y(z),z\big{\rangle}-\psi\big{(}Y(z)\big{)}\Big{)}\,q(dz),$
(2.21)

where $L^{2}(q;\mathds{R}^{d})$ denotes the space of $\mathds{R}^{d}$-valued
Borel measurable functions on $\mathds{R}^{d}$, which are square-integrable
under $q$. Clearly, for any $Y\in L^{2}(q;\mathds{R}^{d})$ we have

$\int\Big{(}\big{\langle}Y(z),z\big{\rangle}-\psi\big{(}Y(z)\big{)}\Big{)}\,q(dz)\leqslant\int\sup_{y\in\mathds{R}^{d}}\big{(}\langle
y,z\rangle-\psi(y)\big{)}\,q(dz)=\int\psi^{\ast}(z)\,q(dz),$

which shows the inequality $\phi^{\psi}\leqslant\int\psi^{\ast}\,dq$. In order
to see the reverse inequality, we define the auxiliary problem

$\phi_{\infty}^{\psi}\coloneqq\sup_{Y\in
L^{\infty}(q;\mathds{R}^{d})}\int\Big{(}\big{\langle}Y(z),z\big{\rangle}-\psi\big{(}Y(z)\big{)}\Big{)}\,q(dz),$
(2.22)

where $L^{\infty}(q;\mathds{R}^{d})$ denotes the space of
$\mathds{R}^{d}$-valued Borel measurable functions on $\mathds{R}^{d}$, which
are bounded $q$-a.e. Comparing (2.21) with (2.22), we obviously have
$\phi^{\psi}\geqslant\phi_{\infty}^{\psi}$. Now we claim that

$\phi_{\infty}^{\psi}\geqslant\int\sup_{y\in\mathds{R}^{d}}\big{(}\langle
y,z\rangle-\psi(y)\big{)}\,q(dz)=\int\psi^{\ast}(z)\,q(dz),$ (2.23)

which will finish the proof of (2.19). To see this, we first write

$\phi_{\infty}^{\psi}=\lim_{N\rightarrow\infty}\,\sup_{\begin{subarray}{c}Y\in
L^{\infty}(q;\mathds{R}^{d}),\\\ |Y|\leqslant
N\end{subarray}}\,\int\Big{(}\big{\langle}Y(z),z\big{\rangle}-\psi\big{(}Y(z)\big{)}\Big{)}\,q(dz).$

Using a measurable selection argument, we obtain

$\sup_{\begin{subarray}{c}Y\in L^{\infty}(q;\mathds{R}^{d}),\\\ |Y|\leqslant
N\end{subarray}}\,\int\Big{(}\big{\langle}Y(z),z\big{\rangle}-\psi\big{(}Y(z)\big{)}\Big{)}\,q(dz)\geqslant\int\sup_{\begin{subarray}{c}y\in\mathds{R}^{d},\\\
|y|\leqslant N\end{subarray}}\,\big{(}\langle
y,z\rangle-\psi(y)\big{)}\,q(dz).$

Since $\psi$ is proper we can choose
$y_{0}\in\operatorname{dom}\psi\neq\varnothing$. Then for $N$ large enough we
have

$\sup_{\begin{subarray}{c}y\in\mathds{R}^{d},\\\ |y|\leqslant
N\end{subarray}}\,\big{(}\langle y,z\rangle-\psi(y)\big{)}\geqslant\langle
y_{0},z\rangle-\psi(y_{0}),$

with the right-hand side being integrable with respect to $q(dz)$. Hence we
can apply the monotone convergence theorem and deduce that

$\lim_{N\rightarrow\infty}\int\sup_{\begin{subarray}{c}y\in\mathds{R}^{d},\\\
|y|\leqslant N\end{subarray}}\,\big{(}\langle
y,z\rangle-\psi(y)\big{)}\,q(dz)=\int\sup_{y\in\mathds{R}^{d}}\big{(}\langle
y,z\rangle-\psi(y)\big{)}\,q(dz),$

which completes the proof of (2.23). ∎

Lemma 2.6 is an important auxiliary result for the proof of Theorem 1.1, more
precisely, for the proof of the identity (2.16). We first show a simpler
variant of this identity, where we optimize over the class of convex functions
$\psi$ in $C_{2}(\mathds{R}^{d})$ (i.e., which have quadratic growth), as in
(2.24) of Proposition 2.7 below.

######  Proposition 2.7.

The value $\tilde{D}^{q}(\mu,\nu)$ of the dual problem (2.1) is equal to

$D_{2}^{q}(\mu,\nu)\coloneqq\inf_{\begin{subarray}{c}\psi\in
C_{2}(\mathds{R}^{d}),\\\ \textnormal{$\psi$
convex}\end{subarray}}\Big{(}\int\psi\,d\nu-\int(\psi^{\ast}\star
q)^{\ast}\,d\mu\Big{)}.$ (2.24)

###### Proof.

By Proposition 2.3 we have

$\tilde{D}^{q}(\mu,\nu)=\inf_{\begin{subarray}{c}\psi\in
C_{2}(\mathds{R}^{d}),\\\ \textnormal{$\psi$
convex}\end{subarray}}\Big{(}\int\psi\,d\nu-\int\varphi^{\psi}\,d\mu\Big{)}.$

Hence it remains to show that $\varphi^{\psi}=(\psi^{\ast}\star q)^{\ast}$,
for every convex function $\psi\in C_{2}(\mathds{R}^{d})$. In order to do
this, we will first prove that $(\varphi^{\psi})^{\ast}=\psi^{\ast}\star q$.
By definition of the convex conjugate (1.5) and of the maximization problem
(2.17), for $y\in\mathds{R}^{d}$, we have

$\displaystyle(\varphi^{\psi})^{\ast}(y)$
$\displaystyle=\sup_{x\in\mathds{R}^{d}}\big{(}\langle
x,y\rangle-\varphi^{\psi}(x)\big{)}$ (2.25)
$\displaystyle=\sup_{x\in\mathds{R}^{d}}\sup_{p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(p,q)-\int\psi_{y}\,dp\Big{)}$
(2.26)
$\displaystyle=\sup_{p\in\mathscr{P}_{2}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(p,q)-\int\psi_{y}\,dp\Big{)},$
(2.27)

where the function $\psi_{y}$ is defined by
$\psi_{y}(z)\coloneqq\psi(z)-\langle y,z\rangle$, for $z\in\mathds{R}^{d}$.
Now applying Lemma 2.6 to the proper convex function $\psi_{y}$ yields

$(\varphi^{\psi})^{\ast}(y)=\phi^{\psi_{y}}=\int\psi_{y}^{\ast}\,dq=\int\psi^{\ast}(y+z)\,dq(z)=(\psi^{\ast}\star
q)(y),$ (2.28)

where for the last equality we recall the definition (1.6). To complete the
proof, we must justify that $(\varphi^{\psi})^{\ast\ast}=\varphi^{\psi}$,
i.e., that the Fenchel–Moreau theorem is applicable. To see this, we first
note that the function $x\mapsto\varphi^{\psi}(x)$ is convex and we have the
upper bound $\varphi^{\psi}\leqslant\psi<+\infty$. Furthermore, for any
$p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$ we have the inequalities

$\textnormal{MCov}(p,q)\leqslant\tfrac{1}{2}\int|y|^{2}\,dp(y)+\tfrac{1}{2}\int|z|^{2}\,dq(z)<+\infty$

as well as

$\int\psi\,dp\geqslant\ell+\tfrac{1}{2}\int|y|^{2}\,dp(y)>-\infty,$

the latter following from the fact that $\psi\in C_{2}(\mathds{R}^{d})$. As a
consequence, we get the lower bound

$\varphi^{\psi}\geqslant\ell-\tfrac{1}{2}\int|z|^{2}\,dq(z)>-\infty.$

Altogether, $\varphi^{\psi}$ is a convex function which is finite everywhere
on $\mathds{R}^{d}$, thus it is continuous and we indeed have
$(\varphi^{\psi})^{\ast\ast}=\varphi^{\psi}$. ∎

Before we finally turn to the proof of Theorem 1.1 at the end of this section,
we formulate a “relaxed” version of (1.4), (2.24). Similarly as in Lemma 2.4
we now admit convex functions
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ with
$\mu(\operatorname{dom}\psi)=1$ as dual variables, see (2.29) of Proposition
2.8 below.

######  Proposition 2.8.

The value $\tilde{D}^{q}(\mu,\nu)$ of the dual problem (2.1) is equal to

$D_{\textnormal{rel}}^{q}(\mu,\nu)\coloneqq\inf_{\begin{subarray}{c}\mu(\operatorname{dom}\psi)=1,\\\
\textnormal{$\psi$ convex}\end{subarray}}\mathcal{E}(\psi),$ (2.29)

where the dual function $\psi\mapsto\mathcal{E}(\psi)$ is defined by

$\mathcal{E}(\psi)\coloneqq\int\Big{(}\int\psi(y)\,\pi_{x}(dy)-(\psi^{\ast}\star
q)^{\ast}(x)\Big{)}\,\mu(dx),$ (2.30)

with $\pi$ being an arbitrary fixed element of $\mathsf{MT}(\mu,\nu)$.

###### Proof.

We start with showing that the dual function $\mathcal{E}(\,\cdot\,)$ given in
(2.30) is well-defined for every convex function
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ which is $\mu$-a.s.
finite. To this end, we first prove the inequality $(\psi^{\ast}\star
q)^{\ast}\leqslant\psi$. Since $\psi$ is convex with
$\mu(\operatorname{dom}\psi)=1$, for every $y\in\mathds{R}^{d}$, the function

$\mathds{R}^{d}\ni z\longmapsto\psi_{y}(z)\coloneqq\psi(z)-\langle y,z\rangle$
(2.31)

is a proper convex function, so that as in the proof of Proposition 2.7 above
we can apply Lemma 2.6. In particular, recalling the equation (2.28), we again
have $\psi^{\ast}\star q=(\varphi^{\psi})^{\ast}$. Taking the convex conjugate
and using that $\varphi^{\psi}\leqslant\psi$, we obtain

$(\psi^{\ast}\star
q)^{\ast}=(\varphi^{\psi})^{\ast\ast}\leqslant\varphi^{\psi}\leqslant\psi,$
(2.32)

as required. Applying Jensen’s inequality as in the proof of Lemma 2.4, this
implies that the integrand in (2.30) is $\mu$-a.s. non-negative, and hence
$\mathcal{E}(\psi)$ is well-defined and takes values in the interval
$[0,+\infty]$.

Now let us turn to the proof of
$\tilde{D}^{q}(\mu,\nu)=D_{\textnormal{rel}}^{q}(\mu,\nu)$. Recalling (2.29),
(2.24), and using Proposition 2.7, we have

$D_{\textnormal{rel}}^{q}(\mu,\nu)\leqslant
D_{2}^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu),$

so that we need to show the inequality $\tilde{D}^{q}(\mu,\nu)\leqslant
D_{\textnormal{rel}}^{q}(\mu,\nu)$. To this end, let
$\pi\in\mathsf{MT}(\mu,\nu)$ and
$\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be convex with
$\mu(\operatorname{dom}\psi)=1$. Since $\tilde{D}^{q}(\mu,\nu)=P^{q}(\mu,\nu)$
(recall Proposition 2.1 and (2.3)), we have to verify the inequality

$\int\textnormal{MCov}(\pi_{x},q)\,d\mu(x)\leqslant\int\Big{(}\int\psi(y)\,\pi_{x}(dy)-(\psi^{\ast}\star
q)^{\ast}(x)\Big{)}\,\mu(dx).$

It is sufficient to prove, for $\mu$-a.e. $x\in\mathds{R}^{d}$, that

$\textnormal{MCov}(\pi_{x},q)\leqslant\int\psi\,d\pi_{x}-(\psi^{\ast}\star
q)^{\ast}(x).$ (2.33)

We express the convex conjugate on the right-hand side of (2.33) as

$-(\psi^{\ast}\star
q)^{\ast}(x)=\inf_{y\in\mathds{R}^{d}}\big{(}(\psi^{\ast}\star q)(y)-\langle
x,y\rangle\big{)}=\inf_{y\in\mathds{R}^{d}}\Big{(}\int\psi_{y}^{\ast}\,dq-\langle
x,y\rangle\Big{)};$

for the last equality we again relied on the equation (2.28), and the function
$\psi_{y}$ is defined as in (2.31). Substituting back into (2.33) yields

$\textnormal{MCov}(\pi_{x},q)\leqslant\inf_{y\in\mathds{R}^{d}}\Big{(}\int\psi_{y}\,d\pi_{x}+\int\psi_{y}^{\ast}\,dq\Big{)}.$
(2.34)

This is what we need to show to complete the proof of Proposition 2.8. Now
observe that by the Fenchel–Young inequality we have

$\textnormal{MCov}(\pi_{x},q)\leqslant\int f\,d\pi_{x}+\int f^{\ast}\,dq,$

for every proper convex function
$f\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$. In particular, for every
$y\in\mathds{R}^{d}$, it holds that

$\textnormal{MCov}(\pi_{x},q)\leqslant\int\psi_{y}\,d\pi_{x}+\int\psi_{y}^{\ast}\,dq,$

which implies (2.34), as required. ∎

Finally, we give the proof of Theorem 1.1, where we optimize over all
$\nu$-integrable convex functions $\psi$ as in (1.4), (2.16).

###### Proof of Theorem 1.1.

We first show that the integral

$\int(\psi^{\ast}\star q)^{\ast}\,d\mu$ (2.35)

appearing on the right-hand side of (1.4), (2.16) is well-defined for every
convex function $\psi\in L^{1}(\nu)$. To see this, recall from (2.32) above
that $(\psi^{\ast}\star q)^{\ast}\leqslant\psi$. On the other hand, since
$\mu\preceq_{\textnormal{c}}\nu$ and $\psi\in L^{1}(\nu)$ is convex, the
integral of $\psi$ with respect to $\mu$ exists. We conclude that also the
integral (2.35) exists.

Now let us turn to the actual proof of Theorem 1.1. Recalling the beginning of
Subsection 2.3, we have to prove the equality (2.16). But this assertion
follows immediately from Proposition 2.8 and Proposition 2.7 by a “sandwich
argument”. Indeed, we have

$\tilde{D}^{q}(\mu,\nu)=D_{\textnormal{rel}}^{q}(\mu,\nu)\leqslant
D^{q}(\mu,\nu)\leqslant D_{2}^{q}(\mu,\nu)=\tilde{D}^{q}(\mu,\nu).$ (2.36)

The inequalities in (2.36) are due to the inclusions

$\\{\psi\in C_{2}(\mathds{R}^{d})\colon\textnormal{$\psi$
convex}\\}\subseteq\\{\psi\in L^{1}(\nu)\colon\textnormal{$\psi$
convex}\\}\subseteq\\{\textnormal{$\psi$
convex}\colon\mu(\operatorname{dom}\psi)=1\\};$

the equalities on the left-hand side and on the right-hand side of (2.36) are
justified by Proposition 2.8 and Proposition 2.7, respectively.

Finally, we recall that existence and uniqueness of the optimizer
$\hat{\pi}\in\mathsf{MT}(\mu,\nu)$ as well as finiteness of $P^{q}(\mu,\nu)$
have already been shown in Proposition 2.1. ∎

##  3 Existence and properties of $q$-Bass martingales

Throughout this chapter we fix $\mu,q\in\mathscr{P}_{2}(\mathds{R}^{d})$ and
assume that $q$ does not give mass to small sets. Our goal is to prove our
second main result, Theorem 1.5. Its proof is based on the work [BVBST23],
where the Gaussian case $q=\gamma$ is studied in detail. First, we need some
auxiliary results.

######  Lemma 3.1.

Let $\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a proper, lower
semicontinuous convex function.

  1. (i)

The function $x\mapsto\varphi^{\psi}(x)$ defined in (2.2) is convex on
$\operatorname{dom}\psi$.

  2. (ii)

If $\varphi^{\psi}(x)>-\infty$, for some
$x\in\operatorname{int}(\operatorname{dom}\psi)$, then
$\varphi^{\psi}>-\infty$ on $\operatorname{int}(\operatorname{dom}\psi)$.

###### Proof.

(i) To see that the function $x\mapsto\varphi^{\psi}(x)$ is convex on
$\operatorname{dom}\psi$, we let $x\coloneqq\lambda x_{1}+(1-\lambda)x_{2}$,
for $x_{1},x_{2}\in\operatorname{dom}\psi$ and $\lambda\in(0,1)$. If
$\varphi^{\psi}(x_{1})=-\infty$, then there are
$p_{x_{1}}^{(n)}\in\mathscr{P}_{2}^{x_{1}}(\mathds{R}^{d})$ with

$\int\psi\,dp_{x_{1}}^{(n)}-\textnormal{MCov}(p_{x_{1}}^{(n)},q)\rightarrow-\infty.$

We observe that $p_{x}^{(n)}\coloneqq\lambda
p_{x_{1}}^{(n)}+(1-\lambda)\delta_{x_{2}}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$,
and consequently

$\displaystyle\varphi^{\psi}(x)$
$\displaystyle\leqslant\lambda\int\psi\,dp_{x_{1}}^{(n)}+(1-\lambda)\psi(x_{2})-\textnormal{MCov}(p_{x}^{(n)},q)$
$\displaystyle\leqslant\lambda\Big{(}\int\psi\,dp_{x_{1}}^{(n)}-\textnormal{MCov}(p_{x_{1}}^{(n)},q)\Big{)}+(1-\lambda)\psi(x_{2}),$

where we have used the convexity of $\mathscr{P}_{2}(\mathds{R}^{d})\ni
p\longmapsto-\textnormal{MCov}(p,q)$ in the last inequality. We conclude that
$\varphi^{\psi}(x)=-\infty$. The case $\varphi^{\psi}(x_{2})=-\infty$ is
treated similarly. If, on the other hand, both $\varphi^{\psi}(x_{1})>-\infty$
and $\varphi^{\psi}(x_{2})>-\infty$, then

$\varphi^{\psi}(x)\leqslant\lambda\varphi^{\psi}(x_{1})+(1-\lambda)\varphi^{\psi}(x_{2})$

follows by standard arguments. Altogether we see that $\varphi^{\psi}$ is
convex on $\operatorname{dom}\psi$.

(ii) If $\varphi^{\psi}(x)>-\infty$ for one
$x\in\operatorname{int}(\operatorname{dom}\psi)$, then
$\varphi^{\psi}(\tilde{x})>-\infty$ for all
$\tilde{x}\in\operatorname{int}(\operatorname{dom}\psi)$, as can be seen
directly by convexity. ∎

Our next result is the analogue of Lemma 2.6. But now, instead of maximizing
over all $p\in\mathscr{P}_{2}(\mathds{R}^{d})$ as in (2.18), we maximize over
all $p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$, with a fixed barycenter
$x\in\mathds{R}^{d}$, as in (2.17).

######  Lemma 3.2.

Let $\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a lower
semicontinuous convex function and assume that $\varphi^{\psi}(x)>-\infty$ for
some $x\in\operatorname{int}(\operatorname{dom}\psi)$. Then we have the
duality formula

$\varphi^{\psi}(x)=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\int\psi^{\ast}(y+z)\,dq(z)\Big{)}.$ (3.1)

###### Proof.

We first show that $(\varphi^{\psi})^{\ast}=\psi^{\ast}\star q$. To see this,
we recall equations (2.25) – (2.28) of Proposition 2.7 and note that these are
valid for any proper, lower semicontinuous convex function $\psi$. Therefore,
for each $y\in\mathds{R}^{d}$, we have

$(\varphi^{\psi})^{\ast}(y)=\sup_{p\in\mathscr{P}_{2}(\mathds{R}^{d})}\Big{(}\textnormal{MCov}(p,q)-\int\psi_{y}\,dp\Big{)}=\int\psi^{\ast}(y+z)\,dq(z),$
(3.2)

where the function $\psi_{y}$ is defined by
$\psi_{y}(z)\coloneqq\psi(z)-\langle y,z\rangle$, for $z\in\mathds{R}^{d}$.

Since $\varphi^{\psi}(x)>-\infty$ for some
$x\in\operatorname{int}(\operatorname{dom}\psi)$ and $\varphi^{\psi}$ is
convex by Lemma 3.1, the function $\varphi^{\psi}$ is finite and thus also
continuous in a neighbourhood of $x$. Hence we can apply the variant [BC17,
Proposition 13.44] of the Fenchel–Moreau theorem and deduce from (3.2) that

$\varphi^{\psi}(x)=(\varphi^{\psi})^{\ast\ast}(x)=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\int\psi^{\ast}(y+z)\,dq(z)\Big{)},$ (3.3)

which proves the duality formula (3.1). ∎

The duality formula (3.1) established in Lemma 3.2 enables us to generalize
Lemma 2.5 and identify the structure of the optimizer in the minimization
problem (2.2).

######  Lemma 3.3.

Let $\psi\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ be a lower
semicontinuous convex function.

  1. (i)

If there are $x\in\mathds{R}^{d}$ and $\hat{y}(x)\in\mathds{R}^{d}$ such that

$\hat{p}_{x}\coloneqq\nabla\psi^{\ast}\big{(}\hat{y}(x)+\,\cdot\,\big{)}(q)\in\mathscr{P}_{2}^{x}(\mathds{R}^{d}),$
(3.4)

then this is actually the unique optimizer of the infimum in (2.2).

  2. (ii)

If additionally $\varphi^{\psi}(x)>-\infty$ and
$x\in\operatorname{int}(\operatorname{dom}\psi)$, then $\hat{y}(x)$ is the
optimizer of the supremum in (3.1).

###### Proof.

(i) For fixed $y\in\mathds{R}^{d}$ we define the function $\psi_{y}$ by
$\psi_{y}(z)\coloneqq\psi(z)-\langle y,z\rangle$, for $z\in\mathds{R}^{d}$, so
that $\psi_{y}^{\ast}(z)=\psi^{\ast}(y+z)$. With this notation, we can express
the probability measure defined in (3.4) as
$\hat{p}_{x}=(\nabla\psi_{\hat{y}(x)}^{\ast})(q)$. We denote by
$T_{q}^{p_{x}}$ the Brenier map from $q$ to
$p_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$ and note that
$T_{q}^{\hat{p}_{x}}=\nabla\psi_{\hat{y}(x)}^{\ast}$ by Brenier’s theorem. For
arbitrary $p_{x}\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})$ we compute

$\displaystyle\int\psi\,dp_{x}-\textnormal{MCov}(p_{x},q)$
$\displaystyle=\int\Big{(}\psi_{\hat{y}(x)}\big{(}T_{q}^{p_{x}}(z)\big{)}+\big{\langle}T_{q}^{p_{x}}(z),\hat{y}(x)\big{\rangle}-\big{\langle}T_{q}^{p_{x}}(z),z\big{\rangle}\Big{)}\,dq(z)$
$\displaystyle\geqslant\big{\langle}x,\hat{y}(x)\big{\rangle}+\int\inf_{y\in\mathds{R}^{d}}\big{(}\psi_{\hat{y}(x)}(y)-\langle
y,z\rangle\big{)}\,dq(z)$
$\displaystyle=\big{\langle}x,\hat{y}(x)\big{\rangle}-\int\psi^{\ast}(\hat{y}(x)+z)\,dq(z)$
$\displaystyle=\int\psi\,d\hat{p}_{x}-\textnormal{MCov}(\hat{p}_{x},q),$

with equality if and only if $T_{q}^{p_{x}}(z)=T_{q}^{\hat{p}_{x}}(z)$, for
$q$-a.e. $z\in\mathds{R}^{d}$. This in turn is the case if and only if
$p_{x}=\hat{p}_{x}$. We conclude that $\hat{p}_{x}$ is the unique optimizer of
the infimum in (2.2), i.e.,

$\varphi^{\psi}(x)=\int\psi\,d\hat{p}_{x}-\textnormal{MCov}(\hat{p}_{x},q)=\big{\langle}x,\hat{y}(x)\big{\rangle}-\int\psi^{\ast}\big{(}\hat{y}(x)+z\big{)}\,dq(z).$
(3.5)

(ii) If $\varphi^{\psi}(x)>-\infty$ and
$x\in\operatorname{int}(\operatorname{dom}\psi)$, then by Lemma 3.2 we have
that

$\varphi^{\psi}(x)=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\int\psi^{\ast}(y+z)\,dq(z)\Big{)},$

which in light of (3.5) implies that $\hat{y}(x)$ is the optimizer of the
supremum in (3.1). ∎

###  3.1 Proof of Theorem 1.5

######  Definition 3.4.

We say that a lower semicontinuous convex function
$\hat{\psi}\colon\mathds{R}^{d}\rightarrow(-\infty,+\infty]$ satisfying
$\mu(\operatorname{int}(\operatorname{dom}\hat{\psi}))=1$ is an optimizer of
the dual problem (1.4) if $D^{q}(\mu,\nu)=\mathcal{E}(\hat{\psi})$, for the
dual function $\mathcal{E}(\,\cdot\,)$ as defined in (2.30). In short, we say
that $\hat{\psi}$ is a dual optimizer.

###### Proof of Theorem 1.5.

(1) Let $\hat{v}\colon\mathds{R}^{d}\rightarrow\mathds{R}$ be $q$-Bass
martingale generating with initial marginal $\mu$. Then, according to
Definition 1.4, the function $\hat{v}$ is convex, its convex conjugate
$\hat{\psi}=\hat{v}^{\ast}$ satisfies
$\mu(\operatorname{int}(\operatorname{dom}\hat{\psi}))=1$, and the finite-
valued function $\hat{v}\star q$ is strictly convex with gradient

$\nabla(\hat{v}\star q)=(\nabla\hat{v})\star q.$ (3.6)

By [Roc70, Theorem 26.5], it follows that the convex conjugate $(\hat{v}\star
q)^{\ast}$ is differentiable and strictly convex on
$\operatorname{int}(\operatorname{dom}(\hat{v}\star q)^{\ast})$. Furthermore,
the function

$\nabla(\hat{v}\star
q)\colon\mathds{R}^{d}\longrightarrow\operatorname{int}(\operatorname{dom}(\hat{v}\star
q)^{\ast})$

is a bijection with inverse

$(\nabla\hat{v}\star q)^{-1}=\nabla(\hat{v}\star q)^{\ast}.$ (3.7)

Next, we show that $\mu(\operatorname{int}(\operatorname{dom}(\hat{v}\star
q)^{\ast}))=1$. To this end, we will prove the inclusion

$\operatorname{int}(\operatorname{dom}\hat{\psi})\subseteq\operatorname{int}(\operatorname{dom}(\hat{v}\star
q)^{\ast}).$

Indeed, for $x\in\operatorname{dom}\hat{\psi}$, we have

$\displaystyle(\hat{v}\star q)^{\ast}(x)$
$\displaystyle=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\int\hat{v}(y+z)\,dq(z)\Big{)}\leqslant\int\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\hat{v}(y+z)\Big{)}\,dq(z)$
$\displaystyle=\int\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y-z\rangle-\hat{v}(y)\Big{)}\,dq(z)=\int\Big{(}\hat{v}^{\ast}(x)-\langle
x,z\rangle\Big{)}\,dq(z)$ $\displaystyle\leqslant\hat{v}^{\ast}(x)+|x|\
\sqrt{\int|z|^{2}\,dq(z)}<+\infty,$

so that $\operatorname{dom}\hat{\psi}\subseteq\operatorname{dom}(\hat{v}\star
q)^{\ast}$.

Now we are ready to show that the pair $(\hat{v},\hat{\alpha}^{\hat{v}})$ is a
$q$-Bass martingale with initial marginal $\mu$ and with terminal marginal
$\nu^{\hat{v}}$, i.e.,

$(\nabla\hat{v}\star q)(\hat{\alpha}^{\hat{v}})=\mu\qquad\textnormal{ and
}\qquad\nabla\hat{v}(\hat{\alpha}^{\hat{v}}\ast q)=\nu^{\hat{v}}.$ (3.8)

According to Definition 1.4, we have
$\hat{\alpha}^{\hat{v}}=\nabla(\hat{v}\star q)^{\ast}(\mu)$ and
$\nu^{\hat{v}}=\nabla\hat{v}(\hat{\alpha}^{\hat{v}}\ast q)$. Together with
(3.7), we conclude the relations (3.8).

(2) Recalling the second equality in (3.3), we have

$\varphi^{\hat{\psi}}(x)\geqslant(\varphi^{\hat{\psi}})^{\ast\ast}(x)=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-\int\hat{v}(y+z)\,dq(z)\Big{)}.$ (3.9)

As the right-hand side of (3.9) is greater than $-\infty$, for every
$x\in\mathds{R}^{d}$, we can apply Lemma 3.2. Consequently, for every
$x\in\operatorname{int}(\operatorname{dom}\hat{\psi})$, we obtain an equality
in (3.9), i.e.,

$\varphi^{\hat{\psi}}(x)=\sup_{y\in\mathds{R}^{d}}\Big{(}\langle
x,y\rangle-(\hat{v}\star q)(y)\Big{)}.$ (3.10)

Recall that the finite-valued function $\hat{v}\star q$ is strictly convex and
differentiable with gradient (3.6). The first order condition for an optimizer
$\hat{y}(x)\in\mathds{R}^{d}$ of the right-hand side of (3.10) reads

$(\nabla\hat{v}\star q)\big{(}\hat{y}(x)\big{)}=x.$ (3.11)

From (3.7) we deduce that the right-hand side of (3.10) admits a unique
maximizer $\hat{y}(x)$ given by

$\hat{y}(x)=(\nabla\hat{v}\star q)^{-1}(x)=\nabla(\hat{v}\star q)^{\ast}(x),$

for all $x\in\operatorname{int}(\operatorname{dom}\hat{\psi})$.

For each $x\in\operatorname{int}(\operatorname{dom}\hat{\psi})$, we define a
probability measure

$\pi_{x}^{\hat{v}}\coloneqq\nabla\hat{v}\big{(}\hat{y}(x)+\,\cdot\,\big{)}(q).$

Since $\nu^{\hat{v}}$ has finite second moment, we observe that

$\int|\nabla\hat{v}(y+z)|^{2}\,dq(z)<+\infty,$ (3.12)

for $\hat{\alpha}^{\hat{v}}$-a.e. $y\in\mathds{R}^{d}$. By (3.11) and (3.12),
$\pi_{x}^{\hat{v}}$ is an element of $\mathscr{P}_{2}^{x}(\mathds{R}^{d})$,
for $\mu$-a.e. $x\in\mathds{R}^{d}$. Hence we can apply part (i) of Lemma 3.3,
showing that the infimum

$\varphi^{\hat{\psi}}(x)=\inf_{p\in\mathscr{P}_{2}^{x}(\mathds{R}^{d})}\Big{(}\int\hat{\psi}\,dp-\textnormal{MCov}(p,q)\Big{)}$

is attained by $\pi_{x}^{\hat{v}}$, for $\mu$-a.e. $x\in\mathds{R}^{d}$.
Therefore, from the representation (2.12) of the dual function (2.5) and the
fact that $\pi_{x}^{\hat{v}}(dy)\,\mu(dx)\in\mathsf{MT}(\mu,\nu^{\hat{v}})$,
we obtain

$\tilde{\mathcal{E}}(\hat{\psi})=\int\Big{(}\int\hat{\psi}(y)\,\pi^{\hat{v}}_{x}(dy)-\varphi^{\hat{\psi}}(x)\Big{)}\,\mu(dx)=\int\textnormal{MCov}(\pi_{x}^{\hat{v}},q)\,\mu(dx).$
(3.13)

On the other hand, from (1.3), Proposition 2.1 and (2.13) we have

$\int\textnormal{MCov}(\pi_{x}^{\hat{v}},q)\,\mu(dx)\leqslant
P^{q}(\mu,\nu^{\hat{v}})=\tilde{D}^{q}(\mu,\nu^{\hat{v}})\leqslant\tilde{\mathcal{E}}(\hat{\psi}),$
(3.14)

which in light of (3.13) implies that

$P^{q}(\mu,\nu^{\hat{v}})=\int\textnormal{MCov}(\pi_{x}^{\hat{v}},q)\,\mu(dx).$

By uniqueness of the optimizer $\hat{\pi}\in\mathsf{MT}(\mu,\nu^{\hat{v}})$,
we conclude that $\pi_{x}^{\hat{v}}=\hat{\pi}_{x}$, for $\mu$-a.e.
$x\in\mathds{R}^{d}$.

(3) According to (3.10), for every
$x\in\operatorname{int}(\operatorname{dom}\hat{\psi})$, we have that

$\varphi^{\hat{\psi}}(x)=(\hat{v}\star q)^{\ast}(x).$

Recalling the definition of the dual function (2.30), we observe that

$\displaystyle\mathcal{E}(\hat{\psi})$
$\displaystyle=\int\Big{(}\int\hat{\psi}(y)\,\pi_{x}(dy)-(\hat{\psi}^{\ast}\star
q)^{\ast}(x)\Big{)}\,\mu(dx)$
$\displaystyle=\int\Big{(}\int\hat{\psi}(y)\,\pi^{\hat{v}}_{x}(dy)-\varphi^{\hat{\psi}}(x)\Big{)}\,\mu(dx)=\tilde{\mathcal{E}}(\hat{\psi}).$

Finally, from (3.13) and (3.14) we deduce that $\hat{\psi}$ is a dual
optimizer in the sense of Definition 3.4. ∎

## References

  * [AG13] L. Ambrosio and N. Gigli.  A User’s Guide to Optimal Transport.  In Modelling and Optimisation of Flows on Networks, volume $\bm{2062}$ of Lecture Notes in Math., pages 1–155. Springer, Berlin, Heidelberg, 2013.
  * [AMP23] B. Acciaio, A. Marini, and G. Pammer.  Calibration of the Bass Local Volatility model.  arXiv:2311.14567, 2023.
  * [AP22] B. Acciaio and G. Pammer.  A short proof of the characterisation of convex order using the $2$-Wasserstein distance.  arXiv:2207.02096, 2022.
  * [Bas83] R.F. Bass.  Skorokhod imbedding via stochastic integrals.  In J. Azéma and M. Yor, editors, Sémin. Probab. XVII 1981/82 — Proceedings, volume $\bm{986}$ of Lecture Notes in Math., pages 221–224. Springer, Berlin, Heidelberg, 1983.
  * [BB99] J.-D. Benamou and Y. Brenier.  A numerical method for the optimal time-continuous mass transport problem and related problems.  In L.A. Caffarelli and M. Milman, editors, Monge Ampère Equation: Applications to Geometry and Optimization (Deerfield Beach, FL, 1997), volume $\bm{226}$ of Contemp. Math., pages 1–11. American Mathematical Society, Providence, RI, 1999.
  * [BC17] H.H. Bauschke and P.L. Combettes.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces.  CMS Books in Mathematics. Springer International Publishing, Switzerland, second edition, 2017.
  * [BCH17] M. Beiglböck, A.M.G. Cox, and M. Huesmann.  Optimal transport and Skorokhod embedding.  Invent. Math., $\bm{208}$(2):327–400, 2017.
  * [BHLP13] M. Beiglböck, P. Henry-Labordère, and F. Penkner.  Model-independent bounds for option prices — a mass transport approach.  Finance Stoch., $\bm{17}$(3):477–501, 2013.
  * [BHLT17] M. Beiglböck, P. Henry-Labordère, and N. Touzi.  Monotone martingale transport plans and Skorokhod embedding.  Stochastic Process. Appl., $\bm{127}$(9):3005–3013, 2017.
  * [BJ16] M. Beiglböck and N. Juillet.  On a problem of optimal transport under marginal martingale constraints.  Ann. Probab., $\bm{44}$(1):42–106, 2016.
  * [BN15] B. Bouchard and M. Nutz.  Arbitrage and duality in nondominated discrete-time models.  Ann. Appl. Probab., $\bm{25}$(2):823–859, 2015.
  * [BNS22] M. Beiglböck, M. Nutz, and F. Stebegg.  Fine properties of the optimal Skorokhod embedding problem.  J. Eur. Math. Soc., $\bm{24}$(4):1389–1429, 2022.
  * [BNT17] M. Beiglböck, M. Nutz, and N. Touzi.  Complete duality for martingale optimal transport on the line.  Ann. Probab., $\bm{45}$(5):3038–3074, 2017.
  * [Bre87] Y. Brenier.  Décomposition polaire et réarrangement monotone des champs de vecteurs.  C. R. Acad. Sci. Paris Sér. I Math., $\bm{305}$(1):805–808, 1987.
  * [Bre91] Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions.  Comm. Pure Appl. Math., $\bm{44}$(4):375–417, 1991.
  * [BVBHK20] J. Backhoff-Veraguas, M. Beiglböck, M. Huesmann, and S. Källblad.  Martingale Benamou–Brenier: A probabilistic perspective.  Ann. Probab., $\bm{48}$(5):2258–2289, 2020.
  * [BVBP19] J. Backhoff-Veraguas, M. Beiglböck, and G. Pammer.  Existence, duality, and cyclical monotonicity for weak transport costs.  Calc. Var. Partial Differential Equations, $\bm{58}$(6):Article number: 203, 2019.
  * [BVBST23] J. Backhoff-Veraguas, M. Beiglböck, W. Schachermayer, and B. Tschiderer.  The structure of martingale Benamou–Brenier in $\mathbb{R}^{d}$.  arXiv:2306.11019, 2023.
  * [BVST23] J. Backhoff-Veraguas, W. Schachermayer, and B. Tschiderer.  The Bass functional of martingale transport.  arXiv:2309.11181, 2023.
  * [CHL21] A. Conze and P. Henry-Labordère.  Bass Construction with Multi-Marginals: Lightspeed Computation in a New Local Volatility Model.  SSRN:3853085, 2021.
  * [CKPS21] P. Cheridito, M. Kiiski, D.J. Prömel, and H.M. Soner.  Martingale optimal transport duality.  Math. Ann., $\bm{379}$(3–4):1685–1712, 2021.
  * [CLM17] L. Campi, I. Laachir, and C. Martini.  Change of numeraire in the two-marginals martingale transport problem.  Finance Stoch., $\bm{21}$(2):471–486, 2017.
  * [COT19] A.M.G. Cox, J. Obłój, and N. Touzi.  The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach.  Probab. Theory Relat. Fields, $\bm{173}$(1–2):211–259, 2019.
  * [DMT19] H. De March and N. Touzi.  Irreducible convex paving for decomposition of multidimensional martingale transport plans.  Ann. Probab., $\bm{47}$(3):1726–1774, 2019.
  * [DS14] Y. Dolinsky and H.M. Soner.  Martingale optimal transport and robust hedging in continuous time.  Probab. Theory Relat. Fields, $\bm{160}$(1–2):391–427, 2014.
  * [GHLT14] A. Galichon, P. Henry-Labordère, and N. Touzi.  A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options.  Ann. Appl. Probab., $\bm{24}$(1):312–336, 2014.
  * [GKL19] N. Ghoussoub, Y.-H. Kim, and T. Lim.  Structure of optimal martingale transport plans in general dimensions.  Ann. Probab., $\bm{47}$(1):109–164, 2019.
  * [GKL20] N. Ghoussoub, Y.-H. Kim, and T. Lim.  Optimal Brownian Stopping When the Source and Target Are Radially Symmetric Distributions.  SIAM J. Control Optim., $\bm{58}$(5):2765–2789, 2020.
  * [GKP19] N. Ghoussoub, Y.-H. Kim, and A.Z. Palmer.  PDE methods for optimal Skorokhod embeddings.  Calc. Var. Partial Differential Equations, $\bm{58}$(3):Article number: 113, 2019.
  * [GL21] I. Guo and G. Loeper.  Path dependent optimal transport and model calibration on exotic derivatives.  Ann. Appl. Probab., $\bm{31}$(3):1232–1263, 2021.
  * [GLW19] I. Guo, G. Loeper, and S. Wang.  Local Volatility Calibration by Optimal Transport.  In J. de Gier, C.E. Praeger, and T. Tao, editors, 2017 MATRIX Annals, volume $\bm{2}$ of MATRIX Book Series, pages 51–64. Springer, Cham, 2019.
  * [GRST17] N. Gozlan, C. Roberto, P.-M. Samson, and P. Tetali.  Kantorovich duality for general transport costs and applications.  J. Funct. Anal., $\bm{273}$(11):3327–3405, 2017.
  * [HLOST16] P. Henry-Labordère, J. Obłój, P. Spoida, and N. Touzi.  The maximum maximum of a martingale with given $n$ marginals.  Ann. Appl. Probab., $\bm{26}$(1):1–44, 2016.
  * [HN12] D. Hobson and A. Neuberger.  Robust bounds for forward start options.  Math. Finance, $\bm{22}$(1):31–56, 2012.
  * [HT19] M. Huesmann and D. Trevisan.  A Benamou–Brenier formulation of martingale optimal transport.  Bernoulli, $\bm{25}$(4A):2729–2757, 2019.
  * [Kan42] L.V. Kantorovich.  On the translocation of masses.  Dokl. Akad. Nauk. USSR (NS), $\bm{37}$:199–201, 1942.
  * [KS98] I. Karatzas and S.E. Shreve.  Brownian Motion and Stochastic Calculus, volume $\bm{113}$ of Grad. Texts in Math. Springer, New York, second edition, 1998.
  * [KTT17] S. Källblad, X. Tan, and N. Touzi.  Optimal Skorokhod embedding given full marginals and Azéma-Yor peacocks.  Ann. Appl. Probab., $\bm{27}$(2):686–719, 2017.
  * [Loe18] G. Loeper.  Option pricing with linear market impact and nonlinear Black–Scholes equations.  Ann. Appl. Probab., $\bm{28}$(5):2664–2726, 2018.
  * [Loe23] G. Loeper.  The Measure Preserving Martingale Sinkhorn Algorithm.  arXiv:2310.13797, 2023.
  * [McC94] R.J. McCann.  A convexity theory for interacting gases and equilibrium crystals.  Princeton University ProQuest Dissertations Publishing, $\bm{9505121}$, 1994.
  * [McC95] R.J. McCann.  Existence and uniqueness of monotone measure-preserving maps.  Duke Math. J., $\bm{80}$(2):309–323, 1995.
  * [Mon81] G. Monge.  Mémoire sur la théorie des déblais et des remblais.  Mem. Math. Phys. Acad. Royale Sci., pages 666–704, 1781.
  * [OS17] J. Obłój and P. Siorpaes.  Structure of martingale transports in finite dimensions.  arXiv:1702.08433, 2017.
  * [OST15] J. Obłój, P. Spoida, and N. Touzi.  Martingale Inequalities for the Maximum via Pathwise Arguments.  In C. Donati-Martin, A. Lejay, and A. Rouault, editors, In Memoriam Marc Yor — Sémin. Probab. XLVII, volume $\bm{2137}$ of Lecture Notes in Math., pages 227–247. Springer, Cham, 2015.
  * [Roc70] R.T. Rockafellar.  Convex analysis, volume $\bm{28}$ of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1970.
  * [San15] F. Santambrogio.  Optimal Transport for Applied Mathematicians — Calculus of Variations, PDEs, and Modeling, volume $\bm{87}$ of Progress in Nonlinear Differential Equations and Their Applications.  Birkhäuser, Switzerland, 2015.
  * [TT13] X. Tan and N. Touzi.  Optimal transportation under controlled stochastic dynamics.  Ann. Probab., $\bm{41}$(5):3201–3240, 2013.
  * [Vil03] C. Villani.  Topics in Optimal Transportation, volume $\bm{58}$ of Grad. Stud. Math. American Mathematical Society, Providence, RI, 2003.
  * [Vil09] C. Villani.  Optimal Transport — Old and New, volume $\bm{338}$ of Grundlehren Math. Wiss. Springer, Berlin, Heidelberg, 2009.
  * [WZ23] J. Wiesel and E. Zhang.  An optimal transport-based characterization of convex order.  Depend. Model., $\bm{11}$(1):20230102, 2023.