OT Divergences induced by Scoring FunctionsPesenti and Vanduffel
assumptionAssumption remarkRemark hypothesisHypothesis claimClaim
optimisationOptimisation

# Optimal Transport Divergences induced by Scoring Functions††thanks: The
authors thank Tobias Fissler for fruitful and stimulating discussions on the
topic. SP gratefully acknowledges the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC) with funding reference numbers
DGECR-2020-00333 and RGPIN-2020-04289. SV is grateful to Research Foundation
Flanders (FWO) for financial support with funding reference number FWO SBO
S006721N

Silvana M. Pesenti Department of Statistical Sciences, University of Toronto,
Canada<EMAIL_ADDRESS>Steven Vanduffel Department of
Economics and Political Science, Vrije Universiteit Brussel, Belgium ().
<EMAIL_ADDRESS>
###### Abstract

We employ scoring functions, used in statistics for eliciting risk
functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport
problem. This gives raise to a rich variety of novel asymmetric MK
divergences, which subsume the family of Bregman-Wasserstein divergences. We
show that for distributions on the real line, the comonotonic coupling is
optimal for the majority the new divergences. Specifically, we derive the
optimal coupling of the MK divergences induced by functionals including the
mean, generalised quantiles, expectiles, and shortfall measures. Furthermore,
we show that while any elicitable law-invariant convex risk measure gives
raise to infinitely many MK divergences, the comonotonic coupling is
simultaneously optimal.
The novel MK divergences, which can be efficiently calculated, open an array
of applications in robust stochastic optimisation. We derive sharp bounds on
distortion risk measures under a Bregman-Wasserstein divergence constraint,
and solve for cost-efficient portfolio strategies under benchmark constraints.

###### keywords:

Optimal transport, risk measures, scoring functions, elicitability, coupling,
Wasserstein distance, asymmetric Optimal Transport

##  1 Introduction

This work connects optimal transport with the statistics of point forecasts
and risk measure theory to derive asymmetric optimal transport divergences.
Initially developed by Gaspard Monge in the 18-th century to establish the
most efficient way of moving piles of soil from one location to another,
optimal transport (OT) theory has evolved as a flourishing mathematical
discipline that studies the optimal transportation of one distribution to
another; see e.g. the monographs [19] and [23]. The theory of OT has numerous
applications in various distinct disciplines including economics, engineering,
and the natural sciences. For instance, economists use OT to model the flow of
goods and resources in financial markets. It find itself in causal inference,
partial identification, and the analysis of distributional treatment effects
in e.g., biostatistics. In image processing and computer vision, OT is used
for solving matching and registration problems whereas in the domain of fluid
dynamics it optimises the flow of gases.

The most commonly used and best understood OT distance is the (2-)Wasserstein
distance, that arises as the minimiser of the Monge-Kantorovich OT problem
with the symmetric cost function $c(z_{1},z_{2})=(z_{1}-z_{2})^{2}$. In
various problems of interest, however, asymmetry may be desired. A case in
point is the quantification of model ambiguity in optimal portfolio
strategies, where a decision maker assigns a larger costs to potential losses
than to equally large gains (if any). The literature on asymmetric cost
function is scarce, with the exception of the recently introduced Bregman-
Wasserstein (BW) divergences [5]. A BW divergence is the minimiser of the
Monge-Kantorovich OT problem in which the cost function is a Bregman
divergence generated by a strictly convex function $\phi(\cdot)$; we refer to
[20] for discussions and information geometric interpretations. The BW
divergence is an asymmetric generalisation of the Wasserstein distance,
recovered for $\phi(x)=x^{2}$, that allows the modelling of dissimilarities.

The study of elicitability is a fast growing field in statistics and at its
core are scoring functions that incentivise truthful predictions and allow for
forecast comparison, model comparison (backtesting), and model calibration
[14, 9]. In sensitivity analysis, scoring functions are utilised for defining
sensitivity measures which quantify the sensitivity of an elicitable risk
measure to perturbations in the model’s input factors [10]. The most well-
known family of scoring functions are the Bregman divergences that elicit the
mean, where a functional is called elicitable if it is a minimiser of an
expected score, see Definition 2.2. Other elicitable functionals are
quantiles, expectiles, and shortfall risk measures; tools used in risk
management. Scoring functions are by nature asymmetric, making them ideal
candidates for asymmetric cost functions in the Monge-Kantorovich OT problem.
Indeed, we propose novel asymmetric Monge-Kantorovich (MK) divergences where
the OT cost functions are statistical scoring functions. As a Bregman
divergence elicits the mean and gives raise to a BW divergence, our new MK
divergences can be seen as generalisations of BW divergences, and thus the
Wasserstein distance. In addition to scoring functions that elicits the mean,
we study scoring functions that elicit the quantile, the expectile, and law-
invariant convex risk measures. Interestingly, we find that most of the
introduced MK divergences are attained by the comonotonic coupling.
Furthermore, as an elicitable functional possesses infinitely many scoring
functions, and thus gives raise to infinitely many MK divergences, the
comonotonic optimal coupling is typically simultaneously optimal. Using the
celebrated Osband’s principle in statistics, we propose ways to create novel
MK divergences that are attained by the anti- or comonotonic coupling.
Furthermore, we prove that MK divergences induces by any law-invariant
elicitable convex risk measure are attained by the comonotonic coupling.
Finally, we discuss two applications to robust stochastic optimisation. First,
we derive sharp bounds on distortion risk measures when admissible
distributions belong to a BW-ball around a reference distribution, thus
significantly generalising recent results of [2], who solve this problem for
the special case of a Wasserstein ball. Second, we find the cheapest portfolio
strategy under the constraint that the distribution of terminal wealth lie
within a BW-ball around a benchmark distribution.

This paper is organised as follows. Section 2 introduces the MK divergences
after reviewing the statistical concepts elicitability and scoring functions
and the relevant topics in OT. Section 3 is devoted to MK divergences induced
by elicitable risk functionals such as the quantile, expectile, and shortfall
risk measure. We find that for distributions on the real line the majority of
the new MK divergences are attained by the comonotonic coupling. Applications
of the new divergences to risk measure bounds, significantly generalising
recent results by [2], and portfolio management are provided in Section 4.

##  2 Monge-Kantorovich divergences induced by scoring functions

###  2.1 Elicitability and scoring functions

We first review the statistical concepts of elicitability and scoring
functions by following the traditional statistical notation and decision
theoretic setup, see e.g., [14] and [9]. For this let
$(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a complete probability space and
denote by ${\mathcal{L}}:={\mathcal{L}}(\Omega,{\mathcal{F}},{\mathds{R}})$
the space of all random variables. The cumulative distribution function (cdf)
of a random variable $X\in{\mathcal{L}}$ is denoted by
$F_{X}(\cdot):={\mathbb{P}}(X\leq\cdot)$, and we write
${\mathcal{M}^{0}}:={\mathcal{M}^{0}}({\mathds{R}})$ to denote the space of
all cdfs on ${\mathds{R}}$. For a cdf $F\in{\mathcal{M}^{0}}$, we define its
corresponding (left-continuous) quantile function by
${\breve{F}}(u):=\inf\\{y\in{\mathds{R}}~{}|~{}F(y)\geq u\\}$, $u\in[0,1]$,
with the convention that $\inf\emptyset=+\infty$. Throughout, we will use the
notation ${\mathcal{M}}\subseteq\bar{{\mathcal{M}}}\subseteq{\mathcal{M}^{0}}$
to denote sub-classes of cdfs.

######  Definition 2.1 (Scoring function).

A scoring function (or score) is a measurable map
$S\colon{\mathsf{A}}\times{\mathds{R}}\to[0,\infty]$, where
${\mathsf{A}}\subseteq{\mathds{R}}$ is called the action domain. For a given
functional $T\colon\bar{{\mathcal{M}}}\to{\mathsf{A}}$ and a sub-class
${\mathcal{M}}\subseteq\bar{{\mathcal{M}}}$, the scoring function $S$ may
satisfy the following properties:

  1. $(i)$

$S$ is _${\mathcal{M}}$ -consistent_ for $T$, if for all $F\in{\mathcal{M}}$
and for all $z\in{\mathsf{A}}$

(1) $\int S\big{(}T(F),y\big{)}\,{\mathrm{d}}F(y)\;\leq\;\int
S(z,y)\,{\mathrm{d}}F(y)\,.$

  2. $(ii)$

$S$ is _strictly_ ${\mathcal{M}}$-consistent for $T$, if it is
${\mathcal{M}}$-consistent for $T$ and if the inequality in (1) is strict for
all $z\neq T(F)$.

Throughout, we make the following non-restrictive assumptions on the
considered scoring functions. [Normalisation of scores] Let $S$ be an
${\mathcal{M}}$-consistent score for $T$ and denote by $\delta_{y}$,
$y\in{\mathds{R}}$, point measures. Then it holds that

  1. $(i)$

$S\big{(}T(\delta_{y}),y\big{)}<S(z,y)$ for all $z\neq T(\delta_{y})$ and
$y\in{\mathds{R}}$, and

  2. $(ii)$

$S\big{(}T(\delta_{y}),y\big{)}=0$ for all $y\in{\mathds{R}}$.

Definition 2.1 means that the scores are strictly consistent on the space of
point measures and that the scores are normalised to $S(y,y)=0$. Note that any
score $S$ satisfying $(i)$ can be normalised to fulfil $(ii)$, by setting
$\tilde{S}(z,y):=S(z,y)-S(T(\delta_{y}),y)$. Moreover, Definition 2.1 leads to
a unique characterisation of the families of scores of elicitable functionals,
see e.g., Propositions 3.1, 3.4, and 3.9.

######  Definition 2.2 (Elicitability).

A functional $T\colon\bar{{\mathcal{M}}}\to{\mathsf{A}}$ is 1-elicitable 111 A
functional $T$ is called $k$-elicitable, $k\in{\mathds{N}}$, if there exists a
strictly ${\mathcal{M}}$-consistent scoring function
$S\colon{\mathsf{A}}^{k}\times{\mathds{R}}\to{\mathds{R}}$,
${\mathsf{A}}^{k}\subseteq{\mathds{R}}^{k}$ and a
${\boldsymbol{z}}^{*}\in{\mathds{R}}^{k-1}$, such that
$({\boldsymbol{z}}^{*},T(F))=\operatorname*{argmin}_{{\boldsymbol{z}}\in{\mathsf{A}}^{k}}\int
S({\boldsymbol{z}},y){\mathrm{d}}F(y)$, for all $F\in{\mathcal{M}}$. There are
many functionals that are $k$-elicitable but not $1$-elicitable, see e.g.,
[11].  on ${\mathcal{M}}\subseteq\bar{{\mathcal{M}}}$, if there exists a
strictly ${\mathcal{M}}$-consistent scoring function $S$ for $T$. Moreover,
the functional $T$ has the following representation on ${\mathcal{M}}$

(2) $T(F)=\operatorname*{argmin}_{z\in{\mathsf{A}}}\int
S(z,y)\,{\mathrm{d}}F(y)\,,\quad\forall\;F\in{\mathcal{M}}\,.$

By Equation (2), a 1-elicitable functional is a Bayes act and a minimiser of
an expected score [14]. It is well known that the squared loss
$S(z,y)=(z-y)^{2}$ elicits the mean. The squared Euclidean distance, however,
is not the only strictly consistent score for the mean. Indeed, from (2) we
see that a 1-elicitable functional $T$ has infinitely many strictly consistent
scores. In particular, the class of strictly consistent scoring functions for
the mean are given by the so-called Bregman divergences, which we recall next.

######  Definition 2.3 (Bregman divergence).

Let $\phi\colon{\mathds{R}}\to{\mathds{R}}$ be a convex function. The Bregman
divergence associated with $\phi$ is defined as

$B_{\phi}\big{(}z_{1},z_{2}\big{)}:=\phi(z_{1})-\phi(z_{2})-\phi^{\prime}(z_{2})(z_{1}-z_{2})\,,\quad
z_{1},z_{2}\in{\mathds{R}}\,,$

where $\phi^{\prime}(z):=\frac{d}{dz}\phi(z)$ denotes the derivative of
$\phi$.

A Bregman divergence $B_{\phi}(z_{1},z_{2})$ can be seen as a measure of the
deviation of $z_{2}$ from $z_{1}$. Note that in order to have a mathematical
divergence (i.e., $B_{\phi}(z_{1},z_{2})=0$ if and only if $z_{1}=z_{2}$),
$\phi$ needs to be strictly convex. While for the choice $\phi(z)=z^{2}$ the
Bregman divergence coincides with the squared Euclidean distance, i.e.,
$B_{\phi}(z_{1},z_{2})=(z_{1}-z_{2})^{2}$, in general, the Bregman divergence
is not symmetric.

###  2.2 Monge-Kantorovich divergences induced by scoring functions

Next, we use scoring functions as cost functions in the Monge-Kantorovich
optimal transport (OT) problem. In what follows, we call a function
$c\colon{\mathds{R}}^{2}\to{\mathds{R}}_{+}$, ${\mathds{R}}_{+}:=[0,\infty)$,
that is lower-semi-continuous, a cost function. We recall the traditional
Monge-Kantorovich optimisation problem and refer the reader to the books [21,
23] for further details.

######  Definition 2.4 (Monge-Kantorovich optimal transport problem).

Let $c\colon{\mathds{R}}^{2}\to{\mathds{R}}_{+}$ be a cost function. Then the
Monge-Kantorovich optimisation problem with respect to the cdfs
$F_{1}\in{\mathcal{M}^{0}}$ and $F_{2}\in{\mathcal{M}^{0}}$ is given by

(3)
$\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}c\big{(}z_{1},z_{2}\big{)}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\},$

where $\Pi(F_{1},F_{2})$ denotes the set of all bivariate cdfs with marginal
cdfs $F_{1}$ and $F_{2}$, respectively. A bivariate cdf that attains the
infimum in (3) is called an optimal coupling, which exists for any choice of
cost function, see e.g., Theorem 1.7 in [21].

For the cost function $c(z_{1},z_{2}):=(z_{1}-z_{2})^{p}$, $p\geq 1$, we
obtains the well-known $p$-Wasserstein distance

$\displaystyle W_{p}(F_{1},F_{2}):$
$\displaystyle=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\left(\,\int_{{\mathds{R}}^{2}}(z_{1}-z_{2})^{p}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right)^{\frac{1}{p}}\right\\}$
$\displaystyle=\left(\int_{0}^{1}\left|{\breve{F}}_{1}(u)-{\breve{F}}_{2}(u)\right|^{2}{\mathrm{d}}u\right)^{\frac{1}{2}}\,,$

where ${\breve{F}}_{i}$ is the quantile function of $F_{i}$, $i=1,2$, and
where the last equality follows for cdfs on the real line, indicating that the
comonotonic coupling is optimal [7].

This work introduces new asymmetric _Monge-Kantorovich (MK) divergences_ –
divergences on the space of cdfs – that are defined by optimisation problem
(3), where the cost functions are scoring functions. Thus, we not only
introduce new MK divergences but also provide a novel perspective on scoring
functions as OT cost functions. Specifically, we consider cost functions
$c(z_{1},z_{2}):=S(z_{2},z_{1})$, for consistent scoring functions $S$. Note
that the arguments of $c$ and $S$ are exchanged, this is due to different
notational conventions in statistics and optimal transport, see e.g. Equation
(6). Our choice is justified in that we obtain the Bregman-Wasserstein
divergence when choosing any consistent scoring functions for the mean
functional, see e.g., [20].

We formulate the following definition.

######  Definition 2.5 (Monge-Kantorovich divergence).

Let $S$ be a ${\mathcal{M}}$-consistent scoring function for a functional $T$.
Then the Monge-Kantorovich (MK) divergence induced by $S$ from the cdf
$F_{1}\in{\mathcal{M}^{0}}$ to the cdf $F_{2}\in{\mathcal{M}^{0}}$ is given by

(4)
${\mathscr{S}}(F_{1},F_{2}):=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}S\big{(}z_{2},z_{1}\big{)}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}\,.$

We term a bivariate cdf that attains the infimum in (4) an optimal coupling.

The defined MK divergences may not necessarily be a divergence. They are non-
negative and satisfy ${\mathscr{S}}(F_{1},F_{1})=0$, however, additional
assumptions on the score $S$ are needed to ensure that $F_{1}=F_{2}$ implies
${\mathscr{S}}(F_{1},F_{1})=0$, see e.g., [20] for the BW divergence.
Furthermore, a MK divergence is in general not symmetric which is in contrast
to, e.g., the $p$-Wasserstein distance. Clearly, the MK divergence depends on
the choice of scoring function, however, for conciseness of the exposition, we
refrain from writing ${\mathscr{S}}_{S}$, whenever the scoring function is
clear from the context. The assumption to normalise the score, Definition 2.1
$(ii)$, does not affect the optimal coupling as normalisation is achieved by
subtracting from the score a function of $z_{2}$ only; see also the discussion
after Definition 2.1.

If the cost function is the Bregman divergence, i.e., when in (3) one
considers $c(z_{1},z_{2})=B_{\phi}(z_{1},z_{2})$, we obtain the Bregman-
Wasserstein (BW) divergence [20]

(5)
${\mathscr{B}}_{\phi}(F_{1},F_{2}):=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}B_{\phi}(z_{1},\,z_{2})\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}\,,$

which reduces to the 2-Wasserstein distance for $B_{\phi}$ being the squared
loss, i.e. $\phi(x)=x^{2}$.

##  3 Optimal couplings for MK divergences

This section is devoted to MK divergences induced by scores of elicitable risk
functionals and their optimal couplings. Any elicitable risk functional, such
as the mean, quantiles, expectiles, admits infinitely many consistent score,
and thus gives raise to an infinite family of MK divergences. Furthermore,
while the MK divergences are different for each score that elicits the risk
functional, we show that the optimal couplings are the same.

###  3.1 Bregman score as cost function

The most commonly used 1-elicitable functional is the mean, whose family of
scoring functions are the Bregman scores.

######  Proposition 3.1 (Elicitability of Mean – [14]).

Let ${\mathcal{M}}$ denote the class of cdfs with finite mean. If $\phi$ is a
(strictly) convex function with subgradient $\phi^{\prime}$ and if
$\int|\phi(y)|\,\mathrm{d}F(y)<\infty$ for all $F\in{\mathcal{M}}$, then the
scoring function

(6) $S_{\phi}(z,y):=B_{\phi}(y,z)\,,\quad z,y\in{\mathds{R}}\,,$

is (strictly) ${\mathcal{M}}$-consistent for the mean. Moreover, on the class
of compactly supported measures, any (strictly) consistent score for the mean
which is continuously differentiable in its first argument and which satisfies
$S(y,y)=0$, is necessarily of the form (6).

######  Theorem 3.2 (Optimal coupling for BW-divergence).

The optimal coupling of any BW-divergence is the comonotonic coupling, i.e.,
$\big{(}{\breve{F}}_{1}(U),{\breve{F}}_{2}(U)\big{)}$, for any $U\sim U(0,1)$,
is the optimal coupling.

In the language of OT, the comonotonic coupling implies that the _optimal
transport map_ of (5), i.e., the deterministic function mapping $F_{1}$ to
$F_{2}$, is given by $\alpha(x):={\breve{F}}_{2}\big{(}F_{1}(x)\big{)}$. We
refer the reader to [20], who characterise the optimal transport map of the BW
divergence for multivariate cdfs. Here, we provided for completeness a
constructive proof that is valid for univariate cdfs.

######  Proof 3.3.

Let $\phi$ be convex, then the BW-divergence becomes

$\displaystyle{\mathscr{B}}_{\phi}(F_{1},F_{2})$
$\displaystyle=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\phi(z_{1})-\phi(z_{2})-\phi^{\prime}(z_{2})(z_{1}-z_{2})\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}$
$\displaystyle=\int_{\mathds{R}}\phi(z_{1}){\mathrm{d}}F_{1}(z_{1})+\int_{\mathds{R}}\left(\phi^{\prime}(z_{2})z_{2}-\phi(z_{2})\right){\mathrm{d}}F_{2}(z_{2})$
$\displaystyle\quad-\sup_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\phi^{\prime}(z_{2})z_{1}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}\,.$

Thus, to find the optimal coupling, we only have to solve the supremum.
Rewriting in terms of random variables, we have

(7)
$\sup_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\phi^{\prime}(z_{2})z_{1}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}=\sup\;\,{\mathbb{E}}\left[\phi^{\prime}(Z_{2})Z_{1}\,\right]\,,\quad
Z_{i}\sim F_{i}\,,\;i=1,2\,.$

where the supremum is over all copulae between $(Z_{1},Z_{2})$. Since
$\phi^{\prime}$ is increasing, it is well-known that the above supremum is
attained by the comonotonic coupling. Denoting the quantile function of
$F_{i}$ by ${\breve{F}}_{i}$, $i=1,2$, and since the quantile function of
$\phi^{\prime}(Z_{2})$ is $\phi^{\prime}({\breve{F}}_{2}(\cdot))$, we obtain

$\sup\;\,{\mathbb{E}}\left[YZ_{1}\,\right]={\mathbb{E}}\left[\phi^{\prime}\left({\breve{F}}_{2}(U)\right)\,{\breve{F}}_{1}(U)\right]\,,\quad
U\sim U(0,1)\,,$

which concludes the proof.

###  3.2 Scores of generalised quantiles

Next, we consider quantiles and generalised quantiles. Quantiles are
elicitable with the family of scoring functions called the generalised
piecewise linear scores.

######  Proposition 3.4 (Elicitability of Quantile – [14]).

If $g$ is an increasing function, then the scoring function

(8) $S_{g}(z,y):=\big{(}{\mathds{1}}_{\\{y\leq
z\\}}-\alpha\big{)}\big{(}g(z)-g(y)\big{)}\,,\qquad z,y\in{\mathds{R}}\,,$

is ${\mathcal{M}}$-consistent for the $\alpha$-quantile ${\breve{F}}(\alpha)$
if $\int|g(y)|\,\mathrm{d}F(y)<\infty$ for all $F\in{\mathcal{M}}$. If $g$ is
strictly increasing and if for all $F\in{\mathcal{M}}$,
$F({\textrm{VaR}}_{\alpha}(F)+\epsilon)>\alpha$ for all $\epsilon>0$, then (8)
is strictly ${\mathcal{M}}$-consistent. Moreover, on the class of compactly
supported measures, any consistent scoring function for ${\breve{F}}(\alpha)$
which is continuous in its first argument, which admits a continuous
derivative for all $z\neq y$, and which satisfies $S(y,y)=0$, is necessarily
of the form (8).

Hereafter we show that the comonotonic coupling is also optimal for the
generalised piecewise linear score as cost function, i.e., when in (3) we
choose the cost function $c(z_{1},z_{2}):=S_{g}(z_{2},z_{1})$.

######  Theorem 3.5 (Optimal coupling for generalised piecewise linear
scores).

The optimal coupling of the MK minimisation problem induced by any consistent
generalised piecewise linear score is the comonotonic coupling.

######  Proof 3.6.

Let $g$ be increasing, then the MK divergence induced by the score (8) is

$\displaystyle{\mathscr{S}}(F_{1},F_{2})$
$\displaystyle=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\big{(}{\mathds{1}}_{\\{z_{1}\leq
z_{2}\\}}-\alpha\big{)}\big{(}g(z_{2})-g(z_{1})\big{)}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}$
$\displaystyle=\alpha\int_{\mathds{R}}g(z_{1}){\mathrm{d}}F_{1}(z_{1})-\alpha\int_{\mathds{R}}g(z_{2}){\mathrm{d}}F_{2}(z_{2})$
$\displaystyle\quad+\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}{\mathds{1}}_{\\{z_{1}\leq
z_{2}\\}}\big{(}g(z_{2})-g(z_{1})\big{)}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}\,.$

The infimum rewritten in terms of random variables, is

$\displaystyle\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\big{(}g(z_{2})-g(z_{1})\big{)}_{+}\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}=\inf\;{\mathbb{E}}\left[\,\big{(}g(Z_{2})-g(Z_{1})\big{)}_{+}\,\right]\,,$

where the infimum is over all copulae of $(Z_{1},Z_{2})$. Note that for any
bivariate random vector $(X,Y)$, $X\sim F_{X}$, $Y\sim F_{Y}$, it holds that
[16]

(9) $X^{a}+Y^{a}\prec_{cx}X+Y\prec_{cx}X^{c}+Y^{c}\,,$

where $\prec_{cx}$ denotes inequality in convex order and where
$(X^{a},Y^{a})$ and $(X^{c},Y^{c})$ denote the antitonic resp. comonotonic
pair with marginal distributions $F_{X}$ and $F_{Y}$, that is
$(X^{a},Y^{a}):=(F_{X}^{-1}(U),F_{Y}^{-1}(1-U))$ and
$(X^{c},Y^{c})=(F_{X}^{-1}(U),F_{Y}^{-1}(U))$, $U\sim U(0,1)$. As $g$ is
increasing, $(g(Z_{2}^{c}),-g(Z_{1}^{c}))$ is an antitonic pair and the first
inequality in (9) implies that

$g(Z_{2}^{c})-g(Z_{1}^{c})\prec_{cx}g(Z_{2})-g(Z_{1}).$

Since $f(x)=x_{+}$ is a convex function we obtain that

$\inf\;{\mathbb{E}}\left[\,\big{(}g(Z_{2})-g(Z_{1})\big{)}_{+}\,\right]={\mathbb{E}}\left[\,\left(g\big{(}{\breve{F}}_{2}(U)\big{)}-g\big{(}{\breve{F}}_{1}(U)\big{)}\right)_{+}\,\right]\,,\quad
U\sim U(0,1)\,,$

which concludes the proof.

There exist many generalisations of quantiles, such as $L^{p}$ quantiles, $M$
quantiles and $\Lambda$-quantiles. We show in the sequel that consistent
scores of these generalised quantiles give raise to the comonotonic coupling
being optimal for (4). In this section, we discuss the $\Lambda$-quantile,
while the $L^{p}$ and $M$ quantiles are considered in Section 3.5.

For a monotone and right-continuous function
$\Lambda\colon{\mathds{R}}\to[\underline{\lambda},\,\overline{\lambda}]$,
$0<\underline{\lambda}<\overline{\lambda}<1$, the $\Lambda$-quantile is
defined by [4]

(10)
$T_{\Lambda}(F):=\inf\big{\\{}y\in{\mathds{R}}~{}:~{}F(y)>\Lambda(y)\big{\\}}\,.$

The $\Lambda$-quantile is elicitable with score

(11) $S(z,y)=(z-y)_{+}-\int_{y}^{z}\Lambda(s)\,{\mathrm{d}}s\,,$

on the space of cdfs that admit only one unique crossing point with
$\Lambda(\cdot)$.

######  Theorem 3.7 (Optimal coupling for $\Lambda$-quantile score).

The optimal coupling of the MK minimisation problem induced by the score given
in (11) is the comonotonic coupling.

######  Proof 3.8.

As $\partial_{z}S(z,y)={\mathds{1}}_{\\{y\leq z\\}}-\Lambda(z)$ is decreasing
in $y$, for all $z$, and $\partial_{y}S(z,y)=\Lambda(y)-{\mathds{1}}_{\\{y\leq
z\\}}$ is decreasing in $z,$ for all $y$, the score (11) is submodular, that
is for all $z_{1}\leq z_{2}$, and $y_{1}\leq y_{2}$, it holds that
$S(z_{1},y_{1})+S(z_{2},y_{2})\leq S(z_{1},y_{2})+S(z_{2},y_{1}).$ Since the
comonotonic coupling is optimal for submodular cost functions [19], the result
follows.

###  3.3 Expectile score

The comonotonic coupling is also optimal when the cost function is a score
that elicits the $\alpha$-expectile.

######  Proposition 3.9 (Elicitability of Expectiles – [14]).

Let ${\mathcal{M}}$ denote the class of cdfs with finite mean. If $\phi$ is
(strictly) convex with subgradient $\phi^{\prime}$ and if
$\int|\phi(y)|\,\mathrm{d}F(y)<\infty$ as well as
$\int|y|\,\mathrm{d}F(y)<\infty$ for all $F\in{\mathcal{M}}$, then the scoring
function

(12) $S(z,y)=\big{|}{\mathds{1}}_{\\{y\leq
z\\}}-\alpha\big{|}\,B_{\phi}(y,z)\,,\quad z,y\in{\mathds{R}}\,,$

is (strictly) ${\mathcal{M}}$-consistent for the expectile222The expectile was
first introduced in [17] as the minimiser of (2) for the scoring function (12)
with $B_{\phi}(y,z)=(z-y)^{2}.$ . Moreover, on the class of compactly
supported measures, any (strictly) consistent score for the expectile which is
continuously differentiable in its first argument and which satisfies
$S(y,y)=0$, is necessarily of the form (6).

######  Theorem 3.10 (Optimal coupling for $\alpha$-expectile scores).

The optimal coupling of the MK minimisation problem induced by any consistent
score given in $\eqref{eq:expectile}$ is the comonotonic coupling.

######  Proof 3.11.

The MK divergence induced by the scoring function (12) is given as

$\displaystyle{\mathscr{S}}(F_{1},F_{2})$
$\displaystyle=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\left\\{\,\int_{{\mathds{R}}^{2}}\big{|}{\mathds{1}}_{\\{z_{1}\leq
z_{2}\\}}-\alpha\big{|}\,B_{\phi}(z_{1},z_{2})\,\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\,\right\\}$
$\displaystyle=\inf_{\pi\in\Pi(F_{1},\,F_{2})}\;\bigg{\\{}\int_{{\mathds{R}}^{2}}(1-\alpha)B_{\phi}(z_{1},z_{2}){\mathds{1}}_{\\{z_{1}\leq
z_{2}\\}}\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\ $ $\displaystyle\hskip
75.0pt+\int_{{\mathds{R}}^{2}}\alpha
B_{\phi}(z_{1},z_{2}){\mathds{1}}_{\\{z_{1}>z_{2}\\}}\pi({\mathrm{d}}z_{1},{\mathrm{d}}z_{2})\bigg{\\}}\,.$

Convexity of $\phi$ implies that $B_{\phi}(z_{1},z_{2})$ is submodular. Hence
also the function

$(1-\alpha)B_{\phi}(z_{1},z_{2}){\mathds{1}}_{\\{z_{1}\leq z_{2}\\}}+\alpha
B_{\phi}(z_{1},z_{2}){\mathds{1}}_{\\{z_{1}>z_{2}\\}}$

is submodular. As the comonotonic coupling is optimal for submodular cost
functions, the result follows.

###  3.4 Shortfall score

A popular class of risk measures is the family of shortfall risk measures. We
first recall their definition and the scoring functions that elicit shortfall
risk measures.

######  Definition 3.12 (Shortfall risk measure).

Let $\ell\colon{\mathds{R}}\to{\mathds{R}}$ be an increasing and non-constant
function satisfying $\ell(w)<0$, whenever $w<0$, and $\ell(w)>0$, whenever
$w>0$. Then the shortfall risk measure $T_{\ell}$ is defined by

(13)
$T_{\ell}(F):=\inf\left\\{x\in{\mathds{R}}~{}\Big{|}~{}\int\ell(w-x)\,dF(w)\leq
0\right\\}\,,$

whenever the infimum exists. If furthermore $\ell$ is left-continuous and
strictly increasing on either $(-\infty,{\varepsilon})$ or
$({\varepsilon},+\infty)$, for ${\varepsilon}>0$, then $T_{\ell}$ is
1-elicitable with strictly consistent score [1]

(14) $S(z,y)=\int_{0}^{y-z}\ell(s)\,ds\,.$

######  Theorem 3.13 (Optimal coupling for shortfall scores).

The optimal coupling of the MK minimisation problem induced by the scoring
function given in (14) is the comonotonic coupling.

######  Proof 3.14.

The scoring function in (14) is submodular.

###  3.5 Decomposable scores

Next we study a family of scores that elicits different risk functionals
including quantiles, expectiles, and $M$ and $L^{p}$ quantiles.

######  Theorem 3.15 (Optimal coupling for decomposable scores).

Assume the scoring function is of the form

(15)
$S(z,y)=\phi(|z-y|)\left(\alpha\,{\mathds{1}}_{\\{y>z\\}}+\beta\,{\mathds{1}}_{\\{y\leq
z\\}}\right)\,,$

where $\alpha,\beta\in[0,1]$ and
$\phi\colon{\mathds{R}}_{+}\to{\mathds{R}}_{+}$ is an increasing convex
function satisfying $\phi(0)=0$. Then the optimal coupling of the MK
minimisation problem induced by any score given in (15) is the comonotonic
coupling.

######  Proof 3.16.

We rewrite the scoring function to

$S(z,y)=\phi\big{(}|z-y|\big{)}\left(\alpha\,{\mathds{1}}_{\\{y>z\\}}+\beta\,{\mathds{1}}_{\\{y\leq
z\\}}\right)=\alpha\,\phi\big{(}(y-z)_{+}\big{)}+\beta\,\phi\big{(}(z-y)_{+}\big{)}\,.$

As the function $\phi(x_{+})$ is convex, the results follows using similar
arguments as in the proof of Theorem 3.5.

The scoring function (15) elicits the $\alpha$-quantile for the choice
$\phi(x)=x$ and $\beta=1-\alpha$ and the $\alpha$-expectile with
$\phi(x)=x^{2}$ and $\beta=1-\alpha$. More generally, the score (15) for any
arbitrary convex $\phi$ and $\beta=1-\alpha$ elicit the so-called
$M$-quantiles, defined via (2) whenever they exits, see e.g., [3].
$M$-quantiles subsume $L^{p}$-quantiles, which correspond to $\phi(x)=x^{p}$,
$p\in[1,\infty)$, see [6].

###  3.6 Osband’s transformation of scores

Osband’s principle is highly regarded in statistics to create new elicitable
functionals [14]. Here, we use Osband’s principle to characterise the optimal
coupling induced by consistent scores for monotone transformations of
elicitable functionals.

######  Proposition 3.17 (Osband’s principle for OT).

Let $T$ be a strictly monotone transformation of a 1-elicitable functional
$\tilde{T}$, i.e. $T:=g\circ\tilde{T}$ for
$g\colon{\mathds{R}}\to{\mathds{R}}$ strictly monotone, and denote by
$\tilde{S}$ a (strictly) ${\mathcal{M}}$-consistent scoring function for
$\tilde{T}$. Then $T$ is 1-elicitable with (strictly)
${\mathcal{M}}$-consistent scoring function

(16) $S(z,y):=\tilde{S}\big{(}g^{-1}(z),y\big{)}\,.$

Let the corresponding MK minimisation problem of $\tilde{S}$ be attained by
the comonotonic coupling. If further $g$ is increasing (decreasing) then the
optimal coupling of the MK minimisation problem induced by the score (16) is
the comonotonic (antitonic) coupling.

######  Proof 3.18.

By Osband’s principle, the functional $T:=g\circ\tilde{T}$ for any bijective
function $g\colon{\mathds{R}}\to{\mathds{R}}$ is 1-elicitable with scoring
function given in (16), see e.g., Theorem 4 in [14]. If $g$ is strictly
increasing then,
$\frac{d}{dx}g^{-1}(x)=\big{\\{}g^{\prime}\,\big{(}g^{-1}(x)\big{)}\big{\\}}^{-1}>0$,
where $g^{\prime}(x):=\frac{d}{dx}g(x)$. Thus, the optimal coupling is
comonotonic. If $g$ is strictly decreasing, then $\frac{d}{dx}g^{-1}(x)<0$ and
the antitonic coupling is optimal.

Osband’s principle can be used to create new MK divergences. Indeed any
strictly monotonic transformation of an elicitable risk functional leads to a
new MK divergence, where the optimal coupling follows from Proposition 3.17.
Here, we give an example of the reciprocate of risk functionals.

######  Corollary 3.19 (Inverse of risk functionals).

Let $\tilde{T}$ be a 1-elicitable risk functional and assume that the
corresponding MK minimisation problem is attained by the comonotonic coupling.

Next consider the functional

$T(F):=\frac{1}{\tilde{T}(F)}\,,$

then the MK minimisation problem induced by any consistent score for $T$ is
attained by the antitonic coupling.

######  Proof 3.20.

This follows by Proposition 3.17 with
$T(\cdot)=g\big{(}\tilde{T}(\cdot)\big{)}$, where $g(x)=\frac{1}{x}$ is
decreasing.

The next application of Osband’s principle concern transformations of the
distribution. For this, let $F_{Z}$ denote the distribution of a random
variable $Z$.

######  Proposition 3.21 (Osband’s principle for OT).

Let $\tilde{T}$ be a 1-elicitable risk functional. For a function
$h\colon{\mathds{R}}\to{\mathds{R}}$, consider the risk functional
$T(F_{Y}):=\tilde{T}\left(F_{h(Y)}\right)$. Assume that the MK minimisation
problem induced by a consistent score $\tilde{S}$ for $\tilde{T}$ is attained
by the comonotonic coupling. If $h$ is increasing (decreasing), then the
optimal coupling of the MK minimisation problem induced by a consistent score
of $T$ is the comonotonic (antitonic) coupling.

######  Proof 3.22.

A score $\tilde{S}$ is strictly consistent for $\tilde{T}$ applied to
$F_{h(Y)}$ if and only if it holds for all $z\neq\tilde{T}(F_{h(Y)})$ that

(17)
$\int\tilde{S}\Big{(}\tilde{T}\big{(}F_{h(Y)}\big{)},\tilde{y}\Big{)}\,{\mathrm{d}}F_{h(Y)}(\tilde{y})\;\leq\;\int\tilde{S}\left(z,\tilde{y}\right)\,{\mathrm{d}}F_{h(Y)}(\tilde{y})\,.$

The inequalities (17) are equivalent to

(18)
$\int\tilde{S}\Big{(}T\big{(}F_{Y}\big{)},\,h(y)\Big{)}\,{\mathrm{d}}F_{Y}(y)\leq\int\tilde{S}\big{(}z,\,h(y)\big{)}\,{\mathrm{d}}F_{Y}(y)\,,$

for all $z\neq T(F_{Y})$. Thus, the score
$S(z,y):=\tilde{S}\big{(}z,h(y)\big{)}$ is a consistent score for $T(F_{Y})$.
The reminder follows using similar arguments as in the proof of Proposition
3.17.

An immediate example of Proposition 3.21 is that of the functional
$T(F_{Y})={\mathbb{E}}\left[\frac{1}{Y}\right]$ is elicitable and induces MK
divergences for which the antitonic coupling is simultaneously optimal for any
consistent score of $T$.

Using Osband’s principles for OT, we derive the optimal coupling for the
scores of entropic risk measures. The entropic risk measure with parameter
$\gamma>0$, also known as the exponential premium principle in actuarial
science [13], is defined for $F\in{\mathcal{M}^{0}}$, by

$T^{\gamma}(F):=\frac{1}{\gamma}\,\log\,\int e^{\gamma
y}\,{\mathrm{d}}F(y)\,,$

Any (strictly) ${\mathcal{M}}$-consistent scoring function (under mild
regularity conditions) for the entropic risk measure satisfying $S(y,y)=0$ is
given by

$S(z,y)=\phi\big{(}e^{\gamma y}\big{)}-\phi\big{(}e^{\gamma
z}\big{)}+\phi^{\prime}\big{(}e^{\gamma y}\big{)}\left(e^{\gamma z}-e^{\gamma
y}\right)\,,$

where $\phi\colon{\mathds{R}}\to{\mathds{R}}$ is (strictly) convex and
$\int|\phi\big{(}e^{\gamma y}\big{)}|\,{\mathrm{d}}F(y)<\infty$ for all
$F\in{\mathcal{M}}$; for more detail see Appendix A1 in [10].

######  Corollary 3.23 (Optimal coupling for entropic scores).

The optimal coupling of the MK minimisation problem induced by any consistent
scoring function for the entropic risk measure is the comonotonic coupling.

######  Proof 3.24.

Denote by $\tilde{T}(G):=\int x\,{\mathrm{d}}G(x)$ the expectation functional
with consistent score $S_{\phi}$, the Bregman score, i.e. (6). Then, it holds
that

$T^{\gamma}(F):=g\circ\tilde{T}\big{(}F_{h(Y)}\big{)}\,,$

where $g(x):=\frac{1}{\gamma}\log(x)$ and $h(x):=e^{\gamma x}$. By Osband’s
principles, see the proofs of Proposition 3.17 and Proposition 3.21, the
consistent score for $T^{\gamma}$ is thus given by

$S(z,y)=S_{\phi}\left(g^{-1}(z),\,h(y)\right)=\phi(e^{\gamma
z})-\phi(e^{\gamma y})-\phi^{\prime}(e^{\gamma y})(e^{\gamma z}-e^{\gamma
y})\,,$

which is indeed the strictly consistent scoring function for the entropic risk
measure. Finally, applying Proposition 3.17 and Proposition 3.21 concludes the
proof.

###  3.7 Scores of law-invariant risk measures

In this section we consider MK divergences derived from elicitable convex and
coherent risk measures. We show that the optimal coupling of the MK divergence
induced by any strictly consistent score that elicits a coherent or convex
risk measure is the comonotonic coupling. For this, denote by
$T\colon{\mathcal{M}}^{\infty}\to{\mathds{R}}$ a law-invariant risk measure,
where ${\mathcal{M}}^{\infty}\subseteq{\mathcal{M}}^{0}$ is the space of all
cdfs with bounded support. A law-invariant risk measure can equivalently be
defined on the space of absolutely bounded random variables
${\mathcal{L}}^{\infty}:={\mathcal{L}}^{\infty}(\Omega,{\mathcal{F}},{\mathds{R}})$,
as a functional $T\colon{\mathcal{L}}^{\infty}\to{\mathds{R}}$, by setting
$T[X]:=T(F_{X})$, whenever $X$ has cdf $F_{X}$. We use the notation
$T(\cdot)$, when the risk measure is viewed as a function of cdfs and
$T[\cdot]$, when applied to random variables. We say a law-invariant risk
measure $T$ is

  1. $(i)$

monotone: if $T[X]\leq T[Y]$ whenever $X\leq Y$ ${\mathbb{P}}$-a.s.,
$X,Y\in{\mathcal{L}}^{\infty}$,

  2. $(ii)$

translation invariant: if $T[X+m]=T[X]+m$, for all
$X\in{\mathcal{L}}^{\infty}$ and $m\in{\mathds{R}}$,

  3. $(iii)$

positive homogeneous: if $T[\lambda\,X]=\lambda\,T[X]$, for all
$X\in{\mathcal{L}}^{\infty}$ and $\lambda\geq 0$,

  4. $(iv)$

convex: if
$T\big{[}\lambda\,X+(1-\lambda)\,Y]\leq\lambda\,T[X]+(1-\lambda)\,T[Y]$, for
all $X,Y\in{\mathcal{L}}^{\infty}$ and $\lambda\in[0,1]$,

A law-invariant functional $T$ is called a convex risk measure, if it
satisfies the properties $(i)$, $(ii)$, and $(iv)$, and a coherent risk
measure if it additionally fulfils $(iii)$. For discussions and interpretation
of these properties, we refer the reader to [12] and reference therein.

######  Definition 3.25.

We consider the following additional properties on scoring functions

  1. $(i)$

$S(z,y)=0$ if and only if $z=y$

  2. $(ii)$

$S(z,y)$ is continuous in both $z$ and $y$,

  3. $(iii)$

$S(z,y)$ is increasing in $z$, for $z>y$ and decreasing in in $z$ for $z<y$

  4. $(iv)$

for all $z$ in a neighbourhood of 0, $S(z,y)\leq\psi(y)$, where
$\psi\colon{\mathds{R}}\to[1,+\infty)$ is a continuous gauge function.

Property $(iii)$ is called accuracy rewarding in the statistical literature
[15], since it implies that if $T(F)<z_{1}<z_{2}$ or $T(F)>z_{1}>z_{2}$, then

$\int S\big{(}T(F),\,y\big{)}\,{\mathrm{d}}F(y)<\int
S\big{(}z_{1},\,y\big{)}\,{\mathrm{d}}F(y)<\int
S\big{(}z_{2},\,y\big{)}\,{\mathrm{d}}F(y)\,.$

Thus, for accuracy rewarding scores the further away the estimates $z$ are
from the truth $T(F)$, the larger are their expected scores.

######  Theorem 3.26 (Coherent Risk Measures).

Let $T$ be a 1-elicitable coherent risk measure and $S$ any strictly
${\mathcal{M}}^{\infty}$-consistent score for $T$ that satisfies the
properties in Definition 3.25. Then, the optimal coupling of the MK
minimisation problem induced by the score $S$ is the comonotonic coupling.

######  Proof 3.27.

The class of 1-elicitable coherent risk measures coincides with the
$\alpha$-expectiles where $\alpha\in[0.5,1]$, see Theorem 4.9 in [1], or
Corollary 12 in [22]. Applying Theorem 3.10 concludes the proof.

######  Theorem 3.28 (Convex Risk Measures).

Let $T$ be a 1-elicitable convex risk measure and $S$ any strictly
${\mathcal{M}}^{\infty}$-consistent score for $T$ that satisfies the
properties in Definition 3.25. Then, the optimal coupling of the MK
minimisation problem induced by the score $S$ is the comonotonic coupling.

######  Proof 3.29.

The class of convex risk measures that are 1-elicitable coincides with the
class of shortfall risk measures $T_{\ell}$ with convex loss function $\ell$,
see Theorem 4.6 in [1]. Applying Theorem 3.13 concludes the proof.

##  4 Applications

###  4.1 Worst-case distortion risk measures

Worst-case distortion risk measures are often used in robust stochastic
optimisation, see e.g., [2] and [18]. We call $g\colon[0,1]\to[0,1]$ a
distortion function, if it is non-decreasing and satisfies $g(0)=0$ and
$g(1)=1$. A distortion risk measure evaluated at a cdf $G$ with quantile
function ${\breve{G}}$ is defined as the Choquet integral

$H_{g}({\breve{G}}):=-\int_{-\infty}^{0}1-g(1-G(x))\,\mathrm{d}x+\int_{0}^{+\infty}g\big{(}1-G(x)\big{)}\,\mathrm{d}x\,,$

in which at least one of the two integrals is finite. If $g$ is absolutely
continuous, then the distortion risk measure $H_{g}$ has representation

(19) $\displaystyle H_{g}({\breve{G}})$
$\displaystyle=\int_{0}^{1}\gamma(u){\breve{G}}(u)\,\mathrm{d}u\,,$

with weight function $\gamma(u):=\partial_{-}g(x)|_{x=1-u},~{}0<u-<1$, which
satisfies $\int_{0}^{1}\gamma(u)\mathrm{d}u=1$ and where $\partial_{-}$
denotes the derivative from the left. In the sequel we assume that
representation (19) holds and that
$\int_{0}^{1}|\gamma(u)|^{2}\mathrm{d}u<+\infty$. The class of distortion risk
measures is broad and contains the majority of risk measures used in financial
risk management practice including the quantile (Value-at-Risk) and the
average of upper quantiles (Tail Value-at-Risk).

Let $F$ be a given reference cdf and consider the optimisation problem

(20)
$\max_{{\breve{G}}\in{\breve{\mathcal{M}}}}\;H_{g}\big{(}{\breve{G}}\big{)}\,,\quad
s.t.\quad{\mathscr{B}}_{\phi}\big{(}G,\,F\big{)}\,\leq\,{\varepsilon}\,,$

where ${\breve{\mathcal{M}}}$ is the set of left-continuous quantile functions
on ${\mathds{R}}$. Optimisation problems of the form (20) are of great
interest in financial risk management, as risk measures and the values thereof
drive decision making (e.g., through solvency requirements), but typically are
subject to distributional uncertainty, here quantified via the BW divergence.
Specifically, in the optimisation problem (20) one aims to determine the
worst-case value a distortion risk measure can attain when the underlying cdf
belongs to a set of cdfs that are close, in the BW divergence, to a reference
cdf $F$. The special case of optimisation problem (20) for $\phi(x)=x^{2}$,
i.e. when the BW divergence reduces to the 2-Wasserstein distance, was
recently solved in [2]. Here, we significantly extend this result.

######  Theorem 4.1 (Worst-case Distortion Risk Measures).

Assume that the distortion function $g$ is strictly concave and that $\phi$ is
strictly convex. The optimal quantile function to the optimisation problem
(20) is given by

(21)
${\breve{G}}_{\lambda^{*}}(u):=\left(\phi^{\prime}\right)^{-1}\left(\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}+\frac{1}{\lambda^{*}}\gamma(u)\right)\,,$

where $\lambda^{*}>0$ is the unique solution to
${\mathscr{B}}_{\phi}\big{(}G_{\lambda},\,F\big{)}={\varepsilon}$.

######  Proof 4.2.

A solution to problem (20) must attain a BW divergence ${\varepsilon}_{0}$,
$0\leq{\varepsilon}_{0}\leq{\varepsilon}$, and thus solve for all $\lambda>0,$
the optimisation problem

$\displaystyle\operatorname*{argmax}_{{\breve{G}}\in{\breve{\mathcal{M}}}}\;\int_{0}^{1}{\breve{G}}(u)\gamma(u)-\lambda\left(\phi\big{(}{\breve{G}}(u)\big{)}-\phi\big{(}{\breve{F}}(u)\big{)}-\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}{\breve{G}}(u)\
+\
\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}{\breve{F}}(u)-{\varepsilon}_{0}\right)\,.$

Equivalently, the solution solves for all $\lambda>0$ the optimisation problem

$\operatorname*{argmin}_{{\breve{G}}\in{\breve{\mathcal{M}}}}\;\int_{0}^{1}\phi\big{(}{\breve{G}}(u)\big{)}-k_{\lambda}(u){\breve{G}}(u)du\,,$

where
$k_{\lambda}(u):=\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}+\frac{1}{\lambda}\gamma(u).$
Note that for any given function $k_{\lambda}(\cdot),\lambda>0$, the function
$\phi\big{(}{\breve{G}}(u)\big{)}-k_{\lambda}(u){\breve{G}}(u)$ is convex in
${\breve{G}}(u)$. By direct optimisation and noting that $k_{\lambda}(u)$ is
an increasing function, we thus obtain that a solution to (20) is of the form

(22)
${\breve{G}}_{\lambda}(u):=\left(\phi^{\prime}\right)^{-1}\left(\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}+\frac{1}{\lambda}\gamma(u)\right)\,,$

for some $\lambda>0$. Furthermore, $\lambda_{1}<\lambda_{2}$ implies that
${\breve{G}}_{\lambda_{1}}(u)>{\breve{G}}_{\lambda_{2}}(u)$. As a consequence,
it also holds that
$H_{g}({\breve{G}}_{\lambda_{1}})>H_{g}({\breve{G}}_{\lambda_{2}})$ and
moreover that (recall that $\phi$ is convex)
${\mathscr{B}}_{\phi}(F,G_{\lambda_{1}})>{\mathscr{B}}_{\phi}(F,G_{\lambda_{2}})$.
Hence, for a solution to be optimal, $\lambda^{*}$ must satisfy
${\mathscr{B}}_{\phi}(F,G_{\lambda^{*}})={\varepsilon}$. The existence of
$\lambda^{*}$ follows since ${\mathscr{B}}_{\phi}(F,G_{\lambda})$ is
continuously decreasing in $\lambda>0$, $\lim_{\lambda\to
0}{{\mathscr{B}}_{\phi}(F,G_{\lambda})}=\infty$, and
$\lim_{\lambda\to\infty}{{\mathscr{B}}_{\phi}(F,G_{\lambda})}=0.$ Moreover,
$\lambda^{*}$ is unique.

###  4.2 Cheapest payoffs

Let $S_{T}$ represents the non-negative random value of a risky asset at time
$T>0.$ Further consider a bank account earning the continuously compounded
risk-free interest rate $r\in\mathbb{R}$. We define the set of payoffs

$\mathcal{X}:=\Big{\\{}g(S_{T})~{}|~{}\,g:{\mathds{R}}_{+}\to{\mathds{R}}_{+}\text{
is measurable and
}{\mathbb{E}}_{\mathbb{Q}}\big{[}|g(S_{T})|\big{]}<\infty\Big{\\}}\,,$

where ${\mathbb{Q}}$ is a pricing measure equivalent to ${\mathbb{P}}$, and
${\mathbb{E}}_{\mathbb{Q}}[\cdot]$ denotes the expectation under
${\mathbb{Q}}$. For any payoff $X\in\mathcal{X}$, its initial cost is

(23)
$c(X):=e^{-rT}{\mathbb{E}}_{{\mathbb{Q}}}[X]={\mathbb{E}}_{\mathbb{P}}[\xi
X]\,,$

where the last equality follows by defining the state-price-density
$\xi:=e^{-rT}\frac{d{\mathbb{Q}}}{d{\mathbb{P}}}$. We assume that the state-
price density $\xi$ is continuously distributed and denote its cdf under
${\mathbb{P}}$ by $F_{\xi}$. In the sequel, all cdfs of random variables are
taken with respect to $\mathbb{P}$.

We consider the problem of finding the cheapest payoff under the constraint
that its cdf is close to a benchmark. Specifically, we require that its cdf
lies withing a BW-ball around a benchmark cdf $F$, that is we consider the
optimisation problem

(24) $\min_{X\in\mathcal{X}}\;c(X)\,,\quad
s.t.\quad{\mathscr{B}}_{\phi}\big{(}F_{X},\,F\big{)}\,\leq\,{\varepsilon}\,.$

A special case of (24) is considered in [8] who, instead of the BW divergence,
consider payoffs with fixed cdf $F$. That is, they solve

(25) $\min_{X\in\mathcal{X}}\;c(X)\,,\quad s.t.\quad F_{X}\equiv F\,,$

and show that the unique solution to optimisation problem (25) is given by
$X^{*}:={\breve{F}}\big{(}1-F_{\xi}(\xi)\big{)}$. The payoff $X^{*}$ is called
cost-efficient as it is decreasing in $\xi$, and thus yields the cheapest
payoff with given cdf. Next, we provide the solution to optimisation problem
(24).

######  Theorem 4.3 (Cheapest payoffs).

The solution to optimisation problem (24) is given by

(26) $X^{*}:={{\breve{G}}_{\lambda^{*}}}\big{(}1-F_{\xi}(\xi)\big{)}\,,$

in which

(27)
${\breve{G}}_{\lambda^{*}}(u):=\left(\phi^{\prime}\right)^{-1}\left(\phi^{\prime}\big{(}{\breve{F}}(u)\big{)}-\frac{1}{\lambda^{*}}\,{{\breve{F}_{\xi}}}(1-u)\right)\,,$

where $\lambda^{*}>0$ is the unique solution to
${\mathscr{B}}_{\phi}\big{(}G_{\lambda^{*}},\,F\big{)}={\varepsilon}$.

######  Proof 4.4.

We first show that any solution has to be cost-efficient. To this regard, let
$X$ be a solution to optimisation problem (24) with cdf $G$. Define the random
variable $Y:={\breve{G}}\big{(}1-F_{\xi}(\xi)\big{)}$ which has cdf $G$, and
thus its cdf lies within the BW-ball. Moreover, unless $Y=X$,
${\mathbb{P}}$-a.s., $Y$ is strictly cheaper than $X$. Hence, an optimal
solution has to be cost-efficient. Furthermore, from (23) it follows that the
cost of any cost-efficient payoff $Y$ with cdf $G$ can be expressed as

(28) $c(Y)=\int_{0}^{1}\gamma(u)\,{\breve{G}}(u)\,du\,,$

where $\gamma(u):=-{{\breve{F}_{\xi}}}(1-u)$ is an increasing function. We
observe that solving optimisation problem (24) is equivalent to solving an
optimisation problem of the form (20). Next, we apply Theorem 4.1, as an
inspection of its proof reveals that Theorem 4.1 also holds when the
(increasing) weighting function $\gamma$ is negative.

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