\section{Introduction}

The main focus of this paper is the study of robustness properties of optimization procedures within Quantitative Risk Management (QRM).
For this, we introduce a novel general framework, which at the same time is conceptually intuitive and mathematically challenging.
A key and, as we will highlight in the paper, novel, question concerns the influence of the choice of the underlying objective on the resulting robustness properties in risk optimization.
 In particular, we
are interested in the two most popular regulatory risk measures,
the Value-at-Risk (VaR) and the Expected Shortfall (ES), and their robustness properties in the context of risk optimization.

In QRM,  the concept of robustness for risk measures is traditionally studied at the level of objective functionals without involving optimization problems; see \cite{CDS10}, \cite{KPH13}, \cite{KSZ14, KSZ17}, \cite{EWW15}, and the references therein.
In the literature on robust optimization (see e.g.~\cite{BTGN09}),  model uncertainty is typically incorporated through modifying the objective functional or the constraints.

Our setup is different from the above perspectives; it builds on the simple observation that
a risk measure as a standalone function may be robust, but fail to have desirable robustness properties when this measure is used within an optimization context.
To briefly illustrate our ideas, suppose that a risk factor is represented by a random variable $X$ arising from a stochastic model, later in the paper, X will denote a random vector.
An investor has to optimize her position according to the
best of her knowledge, and hence
we shall refer to $X$ as the best-of-knowledge model, and the true model, denoted by $Z$, is typically unknowable.
Ideally, a good model $X$ is close to $Z$ in  a sense to be made clear later.
Based on the best-of-knowledge model $X$ and an objective functional $\rho$, an optimized position  is chosen as a function $g(X)$ of $X$. Whereas the position $g(X)$ may have a desirable objective value $\rho(g(X))$,  this does not guarantee that $\rho(g(Z))$ is also desirable if  $Z$ is ``slightly" different  from $X$.
In the absence of a perfect model, which almost always is the case in financial applications, this issue becomes crucially important.
In the present paper, we put this observation into a rigorous quantitative framework.

The paper \cite{CDS10}  compares the qualitative robustness of VaR and coherent risk measures; the authors conclude that VaR is better in their context.
Some later papers, e.g.~\cite{EWW15} and \cite{KSZ14, KSZ17},  put the corresponding arguments into a different perspective, showing that ES also has certain desirable  robustness properties.
Both streams of research assumed that both VaR and, say, ES are applied to the same financial position. In reality, however, the regulatory choice of a particular risk measure creates certain incentives, just like any other aspect of regulation. These incentives become effective even before that risk measure has ever been applied in a risk management context. For instance, once a specific risk measure has been chosen, portfolios will be optimized with respect to that risk measure. Thus, in reality, VaR and ES will will typically not  be applied to the same position. Therefore, one cannot decouple the technical properties of a chosen risk measure from the incentives it creates. In our current paper, we make a first attempt of taking the effects created by such incentives into account. As an important, and perhaps somewhat surprising  consequence, we arrive at  completely different conclusions concerning robustness properties than the previous literature.

The contribution and the structure of the paper are outlined below.
In Section \ref{sec:2}, we introduce the theoretical framework of robustness properties of risk measures in the context of optimization problems, referred to as 'robustness in optimization' throughout the paper.
The framework is quite general and it includes many practical problems in various fields of applications, not necessarily confined to finance and insurance.
In Section \ref{sec:24}, some preliminary results connecting robustness and continuity are presented.
Keeping the problem of risk measures in mind, several representative  optimization problems are listed in Section \ref{sec:3}.
As one of our contributions, we provide analytical solutions to these optimization problems   in Sections \ref{sec:4} and \ref{sec:5} for VaR and ES, respectively.
From our results, we shall see that, indeed, for the case of VaR which is argued by many as a robust risk measure, its corresponding optimization is highly non-robust and a small model uncertainty would ruin the optimality of the optimized positions. For the case  of ES the optimized positions are generally robust.
In Section \ref{sec:7}, we present some discussions on the implications of our results for the desirability of specific regulatory risk measures, an on-going debate in the financial industry (\cite{BASEL16}, \cite{IAIS14}).
Our results yield a (further) strong argument against using VaR as a risk measure within banking and insurance regulation;
for a related discussion on robustness in the realm of risk sharing, see \cite{ELW18}.
In the last section, Section \ref{sec:6a}, we discuss our notion of robustness in the context of distributionally robust optimization (e.g.~\cite{NPS08}, \cite{GS10}).
The proofs of all results are put in the appendix.

 \section{Theoretical framework}\label{sec:2}
\subsection{Notation}
We work with an atomless probability space $(\Omega, \mathcal F, \mathbb{P})$. Let $L^q$ be the set of all random variables in $(\Omega, \mathcal F, \mathbb{P})$ with finite $q$-th moment, $q\in  {(0,\infty)}$, {$L^0$ be the space of all $\mathbb{P}$-a.s.~finite random variables, and $L^\infty$ the set of all essentially bounded elements of $L^0$}.

For a positive integer $n$, write $L^q_n=(L^q)^n$.
For a vector $x\in \mathbb{R}^n$, $|x|$ is its Euclidean norm.
Throughout, for any $X\in L^0$, $F_X$ represents the distribution function of $X$.
The mappings  $\mathrm{ess\mbox{-}inf}(\cdot)$ and $\mathrm{ess\mbox{-}sup}(\cdot)$ on $L^0$ stand for the essential infimum and the essential supremum of a random variable, respectively.
  We write $X\buildrel \mathrm{d} \over = Y$ if the random variables $X$ and $Y$   have the same distribution under $\mathbb{P}$.
For $x\in \mathbb{R}$, denote by $\delta_x$ the point-mass probability measure at $x$.

 \subsection{Basic setup of optimization problems}

In this section, we first lay out the basic setup for optimization problems when the relevant information on the underlying economic model is known precisely, that is, the case without model uncertainty.
 Let $X$ be an $n$-dimensional random vector, where $n$ is a positive integer.
The random vector $X$ is called an \emph{economic vector}, which includes all random sources in an economic model under study,
such as potential losses, traded securities, hedging instruments, insurance contracts, macro economic factors, or pricing densities.

Let $\mathcal G_{n}$ be the set of {measurable} functions mapping $\mathbb{R}^{n}$ to $\mathbb{R}$.
A random variable $g(X)$ where $g\in \mathcal G_n$ represents a risky position of an investor, in which positive values represent a loss and negative values represent a gain. This sign convention is in line with the regulatory angle we follow in several applications in our paper.
The investor's problem is to choose among admissible  positions $g(X)$ for some functions $g$ in an admissible set $\mathcal G\subset \mathcal G_n$.

For  a set $\mathcal G\subset \mathcal G_n$,
we formulate the problem
\begin{equation}\mbox{to minimize: } \rho(g(X)) \mbox{~~~~~subject to~} g\in \mathcal G, \label{eq:opt1}\end{equation}
where $\rho$ is an objective functional mapping a set containing $\{g(X):g\in \mathcal G\}$ to $\mathbb R\cup\{+\infty\}$.
Here one prefers a smaller value of the objective functional over a larger value.
Objective functionals  considered {may be}  general; examples  include (up to a sign change) mean-variance functionals, expected utilities,  rank-dependent utility functionals, functionals in cumulative prospect theory, and various risk measures as discussed in \cite{ADEH99} and \cite{FS16}. {Our main interest will be, however, in the risk measures Value-at-Risk (VaR) and Expected Shortfall (ES\footnote{A formal definition of VaR and ES is given in Section \ref{sec:3}. ES is also known as CVaR, CTE, AVaR and TVaR, depending on the context (see e.g.~\cite{GW07}, \cite{MFE15} and \cite{FS16}). We use the term ES to be consistent with the Basel Committee on Banking Supervision (\cite{BASEL16}), because our study is motivated by the comparative advantages of VaR and ES in regulation; see discussions in Section \ref{sec:7}.}).}

The elements in the optimization problem \eqref{eq:opt1} can be summarized by an objective functional $\rho$ and a pair $(X,\mathcal G)$.
We always assume that the domain of the objective functional $\rho$ contains $\{g(X):g\in \mathcal G\}$, otherwise \eqref{eq:opt1} is meaningless.

\begin{example}\label{hedge ex}
An illustrative example is the classic problem of hedging in a financial market. Suppose that an investor currently faces a risk $W$ and would like to hedge against $W$.
She has access to hedging instruments in a set $\{g(Y):g\in \mathcal G'\}$ where $Y$ is an ${(n-1)}$-dimensional economic vector and, say, $\mathcal G'\subset \mathcal G_{{n-1}}$, ${n\ge 2}$.
Typically, the set $\mathcal G'$ involves a budget constraint.
Equivalently, she chooses risky positions in the set $\{W-g(Y): g\in \mathcal G'\}$, which represents all possible hedged positions she may attain.
Note that $\{W-g(Y): g\in \mathcal G'\}=\{f(X): f\in \mathcal G\}$ where $X=(W,Y)$ is an ${n}$-dimensional random vector and
 $$\mathcal G=\{f \in \mathcal G_{{n}}:f(w,y)=w-g(y),~g\in \mathcal{G}',~w\in \mathbb{R},~y\in \mathbb{R}^{{n-1}}\};$$ therefore the hedging problem is a special case of our general setting \eqref{eq:opt1}.
Here we allow $W$ to be arbitrarily dependent on $Y$. If $W$ is a financial risk and $Y$ is the vector of asset prices in a complete financial market, then it may be that $W$ {is a function of} $Y$.
On the other hand, if $W$ represents a property and casualty insurance risk and $Y$ is  the vector of asset prices in a financial market, then it may be reasonable to assume that $W$ and $Y$ are independent. {See Remark \ref{rem:ctfinmarket}
for a discussion on how a set of more specific optimization problems can be interpreted within the context of a complete market model with $d$ assets in continuous time. }
\end{example}

 \begin{example}\label{Markowitz ex}In Markowitz portfolio selection, a investor wishes to find an optimal allocation vector  $w\in\mathbb R^n$ based on a return vector $Y\in L^2_n$ for $n$ stocks. The problem can be described as the minimization of $\mathbb E[g(Y)]$ where $g\in\mathcal G'$ and $\mathcal G'$ consists of all functions $g\in \mathcal G_n$ that are of the form $g(y)=\lambda(w^\top y)^2-w^\top y$ for $w\in\mathbb R^n$ satisfying $\sum_{i=1}^nw_i=x_0$. Here, \textcolor{red}{$\lambda>0$} is a risk-aversion parameter and $x_0$ represents the budget constraint of the investor.
\end{example}

In this paper, we keep the choice of $(X,\mathcal G)$ as generic as possible. Some special cases and examples are studied in Sections \ref{sec:3}-\ref{sec:5}.

\begin{remark}
 One can also consider continuous-time models where optimizers are chosen over a set of stochastic processes (e.g.~admissible trading strategies).
Our discussions apply to such problems, as long as the optimizers are functions of the random source $X$, be it finite dimensional or infinite dimensional.
In fact, in many classic financial models, the continuous-time optimization problem (such as the hedging example above) can be translated into a single-period optimization problem via the martingale approach; see \cite{FSW09}.
For the {sake of simplicity, we will focus in this paper} on the case of optimization over functions of finite dimensional vectors.
\end{remark}

\subsection{Uncertainty and robustness in optimization}

We proceed to put uncertainty into the optimization problem \eqref{eq:opt1} described above.
For  $X\in L_n^0$, $\mathcal G\subset \mathcal G_n$ and an objective functional $\rho$,
denote by $\rho(X;\mathcal G)$ the minimum possible value of $\rho$, namely,
$$\rho(X;\mathcal G)= \inf\{\rho(g(X)): g\in \mathcal G\},$$
and by $\mathcal G_X(\rho)$ the set of \emph{optimizing functions}, that is,
\begin{equation}\label{eq:opt2}
\mathcal G_X(\rho)=\{g\in \mathcal G:\rho(g(X))=\rho(X;\mathcal G)\}.
\end{equation}
Note that $\mathcal G_X(\rho)$ might be an empty set.  Throughout this paper, the notation $g_X$ will refer to a generic element $g_X\in  \mathcal G_X(\rho)$, and
$g_X(X)$ will be called an \emph{optimized position}.

We shall use the random variable $X$ to represent our (perceived) model  for the underlying economic vector.
In practice, the model $X$ is obtained based on stochastic assumptions  and statistical inference, and it may not represent a true model  for the underlying economic vector.
In other words, the  optimization problem \eqref{eq:opt1} is often subject to severe model uncertainty. To reflect this issue, let $\mathcal{Z}\subset L_n^0$ be a set of \emph{possible} economic vectors including $X$; $\mathcal{Z}$ may be interpreted as the set of \emph{alternative} models\footnote{
 For instance, $\mathcal{Z}$ can be a parametric family of risk models and hence corresponds to parameter uncertainty of the underlying risk models.
Each component of the vector $X$ may have a different economic meaning.  Some of them may be subject to more severe model uncertainty whereas others may be free of model uncertainty.
This can be reflected in the choice of $\mathcal{Z}$ which may be contained in a low-dimensional subset of the set of $n$-random vectors.}.
 Suppose that the \emph{real} economic vector  $Z\in \mathcal{Z}$ is different from the \emph{perceived} economic vector $X$.
The information  we have at hand is  about $X$ rather than $Z$,
and we shall  refer to $X$ as the \emph{best-of-knowledge model} and $Z$ as the \emph{true model}, which is unknowable.
We have to make decisions according to the best of our knowledge, that is, as in \eqref{eq:opt2}, choosing $ g_X \in   \mathcal G_X(\rho)$  optimizing our objective $\rho$.
The real \emph{real} but unknown   position $  g_X (Z)$ may be different from the \emph{perceived} optimal
 position $  g_X (X)$.
If $Z$ and $X$ are close to each other according to some {(pseudo-)}metric $\pi$ (e.g.~$L^\infty$-metric on the   space of bounded random vectors),
we would like  $\rho(  g_X (Z))$ to be close to $\rho(  g_X (X))$ in order to make sense of the position $  g_X (Z)$, which may no longer be optimal.
 In other words, we naturally would desire some continuity of the mapping  {$Y\mapsto \rho(  g_X (Y ))$} at $Y=X$.

 {Note that, in our situation, there is no point in analyzing  the problem of optimizing $g(Z)$ over $g\in \mathcal G$, because $Z$ is unknown. } This makes our framework conceptually different from the stream of research on stability of the set of optimizers under model uncertainty; see Remark \ref{pb rem} below for an example from financial regulation where we explicit this very important distinction between our paper and other approaches in the literature.

Putting this into the  {hedging} context  {of Example \ref{hedge ex}}, suppose that
the real economic vector $Z\in \mathcal{Z}$ is different from $X$ and
an investor has the real risk $f(Z)$ to hedge where  $f\in \mathcal G$.
The information she has is  about $X$, and she hedges from a set of instruments ${ \mathcal G'\subset\mathcal G}$.
In this case,   $f(Z)-g(Z)$   is the remaining risk she actually faces after hedging.
Under this setting, assuming the quality of model $X$ is good, $Z$ should be close to $X$ in some sense, and she would naturally desire some continuity of the function $ {Y\mapsto\rho(f(Y)-  g_X (Y))}$  at $Y=X$.

We also note that, in our framework, the admissible set $\mathcal G$ is not subject to model uncertainty; the investor knows which positions she can choose in the optimization problem.
For instance, in the above hedging example, a budget constraint that determines $\mathcal G$ is not affected by the model assumptions made for $X$; it is simply the observed prices for the hedging instruments.

In light of the above consideration,
we endow the set $\mathcal Z$ of all possible economic vectors with a pseudo-metric $\pi$.
Below we give three prominent examples of $\pi$, which will appear throughout the paper.
\begin{enumerate}[(i)]
\item For   a subset $\mathcal Z$  of $L^\infty_n$,
the $L^\infty$-metric $\pi_n^\infty$ is defined as
\begin{equation}\label{eq:linftym}
\pi_n^\infty(X,Y)=||X-Y||_\infty=\mathrm{ess\mbox{-}sup}(|X-Y|),~X,Y \in \mathcal Z.
\end{equation}
\item For $q\in [1,\infty)$ and a subset $\mathcal Z$  of $L^q_n$,
the $L^q$-metric $\pi_n^q$ is defined as
\begin{equation}\label{eq:lqm}
\pi_n^q(X,Y)=||X-Y||_q=(\mathbb{E}[|X-Y|^q])^{\frac 1q},~X,Y \in  \mathcal Z.
\end{equation}
\item For a subset $\mathcal{Z}$ of $L^0_n$,
the pseudo-metric $\pi^W_n$ is defined as
\begin{equation}\label{eq:weakm}
\pi_n^W(X,Y)=\pi_P(F_X,F_Y),~X,Y \in  \mathcal Z.
\end{equation}
where $\pi_P$ is the Prokhorov metric over the set of probability distribution measures\footnote{Precisely, $\pi_P(\mu ,\nu )=\inf \left\{\varepsilon >0:\mu (A)\leq \nu (A_{{\varepsilon }})+\varepsilon \ {\text{and}}\ \nu (A)\leq \mu (A_{{\varepsilon }})+\varepsilon \ {\text{for all}}\ A\in {\mathcal  {B}}(\mathbb{R}^n)\right\}$, where $A_{{\varepsilon }}=\{x\in \mathbb{R}^n:  ||x-y||<\varepsilon \mbox{~for some~}y\in A\}$ and $||\cdot||$ is the Euclidean norm.
}.
In this case, convergence in $\pi_n^W$ is equivalent to convergence in distribution (or weak convergence).
\end{enumerate}
The reason of considering a pseudo-metric instead of a metric is to be able to incorporate cases such as \eqref{eq:weakm} for objective functionals based on the distributions of risks, for instance, law-invariant risk measures.

\begin{definition}\label{def:0}
We call $(\mathcal{G},\mathcal{Z},\pi)$ an \emph{uncertainty triplet} if  $\mathcal{G}\subset \mathcal{G}_n$ and $(\mathcal{Z},\pi)$ is a pseudo-metric space of $n$-random vectors.
For a given uncertainty triplet $( \mathcal G,\mathcal{Z},\pi)$, we say that an objective functional $\rho$ is \emph{compatible} if
$\rho$ maps $\mathcal{G}(\mathcal{Z})=\mathcal\{g(Z):Z\in \mathcal{Z},~g\in \mathcal G\}$ to $\mathbb R\cup\{+\infty\}$, and $\rho(g(Y))=\rho(g(Z))$ for all $g\in \mathcal G$ and  $Y,Z\in \mathcal{Z}$ with $\pi(Y,Z)=0$, i.e.~$Y$ and $Z$ are  indistinguishable under the pseudo-metric $\pi$.\end{definition}

\begin{definition}\label{def:1}
Let $( \mathcal G,\mathcal{Z},\pi)$ be an uncertainty triplet.
A compatible objective functional
$\rho$ is \emph{{robust} at
$X\in \mathcal{Z}$ relative to $(\mathcal{G}, \mathcal{Z},\pi)$}  if there exists $  g_X \in  \mathcal G_X(\rho)$ such that
the function
$ {Y\mapsto \rho(  g_X (Y))}$ is $\pi$-continuous at $Y=X$.
\end{definition}

\begin{comment}
\rw{I would still like to discuss a bit about the pros and cons of this.
An   optimization problem with uncertainty consists of  an uncertainty triplet $( \mathcal G,\mathcal{Z},\pi)$ and a compatible objective functional $\rho$.
\begin{definition}
An optimization problem with uncertainty $(\rho,\mathcal{G},\mathcal{Z},\pi)$ is \emph{robust} at
$X\in \mathcal{Z}$ if there exists $g\in \mathcal G_X(\rho)$ such that
the function
$Z\mapsto \rho(g(Z))$ is $\pi$-continuous at $Z=X$.
\end{definition}
Another way to do this is the following, which is related to the issue of the ``robustified objective". It depends on how far we like to go into that discussion.
\begin{definition}
Let   $(\rho,\mathcal{G},\mathcal{Z},\pi)$ be an optimization problem with uncertainty.
a function $g\in \mathcal{G}$ is $(\rho,\mathcal{Z},\pi)$-\emph{robust} at
$X\in \mathcal{Z}$ if
$Z\mapsto \rho(g(Z))$ is $\pi$-continuous at $Z=X$.
The optimization problem $(\rho,\mathcal{G},\mathcal{Z},\pi)$ is \emph{robust} at
$X\in \mathcal{Z}$ if there exists a $(\rho,\mathcal{Z},\pi)$-robust $g\in \mathcal G_X(\rho)$.
\end{definition}
}
\end{comment}

In this paper, the concept of robustness is always in the sense of Definition \ref{def:1} and it should not be confused with the classic qualitative robustness as studied in e.g.~\cite{CDS10}, \cite{KPH13}, \cite{KSZ14} and \cite{EWW15}.
On the other hand, in contrast to the robust optimization literature (e.g.~\cite{GS10}, \cite{WKS14}),
our focus is the robustness of objective functionals in optimization, instead of \emph{how to solve} particular optimization problems.
For this, we do not assume specific properties (e.g.~convexity) of the objective functionals.
As such, our setup and methodology are also different from classic ones in the optimization literature.

We make a few immediate observations about Definition \ref{def:1}.
Robustness is a  {joint property} of {$(\rho, X,\mathcal{G}, \mathcal{Z},\pi)$}, and only a {$\pi$-neighbourhood} of $X$ in $\mathcal{Z}$ matters in the definition.
If $\rho$  is  robust at
$X$ relative to $(\mathcal{G}, \mathcal{Z},\pi)$, then  $\rho$  is   also robust at $X$  relative to $(\mathcal{G},   \mathcal{Z}',\pi)$ if   {$X\in \mathcal{Z}'\subset \mathcal{Z}$},
and the same holds true for $(\mathcal{G}, \mathcal{Z}, \pi')$ if $ \pi'$ is a {stronger}  pseudo-metric than $\pi$.
On the other hand, if the optimization problem does not admit a solution, that is, $ \mathcal G_X(\rho) =\varnothing$, then  $\rho$ is not {robust} at $X$.

 \begin{remark}\label{pb rem}Let us comment on the relations between our definition of robustness and related notions in the literature on optimization and model uncertainty. To this end,  we consider the problem of solvency capital calculation of a firm  as set forth in the Basel III/IV and Solvency II agreements. Suppose that $Z$ is the true (but unknown) model and $g_Z\in {\mathcal G_Z(}\rho)$; see \eqref{eq:opt2} with $X=Z$. In   solvency capital calculation, the following quantities have different physical meanings:
\begin{itemize}
\item $\rho(g_X(X))$: the \emph{perceived} risk value  (solvency capital requirement)   optimized for $X$ by the firm;
\item $\rho(g_Z(Z))$: the  \emph{idealistic} risk value optimized for $Z$ as if $Z$   {were} known;\
\item  $\rho(g_X(Z))$:  the \emph{actual} risk value    of the model $Z$, but the optimization is made for $X$.
\end{itemize}
Among the above quantities, the idealistic risk  value $\rho(g_Z(Z))$ represents what would be the best-case if the true model were known. Since the true model is not known, this value is not available  and hence irrelevant for the solvency capital calculation.
Therefore, for solvency risk management purposes, we are interested in
the \emph{solvency gap}
\begin{equation}
~~~~\underbrace{\rho(g_{{X}}({Z}))}_{\scriptsize \mbox{actual risk}} ~-~\underbrace{\rho (g_{{X}}({X}))}_{\scriptsize\mbox{ {perceived} risk}}, \label{eq:insolvency}
\end{equation}
not the
\emph{optimality gap}
\begin{equation}
\underbrace{\rho(g_{ Z}( Z))}_{\scriptsize\mbox{idealistic optimum}}  ~-~\underbrace{\rho (g_{ X}( Z)) }_{\scriptsize\mbox{actual risk}}, \label{eq:optimality}
\end{equation}
nor the \emph{optimality shift} \begin{equation}
\underbrace{\rho(g_{ Z}( Z))}_{\scriptsize\mbox{idealistic optimum}}  ~~-\underbrace{\rho (g_{ X}( X)) .}_{\scriptsize\mbox{{perceived} optimum}}  \label{eq:shift}
\end{equation}
Note that both \eqref{eq:optimality} and \eqref{eq:shift} involve $\rho(g_Z(Z))$ which is not relevant for solvency considerations. In the optimization literature, the continuity of the set mapping $Z\mapsto  {\mathcal G_Z(}\rho)$, as well as that of the function $Z\mapsto \rho(g_Z(Z)) $, is referred to as the problem of stability, i.e.~how do the optimal solutions and the optimality shift \eqref{eq:shift} change when the underlying model changes from $X$ to $Z$; see, e.g., \cite{BS00} and the references therein.
\end{remark}

 {Let us further illustrate our notion of robustness by means of the following two examples.}

\begin{example}Suppose that the model $X$ leads to the unique optimal decision  $g_X(x)=x_0\in \mathbb{R}$, which means fully liquidating this asset or a perfect hedge (such a situation will arise, e.g., in the context of Proposition \ref{prop:es-cm}).
In this case, $g_X$ is a constant function, and hence $Z\mapsto \rho(g_X(Z))$ is a constant mapping, thus always robust according to our definition.
In other words, the solvency gap \eqref{eq:insolvency} will be zero,  no matter what the optimizer for the true risk $Z$ is. Hence,  model uncertainty is irrelevant for the calculation of the solvency capital.
On the other hand,
if the true model $Z$ is not equal to $X$ and liquidating the asset is not optimal for  $Z$, then we have $\rho(g_Z(Z))<x_0$; thus
the optimality shift \eqref{eq:shift} will be strictly negative.
Therefore, the  solvency gap in \eqref{eq:insolvency} is the right notion to look at in this scenario, not   the optimality shift.
\end{example}

 \begin{example}Suppose that
$X$ and $Z$ are similar  {in the sense} that
$g_X(X)$ and $g_Z(Z)$ are identically distributed,  but $g_X(Z)$  {and $g_X(X)$ are} not identically distributed.
In this case, we have $\rho(g_X(X)) = \rho(g_Z(Z))$ for any law-invariant risk measure $\rho$, such as $\mathrm{VaR}_p$ or $\mathrm{ES}_p$,
but  $\rho(g_X(Z))>\rho(g_Z(Z))= \rho(g_X(X))$ since $g_X$ is generally not optimal for $Z$.
  Clearly, the solvency gap \eqref{eq:insolvency}
is strictly positive
and the optimality shift \eqref{eq:shift} vanishes.
In this situation, the regulator is   concerned about the insolvency issue regarding model uncertainty. Indeed,  the true risk value $\rho(g_X(Z))$ is larger than the perceived risk value $\rho(g_X(X))$, which means that the solvency capital   is insufficient. Note however  that  there is no gap between  $\rho(g_Z(Z))$ and $\rho(g_X(X))$.
Therefore, also here,  the solvency gap is the right notion to study rather than the optimality gap.
\end{example}

\begin{remark}
There are some alternative ways to formulate the notion of robustness in Definition \ref{def:1}. For instance, one may use topologies instead of metrics, or use uncertainty on the set of probability measures  instead of that on the set of random vectors.
By choosing appropriate metrics and topologies, these formulations become equivalent.
To keep the paper accessible to risk management practice, we choose to work with metrics on the set of random vectors.
Another alternative formulation would be to require all (instead of one)  $  g_X \in  \mathcal G_X(\rho)$  to satisfy that $Z\mapsto \rho(  g_X (Z))$ is $\pi$-continuous at $Z=X$.
This requirement would be stronger than the current Definition \ref{def:1}.
Since a main message of this paper (Theorem \ref{th:varrob}) is that VaR is generally \emph{not robust},
the current formulation in Definition \ref{def:1} yields a stronger argument, as the requirement for robustness is weaker than the above alternative.
\end{remark}

\section{General relations between {robust}ness and continuity}\label{sec:24}

Clearly, the {robust}ness of $\rho$ relies on both some continuity of $\rho$ on $\mathcal{G}(\mathcal{Z})$ and some continuity of functions in $ \mathcal G_X(\rho)$. In particular, if $ \mathcal G_X(\rho)=\varnothing$, then $\rho$ is not {robust} by definition.
In what follows, we give a few general results where  $ \mathcal G_X(\rho)$ contains a continuous function.
 However, later we will see that in many practical problems, $ \mathcal G_X(\rho)$ does not have any continuous elements for commonly used risk measures such as the Value-at-Risk and the Expected Shortfall; their {robust}ness properties will be investigated in detail  in Sections \ref{sec:4} and \ref{sec:5}.

For a bijection $g\in \mathcal G_n$ and a pseudo-metric space $(\mathcal{Z},\pi)$ of $n$-dimensional random vectors,
let $(g(\mathcal{Z}),\pi_{g})$ be another pseudo-metric space defined as
$$\pi_{g}(g(X),g(Y))= \pi(X,Y)~~ \mbox{for $X,Y\in \mathcal{Z}$.}$$

\begin{proposition}\label{prop:31}
Suppose that for an uncertainty triplet $(\mathcal G,\mathcal{Z},\pi)$, $X\in \mathcal{Z}$ and a compatible objective functional $\rho$, $ \mathcal G_X(\rho)$ contains a bijection $g$, and $\rho$ is $\pi_{g}$-continuous on $g(\mathcal{Z})$. Then
$\rho$ is {{robust}} at
$X$ relative to $(\mathcal{G}, \mathcal{Z},\pi)$.
\end{proposition}

Next we look at the basic settings of $\mathcal{Z}=L^\infty_n$, $\mathcal{Z}=L^q_n$, and $\mathcal{Z}=L^0_n$, equipped with the $L^\infty$ metric $\pi_n^\infty$ in \eqref{eq:linftym}, the $L^q$ metric $\pi_n^q$ in \eqref{eq:lqm}, and the pseudo-metric $\pi_n^W$ in \eqref{eq:weakm}, respectively.

\begin{proposition}\label{prop:32}
Let   $\rho$ be a compatible objective functional for the uncertainty triplet $(\mathcal G,\mathcal{Z}, \pi)$ and $X\in \mathcal{Z}$.
\begin{enumerate}[(i)]
\item Suppose $(\mathcal{Z},\pi)=(L^\infty_n, \pi^\infty_n)$. If
 $ \mathcal G_X(\rho)$ contains a continuous function $g:\mathbb{R}^n\to \mathbb{R}$ and $\rho$ is $\pi_1^\infty$-continuous, then
$\rho$ is {{robust}} at
$X$ relative to $(\mathcal{G}, \mathcal{Z}, \pi)$.
\item Suppose $(\mathcal{Z},\pi)=(L^q_n, \pi^q_n)$, $q\in [1,\infty)$.
If  $ \mathcal G_X(\rho)$ contains a  continuous and linearly growing\footnote{A function $g:\mathbb{R}^n\to \mathbb{R}$ is \emph{linearly growing}
if for some $C>0$, $|g(y)|\le C|y|$  for all  $y\in \mathbb{R}^n$ with $|y|>1$. This property is satisfied by, for instance, Lipschitz-continuous functions. } function $g:\mathbb{R}^n\to \mathbb{R}$ and  $\rho$ is $\pi_1^q$-continuous, then
$\rho$ is {{robust}} at
$X$ relative to $(\mathcal{G}, \mathcal{Z},\pi)$.
\item Suppose $(\mathcal{Z},\pi)=(L^0_n, \pi_n^W)$. If
 $ \mathcal G_X(\rho)$ contains a continuous function $g:\mathbb{R}^n\to \mathbb{R}$ and $\rho$ is $\pi_1^W$-continuous, then
$\rho$ is {{robust}} at
$X$ relative to $(\mathcal{G}, \mathcal{Z}, \pi)$.
\end{enumerate}
\end{proposition}

Proposition \ref{prop:32} provides simple criteria for verifying robustness of some objective functionals based on continuity of the optimizing functions in $\mathcal G_X(\rho)$.
As we shall see in Sections \ref{sec:3}-\ref{sec:5}, for the popular risk measures VaR and ES, such criteria may not be very useful, as typically the optimizing functions lack the corresponding continuity.
More detailed analyses are needed to draw meaningful  conclusions for such objectives.

\section{Representative optimization problems}\label{sec:3}

As mentioned before, the {robust}ness in optimization is a property of a functional, depending on an uncertainty triplet.
Therefore, in order to study the {robust}ness of certain functionals, we need to specify the functional as well as  the optimization problem and the corresponding uncertainty model.

Our main {interest is in the risk measures} Value-at-Risk (VaR)  and    Expected Shortfall (ES).
Here, a positive value of $Y$ represents a loss and a negative value represents a gain.
The VaR at confidence level $p\in(0,1)$ is defined as
$$
 \mathrm{VaR}_p(Y)= \inf\{x\in \mathbb{R}: \mathbb{P}(Y\le x)\ge p\}=F_{Y}^{-1}(p),~~Y\in L^0,
$$
and the ES at confidence level $p\in(0,1)$ is defined as
\begin{equation}\label{ES def}
  \mathrm{ES}_p(Y)=\frac{1}{1-p}\int_p^1 \mathrm{VaR}  _q(Y)\mathrm{d} q,~~Y\in L^0.
\end{equation}
Note that $\mathrm{ES}_p(Y)$ may take the value $\infty$ if $Y$ is not integrable.
In addition, we  write
$$\mathrm{ES}_1(Y)=\mathrm{VaR}_1(Y)= \mathrm{ess\mbox{-}sup}(Y)= \sup\{x\in \mathbb{R}: \mathbb{P}(Y\le x)<1\}.$$

For any $p\in (0,1)$, we summarize  {some well known} robustness properties of $\mathrm{VaR}_p$ and $\mathrm{ES}_p$ below.
\begin{enumerate}
\item[1.)]
$\mathrm{VaR}_p$ is continuous with respect to convergence in distribution  (and hence, (Wasserstein) $L^q$-convergence for $q\in [1,\infty]$) at $X$ if and only if the inverse cdf of $X$ is continuous at $p$; see, e.g., Proposition 7.3.1 in \cite{Shorack}.
\item[2.)]  It follows immediately from 1.) and \eqref{ES def} that $\mathrm{ES}_p$ is continuous with respect to convergence in distribution on every uniformly integrable subset of $ L^1$. In particular, $\mathrm{ES}_p$ is continuous with respect to   $L^q$-convergence for $q\in [1,\infty]$. On the other hand, $\mathrm{ES}_p$ is not continuous with respect to convergence in distribution on any set containing $L^\infty_+$.
\end{enumerate}

We  concentrate on some  {simple} optimization problems, which highlight the fundamental differences between the two risk measures above.
Let $n=1$ and $X$ be a random loss.
Suppose that  random losses can be traded on a financial market with pricing density
 $\gamma$.
 By holding a risky position $g(X)$, one receives the monetary amount $\mathbb{E}[\gamma g(X)]$.
Let  $x_0$ represent  the budget of the investor, then the optimization problem is taken over the   functions $g$ satisfying the budget constraint
 $\mathbb{E}[\gamma g(X)]\ge x_0$.

We consider the following three classic setups of optimization problems in a financial market.
Recall that our optimization problem is to minimize
$ \rho(g(X)) \mbox{~subject to~} g\in \mathcal G,$
for some $\mathcal G\subset \mathcal G_1$.
\begin{enumerate}[(a)]
\item An optimal investment or hedging problem in the case of a complete market:
$$ \mathcal G_{\mathrm{cm}}=\{ g\in \mathcal G_{ 1}: \mathbb{E}[\gamma g (X)]\ge x_0\}.$$
\item The problem of optimal investing or hedging with a no short-selling or over-hedging constraint:
$$ \mathcal G_{\mathrm{ns}}=\{ g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge  x_0,~0\le  g( X) \le X \}.$$
In this problem, we assume $0\le x_0<\mathbb{E}[\gamma X]$ to avoid triviality.
\item The problem of optimal investing or hedging with a bounded constraint: for some $m>0$,
$$ \mathcal G_{\mathrm{bd}}=\{ g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge  x_0,~0\le  g( X) \le m \}.$$
In this problem, we assume $0\le x_0<m$ to avoid triviality.
\end{enumerate}
Note that Problem (c) cannot be treated as a special case of Problem (b) as  $X$ in Problem (b) not only  serves as the constraint but also as the source of randomness in the optimized position $g(X)$.

We make the following  assumptions on $X$,  $\gamma$ and our choice of the uncertainty $(\mathcal Z,\pi)$.

\renewcommand\theassumption{A}
\begin{assumption}[Global assumptions on optimization problems] \label{assm:1}~
\begin{enumerate}
\item $X\in \mathcal Z$, $X\ge 0$ and the distribution   of $X$ is has a positive density on its support.
\item  The pricing  density $\gamma=\gamma(X)>0$ is a continuous and   {strictly} positive function of $X$, $\mathbb{E}[\gamma]=1$, and $\mathbb{E}[\gamma X]<\infty$.
 \item
$(\mathcal Z,\pi)$ is one of the classic choices $(L^\infty, \pi^\infty_1)$, $(L^q, \pi^q_1)$ for $q\in [1,\infty)$, and $(L^0, \pi^W_1)$.
\end{enumerate}
\end{assumption}

The assumptions for $X$ and $\gamma$ are
standard; for instance, they are satisfied by the classic Black-Scholes  model of a single asset with terminal price $X$.
Recall that $X$ represents the source of randomness and hence $\gamma$ is naturally a measurable function of $X$.
We will use the notation $\gamma$ for both the random variable $\gamma(X)$ and the function $\gamma:\mathbb{R}\to \mathbb{R}$.
Assumption \ref{assm:1} will be imposed throughout the rest of the paper,
and we do not specify $X$ and $(\mathcal Z,\pi)$ unless necessary.

\section{{Optimization of  Value-at-Risk}}\label{sec:4}

In this section we  discuss the robustness properties of $\mathrm{VaR}_p$, $p\in (0,1)$, with respect to the three optimizations problems (a)-(c).
From the results obtained, we shall see  that VaR is generally not robust for these optimization problems.
Recall that by the definition of robustness (Definition \ref{def:1}), in order to show such non-robustness,  we have to identify all possible forms of the optimizing positions.
We  present the solutions to the optimization problems in Section \ref{sec:41} and then our conclusions on the robustness properties in Section \ref{sec:42}.

\subsection{Solutions to optimization problems}\label{sec:41}

We first  look at Problem (a) for  $\mathrm{VaR}_p$, that is,
\begin{equation}\label{eq:var-cm}
  \mbox{to minimize~}  \mathrm{VaR}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{{1}}: \mathbb{E}[{\gamma} g (X)]\ge x_0.
\end{equation}
It turns out that a VaR is not {robust} for any choice of $(\mathcal Z,\pi)$ since the optimization problem \eqref{eq:var-cm} does not have a solution.

\begin{proposition} \label{prop:var-cm}
$\mathrm{VaR}_p(X;\mathcal G_{\rm cm})=-\infty$.
 Hence,  Problem \eqref{eq:var-cm} admits no solution.
\end{proposition}

Next, we turn toward solving Problem (b) for $\mathrm{VaR}_p$,   that is,
\begin{equation}\label{eq:var-ns}
  \mbox{to minimize~}  \mathrm{VaR}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge x_0,~0\le  g( X) \le X,
\end{equation}
where  $0\le x_0<\mathbb{E}[\gamma X]$.
The optimization problem \eqref{eq:var-ns} is non-trivial to solve but fortunately it admits explicit solutions as we will see below.
In what follows, let
$$
q=\mathrm{VaR}_p(X;\mathcal G_{\rm ns})=\inf\big\{\mathrm{VaR}_p(g(X)):g\in \mathcal G_{\mathrm{ns}}\big\}.
$$
Obviously, $q\ge 0$ since $g(X)$ is non-negative.

For this problem, an optimal solution always exists (see Lemma \ref{lem:varopt3-ns} in Appendix \ref{app:a2}).
We summarize the solutions in the following proposition.
Two cases are distinguished. If $q=0$,  the risk can be completely hedged under the criterion of  $\mathrm{VaR}_p$.
In this case, the solution of Problem \eqref{eq:var-ns} is generally not unique.
This case is arguably less relevant, since there is no risk left according to $\mathrm{VaR}_p$ after optimization.
In the more interesting case $q>0$, we identify the form of all possible solutions to  Problem \eqref{eq:var-ns}.

Before we proceed, we introduce the  {notion of a} \emph{uniform transform} $U$ of a random variable $Y$, which is a uniform $[0,1]$ random variable such that $F^{-1}_Y(U)=Y$ almost surely, where $F^{-1}_{Y}$ is the quantile function of $Y$, defined
as $F^{-1}_Y(t)=\mathrm{VaR}_t(Y)$, $t\in (0,1)$.  For an atomless probability space, the existence of a uniform transform of any $Y$ is guaranteed. See e.g.~Lemma A.32 of \cite{FS16}.

\begin{proposition}\label{prop:var-ns-sol}
Let $U$ be a uniform transform of $ {(X-q)\gamma}$ on the probability space $(\Omega,\sigma(X), \mathbb{P})$\footnote{Recall our convention of identifying $\gamma$ with the random variable $\gamma(X)$. Due to this convention, the random variable $(X-q)\gamma$ is the same as  $(X-q)\gamma(X)$.}.
\begin{enumerate}[(i)]
\item $q=0$ if and only if $\mathrm{ES}_p(\gamma X)\ge   \frac{x_0}{1-p}$.
\item
If $q=0$,  a solution of Problem \eqref{eq:var-ns} is given by
\begin{equation}\label{eq:varopt1-ns}
 {g_X(X)}=X\mathds{1}_{\{U>p\}}.
\end{equation}
\item If $q>0$,  any solution to Problem \eqref{eq:var-ns}  has  the form
\begin{equation} \label{eq:varopt2-ns}
 {g_X(X)}=X\mathds{1}_{\{U>p \}}  + (X\wedge q) \mathds{1}_{\{U\le p\}},~a.s.
\end{equation}
\end{enumerate}
\end{proposition}

The random variable $U$ in the preceding proposition is defined on the probability space $(\Omega,\sigma(X), \mathbb{P})$. It can therefore be represented as a function of $X$, and so the random variables on the right-hand sides of \eqref{eq:varopt1-ns} and \eqref{eq:varopt2-ns} are indeed functions of $X$. To represent these functions in a more explicit form, we  require  the following (very weak) continuity assumption, which holds in particular if $(X-q)\gamma$ has a continuous distribution, which is in turn the case if $\gamma$ is an increasing function.

\renewcommand\theassumption{V1}

\begin{assumption}\label{assm:3}
$q>0$ and $\mathbb{P}({(X-q)\gamma}\le \mathrm{VaR}_p({(X-q)\gamma}))=p$.
\end{assumption}
Under Assumption \ref{assm:3},  we immediately obtain $\{U>p\}= \{{(X-q)\gamma}>\mathrm{VaR}_p({(X-q)\gamma}) \}$ a.s.; we thus obtain the following corollary.
\begin{corollary}\label{coro:1}
Under Assumption \ref{assm:3},  Problem \eqref{eq:var-ns} admits a $\mathbb{P}$-a.s.~unique solution that is of the form
  \begin{equation} \label{eq:varopt4-ns}
 {g_X(X)}=X\mathds{1}_{\{{(X-q)\gamma}> c \}}  + (X\wedge q) \mathds{1}_{\{  (X-q)\gamma\le  c \}},
\end{equation}
where $c=\mathrm{VaR}_p({(X-q)\gamma})$.
\end{corollary}

\begin{remark}
Using \eqref{eq:varopt4-ns} and  the budget condition $\mathbb{E}[\gamma  {g_X(X)}]=x_0$, one can (numerically) solve for the value of $q$.
If there are multiple values of $q$ satisfying $\mathbb{E}[\gamma  {g_X(X)}]=x_0$, then the smallest value is the optimum.
\end{remark}

Finally, we turn to Problem (c) for $\mathrm{VaR}_p$,  that is,
\begin{equation}\label{eq:var-bd}
  \mbox{to minimize~}  \mathrm{VaR}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge x_0,~0\le  g( X) \le m,
\end{equation}
where  $0\le x_0<m$.
The optimization problem \eqref{eq:var-bd} is studied by \cite{HZ11} and it admits a simple solution, which we briefly present below.
In what follows, similarly to Section \ref{sec:42}, let
$$
q'=\mathrm{VaR}_p(X;\mathcal G_{\rm bd})=\inf\big\{\mathrm{VaR}_p(g(X)):g\in \mathcal G_{\mathrm{bd}}\big\}.
$$
Similarly to Problem (b), we present  solutions of Problem (c) in  the cases $q'=0$ and $q'>0$ separately.
\begin{proposition}\label{prop:var-bd-sol}
Let $U$ be a uniform transform of $\gamma$ on the probability space $(\Omega,\sigma(X), \mathbb{P})$.
\begin{enumerate}[(i)]
\item $q'=0$ if and only if ${m\mathrm{ES}_p(\gamma )}\ge \frac{x_0}{1-p}$.
\item If $q'=0$,  a solution of Problem \eqref{eq:var-bd} is given by
$$ {g_X(X)}=m\mathds{1}_{\{U>p\}}.$$
\item
If $q'>0$,  any solution to Problem \eqref{eq:var-bd}  has  the form
$$
 {g_X(X)}=m \mathds{1}_{\{U>p \}}  + q' \mathds{1}_{\{U\le p\}},~a.s.
$$
\end{enumerate}
\end{proposition}

In order to establish the unique form of the optimizing functions for Problem \eqref{eq:var-bd}, we make the following assumption which is analogous to Assumption \ref{assm:3}.

\renewcommand\theassumption{V2}
\begin{assumption}\label{assm:3p}
$q'>0$ and $\mathbb{P}(\gamma\le \mathrm{VaR}_p(\gamma))=p$.
\end{assumption}

 \begin{corollary}\label{coro:2}
Under Assumption \ref{assm:3p},  Problem \eqref{eq:var-bd} admits a $\mathbb{P}$-a.s.~unique solution that is of the form
  \begin{equation} \label{eq:varopt4-bd}
 {g_X(X)}=m\mathds{1}_{\{\gamma > c \}}  + q' \mathds{1}_{\{\gamma  \le  c \}},
\end{equation}
where $c=\mathrm{VaR}_p(\gamma)$.
\end{corollary}

\subsection{Robustness of VaR in optimization}\label{sec:42}

Now, we have gathered enough ingredients to establish the (non-)robustness of VaR.  In the following theorem, $(\mathcal{Z},\pi)$ is any one of the choices specified in Assumption \ref{assm:1}.
The proof of this theorem involves some additional results which are given in Appendix \ref{app:a2}.

 \begin{theorem} \label{th:varrob} For $p\in (0,1)$ and $X\in \mathcal Z $,
 \begin{enumerate}[(i)]
 \item  $\mathrm{VaR}_p$ is not {robust} at $X$ relative to $(\mathcal G_{\rm cm}, \mathcal Z, \pi)$;
\item
 under Assumption \ref{assm:3},
$\mathrm{VaR}_p$ is not {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{ns}},  \mathcal Z,\pi)$;
\item under Assumption \ref{assm:3p},
$\mathrm{VaR}_p$ is not {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{bd}},  \mathcal Z,\pi)$.
\end{enumerate}
\end{theorem}

Theorem \ref{th:varrob} implies that, for all three settings of optimization problems and all choices of commonly used (pseudo-)metrics, $\mathrm{VaR}_p$ is not {robust}, and this result holds for a general continuously distributed random variable $X$.  As a consequence, $\mathrm{VaR}_p$ has the poorest possible robustness in the context of optimization.
This can already be seen from the form of its optimizing functions: the optimizing functions $  g_X $ always have a jump at a $p$-quantile of $ {g_X(X)}$, which makes it  most vulnerable to model uncertainty.

\section{Optimization of Expected Shortfall}\label{sec:5}

In this section we  discuss the robustness properties of $\mathrm{ES}_p$, $p\in (0,1)$, with respect to the three optimizations problems (a)-(c).
From the results obtained, we shall see  that ES is generally robust for these optimization problems, in sharp contrast to the case of VaR.
Problems (a)-(c)  are technically completely different for the two choices of risk measures.
In order to show robustness of ES,  we only need to specify a specific optimizing function and then show its corresponding continuity.
Similarly to the organization of the previous section, we present the solutions to the optimization problems in Section \ref{sec:51} and then our conclusion on robustness in Section \ref{sec:52}. Recall again that Assumption \ref{assm:1} is imposed throughout this section.

\subsection{Solutions to optimization problems}\label{sec:51}

We first study Problem (a) for $\mathrm{ES}_p$, where $0<p<1$, that is,
\begin{equation}\label{eq:es-cm}
  \mbox{to minimize~}  \mathrm{ES}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{{1}}: \mathbb{E}[{\gamma} g (X)]\ge x_0.
\end{equation}
Our first result states that Problem \eqref{eq:es-cm} either does not have a solution or has a trivial solution.
The following condition on the essential supremum $\gamma$ turns out to be crucial for Problem \eqref{eq:es-cm}.

\renewcommand\theassumption{E1}
\begin{assumption}\label{assm:es1}
$
\mathrm{ess\mbox{-}sup} \gamma\le \frac1{1-p}.
$
\end{assumption}

\begin{proposition}\label{prop:es-cm}
Problem \eqref{eq:es-cm} admits a solution  if and only if Assumption \ref{assm:es1} holds,
and if Assumption \ref{assm:es1}  holds,  a solution is given by $  g_X (\cdot)=x_0$.
\end{proposition}

If Assumption \ref{assm:es1} holds, the solution to Problem \eqref{eq:es-cm} is a constant function $  g_X (\cdot)=x_0$, which corresponds to perfect hedging of the original financial position $X$ (e.g.~through purchase of an insurance cover). Since the risk is perfectly hedged, there is no model uncertainty in the optimized position anymore.

Next, we turn toward solving Problem (b) for $\mathrm{ES}_p$, where $0<p<1$, that is,
\begin{equation}\label{eq:es-ns}
  \mbox{to minimize~}  \mathrm{ES}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge x_0,~0\le  g( X) \le X,
\end{equation}
where  $0\le x_0<\mathbb{E}[\gamma X]$.
 Based on Theorem 8.26 of \cite{FS16}, which is a slight generalization of a result by \cite{Sekine04}, the form of an optimized position is obtained explicitly in the following proposition.

\begin{proposition}\label{Sekine lemma1}There exist constants $c>0$, $r\ge0$, and $\lambda\in[0,1]$ such that the function
\begin{equation}\label{eq:es-ns-1}
  g_X (x)=x\mathds{1}_{\{\gamma(x)>c\}}+(x\wedge r)\mathds{1}_{\{\gamma(x)<c\}}+((1-\lambda)x+\lambda (x\wedge r))\mathds{1}_{\{\gamma(x)=c\}}, ~~x\in \mathbb{R},
\end{equation}
solves Problem \eqref{eq:es-ns}. Moreover, $r$ is a $p$-quantile of $ {g_X(X)}$.
\end{proposition}

Finally, we turn to Problem (c) for $\mathrm{ES}_p$, where $0<p<1$, that is,
\begin{equation}\label{eq:es-bd}
  \mbox{to minimize~}  \mathrm{ES}_p(g(X)) \mbox{~~subject to~~} g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge x_0,~0\le  g( X) \le m,
\end{equation}
where  $0\le x_0<m$. The solution to this problem is similar to the case of Problem \eqref{eq:es-ns}.
\begin{proposition}\label{Sekine lemma2}There exist constants $c>0$, $r\in [0,m]$, and $\lambda\in[r,m]$ such that the function
\begin{equation}\label{eq:es-ns-1}
  g_X (x)=m\mathds{1}_{\{\gamma(x)>c\}}+r \mathds{1}_{\{\gamma(x)<c\}}+ \lambda \mathds{1}_{\{\gamma(x)=c\}}, ~~x\in \mathbb{R},
\end{equation}
solves Problem \eqref{eq:es-bd}. Moreover, $r$ is a $p$-quantile of $ {g_X(X)}$.
\end{proposition}

Let us now comment on some possible re-formulations of our optimization problems \eqref{eq:es-cm}, \eqref{eq:es-ns}, and \eqref{eq:es-bd}.

\begin{remark}\label{rem:F-measpositions}Consider the following general version of problems  \eqref{eq:es-cm}, \eqref{eq:es-ns}, and \eqref{eq:es-bd}, in which   risk profiles of the form $g(X)$ are replaced by general $\mathcal F$-measurable random variables:
\begin{equation}\label{eq:es-ns-alt}
  \mbox{minimize~}  \mathrm{ES}_p(Y) \mbox{ over  $\mathcal F$-measurable $Y$ with} \quad\mathbb{E}[\gamma Y]\ge x_0,~0\le Y\le Z,
\end{equation}
where again $0\le x_0<\mathbb{E}[\gamma X]$ and $Z\in\{m,X,+\infty\}$. Suppose that $Y^*$ is any $\mathcal F$-measurable solution. Since $\text{ES}_p$ is monotone with respect to second-order stochastic dominance, we have $\text{ES}_p(Y^*)\ge \text{ES}_p(\mathbb E[Y^*|X])$. Moreover, since $Y^*$ satisfies the constraints in \eqref{eq:es-ns-alt}, then so does $\mathbb E[Y^*|X]$, due to our Assumption 1.2. Therefore, $\mathbb E[Y^*|X]$ is a solution to our optimization problem \eqref{eq:es-ns-alt}. But every $\sigma(X)$-measurable functional is of the form $g(X)$ for some function $g$, and so there must exist a function $g_X\in\mathcal G_1$ satisfying the constraints in \eqref{eq:es-ns}, such that $\mathbb E[Y^*|X]=g_X(X)$ is a solution of both \eqref{eq:es-ns-alt} and the corresponding problem  \eqref{eq:es-cm}, \eqref{eq:es-ns}, or \eqref{eq:es-bd}. This shows that the problem \eqref{eq:es-ns-alt} is not much more general than our representative optimization problems.
The same argument works for every law-invariant convex risk measure in place of  $\text{ES}_p$. \end{remark}

 \begin{remark}\label{rem:ctfinmarket}
Our representative optimization problems  \eqref{eq:es-cm}, \eqref{eq:es-ns}, and \eqref{eq:es-bd}  connect as follows with  continuous-time optimal hedging problems in a complete market with an arbitrary number of primary assets. For instance, suppose that $S=(S_t)_{t\in[0,T]}$ is a $d$-dimensional semimartingale admitting a unique local martingale measure $\mathbb Q$ with density $\gamma=\mathrm{d}\mathbb Q/\mathrm{d}\mathbb P$ on $\mathcal F=\mathcal F^S_T$. We can interpret $S$ as the discounted price process of $d$ risky securities. The fact that $\mathbb Q$ is unique is equivalent to the completeness of the model. In this context, it is a natural task to minimize, e.g.,  $\text{ES}_p(X-V_T)$, where is a $V_T$ is the discounted time-$T$ value of a self-financing trading strategy satisfying a cost constraint on the initial investment $V_0$ and also certain other constraints such as $0\le V_T\le X$. By martingale arguments, the initial investment satisfies $V_0=\mathbb E[\gamma V_T]$. So, by letting $Y:=X-V_T$, we are in the context of the general optimization problem \eqref{eq:es-ns-alt}. On the other hand, market completeness implies that $S$ has the martingale representation property, and so every feasible functional $Y$ can be represented in the form $X-V_T$, where $V$ is again the value process of some self-financing trading strategy. With Remark \ref{rem:F-measpositions}, we thus conclude that our representative optimization problems comprise  corresponding problems of optimizing over multi-dimensional dynamic hedging strategies as a special case. In this context,    our solution function $g_X$ may be interpreted as the payoff of a digital or gap option due to a jump in its graph,  {which can be replicated through a continuously-adjusted trading strategy  {in the assets $S^1,\dots, S^d$}.}
\end{remark}

\subsection{Robustness of ES in optimization}\label{sec:52}

To establish the robustness of ES in optimization, we make the following regularity assumption, which is  satisfied by essentially all practical models of financial markets with continuous model space.

\renewcommand\theassumption{E2}
\begin{assumption}\label{assm:5}
Either $\gamma$ is a constant, or $\gamma$ is a continuous function and $\gamma(X)$ is continuously distributed.
\end{assumption}

 \begin{theorem} \label{th:esrob} For $p\in (0,1)$ and $X\in \mathcal Z $,
 \begin{enumerate}[(i)]
 \item under Assumption \ref{assm:es1},
$\mathrm{ES}_p$ is {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{cm}},  \mathcal Z,\pi)$;
\item under Assumption \ref{assm:5},
$\mathrm{ES}_p$ is {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{ns}},  \mathcal Z,\pi)$ for $(\mathcal Z,\pi)=(L^q,\pi_1^q)$, $q\in [1,\infty]$;
\item under Assumption \ref{assm:5},
$\mathrm{ES}_p$ is {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{bd}},  \mathcal Z,\pi)$.
\end{enumerate}
\end{theorem}

Comparing Theorems \ref{th:varrob} and \ref{th:esrob}, ES has advantages over VaR in terms of robustness for all the settings of Problems (a)-(c).
In particular, the optimized positions for ES in the settings of Problems (b)-(c) are non-trivial but they have much better continuity than the corresponding optimized positions for VaR.
Note that Assumptions \ref{assm:3}, \ref{assm:3p} and \ref{assm:5} impose some continuity properties {on} the underlying random variables. These are realistic for most  practical models.

\begin{remark}
Generally, $\mathrm{ES}_p$ is not {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{ns}},  \mathcal Z, \pi_1^W)$  for a set $\mathcal Z$ that is not uniformly integrable, and this is due to the fact that $\mathrm{ES}_p$ is not continuous with respect to $\pi_1^W$, as mentioned in Section \ref{sec:3}.
On the other hand, if $\mathcal Z$ is uniformly integrable, then $\mathrm{ES}_p$ is {robust} at $X$ relative to  $(\mathcal{G}_{\mathrm{ns}},  \mathcal Z, \pi_1^W)$     under Assumption \ref{assm:5}, using the same proof as in Theorem \ref{th:esrob}.
\end{remark}

 \section{Implications for regulatory risk measures}\label{sec:7}

In both the banking and the insurance sectors, VaR and ES are competing regulatory risk measures for solvency capital calculation; see, for instance, \cite{BASEL16} from the Basel Committee on Banking Supervision and \cite{IAIS14} from the International Association of Insurance Supervisors.
In this paper, with the new notion of robustness, we see that in each of the optimization problems (a)-(c), VaR is generally not robust whereas ES is. This provides  strong support for the use of ES in optimization problems, in addition to its convexity which is very well recognized in the literature. These results   further support the transition from VaR to ES made by \cite{BASEL16} from a novel   theoretical perspective.

Our observations on the VaR vs.~ES issue can be explained intuitively.
From the results in Section \ref{sec:4}, the VaR optimized positions almost always have a jump at the $p$-quantile level of the position, and the optimized position is a bi-atomic distribution which can be interpreted as a portfolio exhibiting a large loss with a small probability (e.g.~selling a large volume of far out-of-the-money call options).
This reflects the fact that ``VaR does not capture the tail risk" as indicated already by many academics and regulators (see e.g.~\cite{DEGKMRS01}, \cite{EPRWB14}, \cite{EKT15} and \cite{BASEL16}).  If there is model uncertainty around this $p$-quantile level, even if small, it ruins completely the optimality of the position.
This can explain the failure of the investment strategies (based on beliefs in small probabilities of default) of of many of the larger banks before the financial crisis; see, for instance, the report by \cite{ACR10} on this matter.
We note that the optimal positions for VaR and ES are very similar in their forms (e.g. compare the optimal positions in Propositions \ref{prop:var-ns-sol}-\ref{prop:var-bd-sol} and Propositions \ref{Sekine lemma1}-\ref{Sekine lemma2}).
By definition, however, ES does not ignore the values of the tail part of the optimal allocation (in contrast to VaR), and this explains why the corresponding   value of the risk measure is not underestimated.

There are extensive discussions on the robustness of  VaR and ES (although not in the context of optimization of this paper), and it may be fair to keep a balanced view. One important issue on the robustness of ES is the difficulty arising from perturbations  which yield probability distributions that may have infinite first moment;
this is why in our analysis, ES is not robust with respect to optimization when the
perturbation (uncertainty)  to the underlying probability model  is measured by the Prokhorov metric $\pi_P$. Therefore, in the minimization of ES from historical data, one needs to always make suitable integrability assumptions, or otherwise minimizing ES  may be as problematic as the case of VaR. Infinite mean models are not of a purely academic nature in risk management; see for instance \cite{NEC2006} in the context of operational risk and \cite{W09} related to the economics of climate change.
For recent academic discussions on various  issues related to the desirability of VaR and ES in banking and insurance regulation, we refer to  \cite{KP16}, \cite{FZ16}, \cite{ELW18}, \cite{AB18} and the references therein.

Our framework can certainly be applied to other settings such as other risk measures (e.g., distortion risk measures)  or expected utility/loss functions, and other settings of optimization problems.
{However,} due to our main interest in the comparative properties of risk measures, we confine our {present} study to the most practical cases of VaR and ES, and several representative optimization problems.
Certainly, there are  many future directions to explore on robustness issues in optimization.

\section{Is distributionally robust optimization robust?}\label{sec:6a}

We conclude this paper by discussing our notion of robustness in the context of \emph{distributionally robust optimization}.
The results in Sections \ref{sec:4}-\ref{sec:5} show that VaR is generally not robust for the optimization problems (a)-(c).
In a classic setting of distributionally robust optimization (e.g.~\cite{QZ08}, \cite{ZF09}, \cite{BM18}),
the objective functional itself is evaluated under the worst-case value over a set of possible models representing uncertainty.
By taking the worst-case value of the objective, model uncertainty is naturally incorporated into the optimization problem.
One   naturally wonders whether the robustness properties of risk measures would be improved with such an approach.

To formulate this consideration mathematically, let $\rho$ be a compatible objective functional for an uncertainty triplet $(\mathcal{G}, \mathcal{Z},\pi)$ and $X\in \mathcal{Z}$.
We look at the following optimization problem, which  is a   robust version of \eqref{eq:opt1},
\begin{equation}\label{eq:robopt}\mbox{to minimize: } \sup_{\pi({Y},X)\le \epsilon}\rho(g({Y})) \mbox{~~~~~subject to~} g\in \mathcal G,
\end{equation}
where  $\epsilon>0$.
Denote by $\mathcal G_X(\rho, \epsilon)$ the set of functions $g\in \mathcal G$ minimizing \eqref{eq:robopt}.
Clearly, if we allow $\epsilon=0$ in \eqref{eq:robopt}, then $\mathcal G_X(\rho, 0)= \mathcal G_X(\rho)$ and we are back in the setting of Section \ref{sec:2}.
In the problem \eqref{eq:robopt}, an investor is interested in the risk measure value $\rho(g(Z))$ of the risky position $g(Z)$, in which $Z$ is the unknowable true model.
Therefore,
similarly to Definition \ref{def:1}, we say that the objective functional $\rho$ is robust for the setting \eqref{eq:robopt} if there exists $g\in \mathcal G_X(\rho,\epsilon)$  such that
the function $Z\mapsto \rho(g(Z))$ is $\pi$-continuous at $Z=X$.

Unfortunately, the minimax  problem \eqref{eq:robopt} is not easy to solve analytically, even for   the representative settings in Section \ref{sec:3} and in the cases of VaR and ES.
Typically, a linear programming approach has to be applied for such problems.
As $\mathrm{ES}_p$ is already shown to be generally robust  in Section \ref{sec:5},  its distributionally robust version is also generally robust;
we \textcolor{red}{thus} focus on the question of whether $\mathrm{VaR}_p$ becomes more robust in this context.

In order to obtain analytic results, we look at a simple setting of Problem (c). Letting $(\mathcal{G},\mathcal{Z},\pi)=(\mathcal{G}_{\rm bd},L^\infty, \pi^\infty_1)$, we can formulate the problem
\begin{equation}\label{eq:robopt2}\mbox{to minimize: } \sup_{\pi^\infty_1({Y},X)\le \epsilon}\mathrm{VaR}_p(g({Y}))\mbox{~~subject to~~} g\in \mathcal G_{1}: \mathbb{E}[\gamma g( X)]\ge x_0,~0\le  g( X) \le m,
\end{equation}
where  $0\le x_0<m$.
Similarly to Section \ref{sec:4},
let $$q=\inf\left\{ \sup_{\pi^\infty_1({Y},X)\le \epsilon}\mathrm{VaR}_p(g({Y})): g\in \mathcal G_{\rm bd} \right\}.$$
In addition to Assumption \ref{assm:1} in Section \ref{sec:3}, we make the following assumption,
which can be seen as a stronger version of Assumption \ref{assm:3p}.

\renewcommand\theassumption{V3}
\begin{assumption}\label{assm:6}
$q>0$, $1/2\le p<1$, $X$ has a  decreasing density on its support and $\gamma$ is an increasing function of $X$\footnote{The monotonicity of $\gamma$ as a function of $X$ has a simple economic meaning. Recall that $X$ represents the loss of an asset. Hence, Assumption \ref{assm:6} requires that the pricing density is larger when the asset has a larger loss.
This requirement is satisfied by classic equilibrium models in the notion of Arrow-Debreu (\cite{AD54}).}.
\end{assumption}

Fortunately,  with Assumption \ref{assm:6}, we are able to obtain an explicit form of the solution to Problem \eqref{eq:robopt2}, allowing us to compare the corresponding robustness property with the results we obtained in  Section \ref{sec:4}.

\begin{proposition}\label{prop:var-rob-new}
Under Assumption \ref{assm:6},  Problem \eqref{eq:robopt2} admits a solution  of the form
  \begin{equation} \label{eq:varopt6-bd}
 g_X (x)=m\mathds{1}_{\{x > c +\epsilon\}}  + q \mathds{1}_{\{x  \le   c +\epsilon\}}, ~~x\in \mathbb{R}, \mbox{~
where $c=\mathrm{VaR}_{p}(X)$.}
\end{equation}
\end{proposition}

With the solution $ g_X $ in Proposition \ref{prop:var-rob-new}, the continuity of VaR mentioned in Section \ref{sec:3} implies that the mapping $Z\mapsto \mathrm{VaR}_p( g_X (Z))$ is $\pi^\infty_1$-continuous at $Z=X$.
As a consequence, $\mathrm{VaR}_p$ is robust for the setting \eqref{eq:robopt}.
This observation is in sharp contrast with Theorem   \ref{th:varrob}, where we see that $\mathrm{VaR}_p$ is not   robust relative to $(\mathcal G_{\rm bd},L^\infty,\pi^\infty_1)$ under some weak assumptions.
Therefore, at least for  the above illustrative example, the modified optimization problem \eqref{eq:robopt2} improves the robustness of VaR.
It is unclear how this result can be generalized to other optimization problems, as analytic results for \eqref{eq:robopt} are rarely available.

Although $\mathrm{VaR}_p$ becomes robust in the setting \eqref{eq:robopt2}, its optimizing function takes a similar form as in Proposition \ref{prop:var-bd-sol}, that is, a bi-atomic distribution which results in a big loss with small probability.
Note that if $\epsilon$ approaches to zero, then the above solution $ g_X $ converges to the solution in Proposition \ref{prop:var-bd-sol}.
This type of optimizing functions is highly undesirable and is subject to considerable model uncertainty if $\epsilon$ is small; see the discussions in Section \ref{sec:7}.