# Robust optimized certainty equivalents and quantiles for loss positions with
distribution uncertainty111The authors appreciated the support of the NSFC
grant (No. 12171471).

Weiwei Li<EMAIL_ADDRESS>Dejian Tian<EMAIL_ADDRESS>School of
Mathematics, China University of Mining and Technology, Xuzhou, P.R. China

###### Abstract

The paper investigates the robust optimized certainty equivalents and analyzes
the relevant properties of them as risk measures for loss positions with
distribution uncertainty. On this basis, the robust generalized quantiles are
proposed and discussed. The robust expectiles with two specific penalization
functions $\varphi_{1}$ and $\varphi_{2}$ are further considered respectively.
The robust expectiles with $\varphi_{1}$ are proved to be coherent risk
measures, and the dual representation theorems are established. In addition,
the effect of penalization functions on the robust expectiles and its
comparison with expectiles are examined and simulated numerically.

###### keywords:

robust optimized certainty equivalents , robust quantiles , robust expectiles
, distribution uncertainty

††journal: arXiv.org

##  1 Introduction

From a risk regulator’s point of view, Artzner et al. (1997) and Artzner et
al. (1999) first propose the axiomatic definition of coherent risk measures,
which satisfy the following properties: monotonicity, translation invariance,
positive homogeneity and subadditivity. Then, Heath (2000), Frittelli and
Rosazza Gianin (2002), Föllmer and Schied (2002) independently introduce the
convex risk measures, which also need to satisfy the convexity.

Based on the degree of risk aversion of financial agents, Ben-Tal and Teboulle
(1986) put forward the optimized certainty equivalent, which is a decision
theoretic criterion on some utility functions. Ben-Tal and Taboulle (2007)
further examine that the optimized certainty equivalents satisfy the axiomatic
definition of convex risk measures, and obtain their dual representation
theorem. The optimized certainty equivalents consider the extremum problem of
a stochastic nonlinear programming.

Risk measures can also be defined by the optimal solution of the extremum
problem such as the VaR (Value at Risk). VaR, a quantile function of the loss
position, is a simple and reasonable risk measures set by risk regulators for
the banking industry, see Duffie and Pan (1997). VaR has good properties, such
as homogeneity, translation invariance and monotonicity. However, as a risk
measure, VaR is not coherent or convex, and it cannot capture the tail risk
and does not pay attention to the scale of loss. Therefore, scholars have
attempted to construct convex risk measures or coherent risk measures based on
VaR from two perspectives.

The first one perspective is based on the VaR itself. For example, Artzner et
al. (1999) and Delbaen (2002) propose WCE (Worst conditional expectation) and
TCE (Tail conditional expectation). Further, Acerbi (2002), Acerbi and Tasche
(2002), Rockafellar and Uryasev (2002), Tasche (2002), Cherny (2006) and other
scholars introduced Conditional VaR, Expected shortfall, Tail VaR, Average
VaR, Weight VaR, etc. Thus, a more reasonable and effective risk measurement
model system is gradually established. The reader can refer to Chapter 4 of
Föllmer and Schied (2016).

The other perspective is based on the equivalent characterization of VaR.
Koenker and Bassett (1978) propose $VaR_{\alpha}(X)$ is the the solution of
the following minimization problem

$\displaystyle\min_{m\in\mathbb{R}}\left\\{\alpha
E_{P}\left[(X-m)^{+}\right]+(1-\alpha)E_{P}\left[(X-m)^{-}\right]\right\\},\alpha\in(0,1).$

Based on this minimization problem, several quantiles have been introduced by
considering more general loss functions. Newey and Powell (1987) introduce the
expectiles as the minimizers of a piecewise quadratic loss function. Breckling
and Chambers (1988) discuss M-quantiles and Chen (1996) considers the power
loss function respectively. Bellini et al. (2014) study the generalized
quantiles, which is the optimal solution of

$\displaystyle\underset{m\in\mathbb{R}}{\operatorname{min}}\left\\{\alpha
E_{P}\left[l_{1}\left((X-m)^{+}\right)\right]+(1-\alpha)E_{P}\left[l_{2}\left((X-m)^{-}\right)\right]\right\\},$

where $l_{1}$, $l_{2}$ are increasing convex functions, and the only
generalized quantiles that are coherent risk measures are the expectiles with
$\alpha>1/2$. Besides, Mao and Cai (2018) propose new generalized quantiles
based on rank-dependent expected utility (RDEU). The generalized quantiles
have been proved to have important applications in risk measurement and
mathematical finance. For example, the reader can refer to Tadese and Drapeau
(2002), Mao and Yang (2015), Chen and Hu (2019) and Xia, Zou and Hu (2023)
etc.

Whether the optimized certainty equivalents or the generalized quantiles, we
assume that we know the exact distribution of the loss position in a given
probability space. However, the distributions of future losses are uncertain.
Recently, Bartl, Drapeau and Tangpi (2020) consider the distribution
uncertainty problems and propose the robust optimized certainty equivalent,
which is defined as222In fact, given a prior distribution $\mu_{0}$, the
original definition of the robust optimized certainty equivalents by Bartl,
Drapeau and Tangpi (2020) is
$\displaystyle\mathcal{OCE}(l):=\inf_{m\in\mathbb{R}}\left\\{m+\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}l(x-m)\mu(dx)-\varphi\left(d_{c}\left(\mu_{0},\mu\right)\right)\right)\right\\}.$
In our situation, we emphasize the loss position $X$.

$\displaystyle\mathcal{OCE}_{l,\varphi}(X):=\inf_{m\in\mathbb{R}}\left\\{m+\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}l(x-m)\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)\right\\},$
(1)

with $l:\mathbb{R}\to\mathbb{R}$ the loss function, $\mathcal{M}(\mathbb{R})$
the set of probability on $\mathbb{R}$, $\varphi$ being a penalization
function, $d_{c}$ the Wasserstein distance with cost function $c(x,y)$,
$\mu_{X}$ a given priori distribution of loss position $X$. We generally
consider $\mu_{X}$ is likely to the true distribution of future losses.

Motivated by the robust optimized certainty equivalents, for any two functions
$l_{1}$ and $l_{2}$, and $h(x)=\alpha
l_{1}\left(x^{+}\right)+(1-\alpha)l_{2}\left(x^{-}\right)$ with
$\alpha\in(0,1)$, this paper proposes the robust generalized quantiles
$q_{\alpha,\varphi}(X)$ for $X$ which satisfy

$q_{\alpha,\varphi}(X)\in\underset{m\in\mathbb{R}}{\operatorname{argmin}}\hskip
3.00003pt\left\\{\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-m)\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)\right\\}.$

The robust generalized quantiles are the natural generalization of quantiles
for loss positions with distribution uncertainty.

The paper contributes to the literature in the following three aspects. First,
in order to investigate the loss positions with distribution uncertainty, the
definition of the robust optimized certainty equivalents is refined. On this
basis, we can study their properties as risk measures and find that they
satisfy translation invariance, monotonicity and convexity (see Proposition
2.1). Furthermore, Proposition 2.2 analyzes the influences of loss function
and penalization function on the robust optimized certainty equivalents. The
loss function reflects the agent’s risk aversion, while the penalization
function reflects the agent’s robust aversion. The larger loss function
implies the stronger risk aversion, and the larger penalization function
implies the weaker robust aversion, which means the higher the reliability of
the priori distribution. A reachability condition is also given for the robust
optimized certainty equivalents.

Second, we propose the robust generalized quantiles, which incorporate with
the distribution uncertainty of the loss positions. Proposition 2.4 provides
the sufficient conditions for the existence of the robust generalized
quantiles. For any $\alpha\in(0,1)$, when $h(x)=\alpha x^{+}+(1-\alpha)x^{-}$
and the cost function $c(x,y)=|x-y|$, Proposition 2.5 shows that the robust
generalized quantiles degenerates into classical VaR for any penalization
function $\varphi$. It indicates that the robust generalized quantiles
displays the robustness for penalization function in this specification.

Third, we consider two kinds of robust generalized quantiles by introducing
two specific penalization functions $\varphi_{1}(x)=\delta_{1}x$ with
$\delta_{1}>max\\{\alpha,1-\alpha\\}$ and $\varphi_{2}(x)=\infty
I_{(\delta_{2},+\infty)}(x)$ with $\delta_{2}>0$. We call them the robust
expectiles with $\varphi_{1}$ and the robust expectiles with $\varphi_{2}$
respectively. By using dual formula, we transform the problem of solving for
the robust expectiles into the minimum problem of finite dimensions, so as to
further study their properties as risk measures and the impact of penalization
parameters on them. We find that the robust expectiles with $\varphi_{1}$ are
coherent risk measures when $\alpha>1/2$, and we establish the dual
representations theorem (see Theorem 3.1 and Theorem 3.2). Robust expectiles
with $\varphi_{2}$ are also studied. Besides, we also provide the comparisons
between the robust expectiles and the expectiles under some specific prior
distributions.

The paper is organized as follows. Section 2 considers the properties of the
robust optimized certainty equivalents and proposes the definition of the
robust generalized quantiles. Section 3 mainly focuses on two kinds of
specific robust expectiles corresponding to two popular penalization
functions. All the proofs are relegated to Section 4. Section 5 concludes the
paper.

##  2 Robust optimized certainty equivalents and generalized quantiles

In this section, after introducing the robust optimized certainty equivalents,
originally proposed by Bartl, Drapeau and Tangpi (2020), we then investigate
its properties and give the definition of the robust generalized quantiles.

###  2.1 Robust optimized certainty equivalents

Let $(\Omega,\mathcal{F})$ be a measurable space, and there exists a priori
probability measure $P$ on it. Let $X$ be a random variable from
$(\Omega,\mathcal{F})$ to $(\mathbb{R},\mathcal{B}{(\mathbb{R})})$. The priori
distribution or law of $X$ is defined as the probability measure on the line
given by

$\mu_{X}(A):=(P\circ X^{-1})(A):=P(X^{-1}(A)),\qquad\forall
A\in\mathcal{B}{(\mathbb{R})}.$

Then, $(\mathbb{R},\mathcal{B}{(\mathbb{R})},\mu_{X})$ is a probability space,
and $\mu_{X}$ is the priori distribution of $X$. Denotes
$\mathcal{M}(\mathbb{R})$ a set of all probabilities on the
$\mathcal{B}{(\mathbb{R})}$. Then, for any measurable and bounded from below
function $l:\mathbb{R}\to\mathbb{R}$, we can define the robust optimized
certainty equivalents with respect to $X$, i.e.,

$\displaystyle\mathcal{OCE}_{l,\varphi}(X):=\inf_{m\in\mathbb{R}}\left\\{m+\mathcal{E}_{\varphi}(l,X,m)\right\\},$
(2)

where the nonlinear functional $\mathcal{E}_{\varphi}(l,X,m)$ is defined as

$\mathcal{E}_{\varphi}(l,X,m):=\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}l(x-m)\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right),$
(3)

where $\varphi$ is a penalization function, $d_{c}$ is a distance with cost
function $c(\cdot,\cdot)$ such as the Wasserstein distance.

We give some specifications as follows:

  * 1.

A loss function $l$: $\mathbb{R}\to\mathbb{R}$, which is measurable and
bounded from below.

  * 2.

A penalization function $\varphi$: $[0,+\infty]\rightarrow[0,+\infty]$, which
is convex, increasing, lower semicontinuous with $\varphi(0)=0$. $\varphi^{*}$
is the convex conjugate of $\varphi$, that is $\varphi^{*}(y)=\sup_{x\geq
0}(xy-\varphi(x))$. $\varphi$ and $\varphi^{*}$ are not constants.

  * 3.

The cost function $c(x,y)=|x-y|^{p}$ with $p\geq 1$, for all
$x,y\in\mathbb{R}$.

  * 4.

The distance $d_{c}$ between $\mu_{X}$ and $\mu$ : for any
$\mu\in\mathcal{M}(\mathbb{R})$,

$d_{c}\left(\mu_{X},\mu\right):=\inf\left\\{\int_{\mathbb{R}\times\mathbb{R}}c(x,y)\pi(dx,dy):\begin{array}[]{l}\pi\in\mathcal{M}(\mathbb{R}\times\mathbb{R})\text{
such that }\\\ \pi(\cdot\times\mathbb{R})=\mu_{X}\text{ and
}\pi(\mathbb{R}\times\cdot)=\mu.\end{array}\right\\}.$

Unlike Bartl, Drapeau and Tangpi (2020), we emphasize the priori distribution
of random variable $X$ in the definition of robust optimized certainty
equivalent. Here, we treat $\mu_{X}$ as the fixed baseline distribution of the
random loss $X$. The reason why the distance $d_{c}$ is a popular choice to
model the ambiguity distribution is that one has
$d_{c}(\mu_{X},\mu_{n})\rightarrow 0$ if and only if $\mu_{n}$ converges
weakly to $\mu_{X}$ and
$\int_{\mathbb{R}}x^{p}\mu_{n}(dx)\to\int_{\mathbb{R}}x^{p}\mu_{X}(dx)$ for
$p\geq 1$, see Villani (2008). This means that we can use
$\varphi(d_{c}(\cdot,\cdot))$ to penalize those distributions that are far
away from the baseline distribution $\mu_{X}$ accurately. Hence, the
definition of robust optimized certainty equivalent can be used to describe
the distribution uncertainty of the random variables.

Fixing a prior distribution $\mu_{0}$, Bartl, Drapeau and Tangpi (2020) mainly
solve the computational problem. They does not stress the loss position $X$
and not take into account the properties of $\mathcal{OCE}_{l,\varphi}(\cdot)$
and not address the effects for different random variables. Motivated by Ben-
Tal and Teboulle (1986) and Ben-Tal and Taboulle (2007), we consider some
properties for the robust optimized certainty equivalents in this paper.

######  Proposition 2.1.

Let a loss function $l:\mathbb{R}\to\mathbb{R}$ be convex and increasing and
$\varphi$ be a penalization function. Then the following properties hold.

  * (a)

Prior distribution invariance: If $X$ and $Y$ have the same prior distribution
under $P$, then $\mathcal{OCE}_{l,\varphi}(X)=\mathcal{OCE}_{l,\varphi}(Y)$.

  * (b)

Translation invariance:
$\mathcal{OCE}_{l,\varphi}(X+C)=\mathcal{OCE}_{l,\varphi}(X)+C$, for any
$C\in\mathbb{R}$.

  * (c)

Monotonicity: If $X\leq Y$, $P$-a.s., then
$\mathcal{OCE}_{l,\varphi}(X)\leq\mathcal{OCE}_{l,\varphi}(Y)$.

  * (d)

Convexity: For any random variables $X$ and $Y$, and any $t\in(0,1)$, one has

$\mathcal{OCE}_{l,\varphi}(tX+(1-t)Y)\leq
t\mathcal{OCE}_{l,\varphi}(X)+(1-t)\mathcal{OCE}_{l,\varphi}(Y).$

######  Remark 2.1.

Although the robust optimized certainty equivalents
$\mathcal{OCE}_{l,\varphi}(\cdot)$ involve the distribution uncertainty of
random variables, $\mathcal{OCE}_{l,\varphi}(\cdot)$ still may satisfy some
good properties such as monotonicity, translation invariance and convexity. It
is worth noting that $\mathcal{OCE}_{l,\varphi}(\cdot)$ does not necessarily
satisfy the property of preserving constants. For example, taking $p=1$,
$l(x)=1+x^{+}$, for any penalization function $\varphi(\cdot)$, it is easy to
verify that $\mathcal{OCE}_{l,\varphi}(0)=1$.

The following proposition can be easily obtained from the definition of the
robust optimized certainty equivalents, which displays the influences of loss
functions and penalization functions.

######  Proposition 2.2.

Let $X$ be a random variable on the prior probability space
$(\Omega,\mathcal{F},P)$. Given some loss functions $f$, $g$ and $l$, and some
penalization functions $\phi$, $\psi$ and $\varphi$. Then the following
properties hold.

  * (a)

If $f\geq g$, then
$\mathcal{OCE}_{f,\varphi}(X)\geq\mathcal{OCE}_{g,\varphi}(X)$.

  * (b)

If $\phi\geq\psi$, then
$\mathcal{OCE}_{l,\psi}(X)\geq\mathcal{OCE}_{l,\phi}(X)\geq OCE_{l}(X)$, where

$OCE_{l}(X)=\inf_{m\in\mathbb{R}}\left\\{m+E_{P}[l(X-m)]\right\\}.$

Proposition 2.2 indicates that the larger loss function leads to the larger
$\mathcal{OCE}$, while the larger penalization function results in the smaller
$\mathcal{OCE}$. On the other hand, the penalty function reflects the agent’s
trust in the baseline distribution. Hence, the larger the penalty function is,
the closer it is to ${OCE}_{l}(X)$ under the priori distribution $\mu_{X}$,
which means the higher the reliability of the priori distribution.

Given a random variable $X$ and a penalization function $\varphi$. Let
$l:\mathbb{R}\to\mathbb{R}$ be a convex and increasing loss function and
$l^{\lambda c}$ be the $\lambda c$-transform of $l$, defined in Lemma 4.1.
Denote

$\Lambda_{l}(X,\varphi):=\left\\{\lambda\geq
0~{}|~{}\exists~{}m\in\mathbb{R},s.t.,~{}E_{P}[l^{\lambda
c}(X-m)]+\varphi^{*}(\lambda)<+\infty\right\\}.$

If $\Lambda_{l}(X,\varphi)=\emptyset$, by the definition of robust optimized
certainty equivalent and Lemma 4.1, then it implies that
$\mathcal{OCE}_{l,\varphi}(X)\equiv+\infty$. Suppose
$\Lambda_{l}(X,\varphi)\neq\emptyset$, and define a class of loss functions as
follows:

$L(X,\varphi):=\left\\{\begin{array}[]{l}\text{loss function }\\\
l:\mathbb{R}\to\mathbb{R}\end{array}:\begin{array}[]{l}l\text{ is convex and
increasing such that }\\\ \text{for any
}\lambda\in\Lambda_{l}(X,\varphi),l^{\lambda c}(x)\geq l^{\lambda
c}(0)+x\text{ for all }x\in\mathbb{R}.\end{array}\right\\}.$

Then the following result shows that we can find the optional solution in the
support of random variable $X$ when we choose an appropriate loss function and
penalization function.

######  Proposition 2.3.

Let $\varphi$ be a penalization function. Let $X$ be random variable with
compact support $suppX=[x_{min},x_{max}]$, $(-\infty<x_{min}\leq
x_{max}<+\infty)$. Then, for all $l\in L(X,\varphi)$,

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)=\min_{m\in
suppX}\left\\{m+\mathcal{E}_{\varphi}(l,X,m)\right\\}.$

###  2.2 Robust generalized quantiles

As what we have expected, robust optimized certainty equivalents based on the
nonlinear functional $\mathcal{E}_{\varphi}(l,X,m)$ have good properties as
risk measures. Motivated by Bellini et al. (2014), we consider the generalized
quantiles under robust distributions with $\mathcal{E}_{\varphi}(l,X,m)$.

Let $l_{1},l_{2}:[0,+\infty)\rightarrow[0,+\infty)$ be two convex and
increasing loss functions. For any $\alpha\in(0,1)$,

$\displaystyle h(x):=\alpha l_{1}(x^{+})+(1-\alpha)l_{2}(x^{-}),~{}~{}\forall
x\in\mathbb{R}.$ (4)

Then $h$ is a convex loss function. For a random variable $X$ and a
penalization function $\varphi$, now we consider the following minimization
problem

$\displaystyle\pi_{\alpha}(X):=\underset{m\in\mathbb{R}}{\operatorname{inf}}~{}\mathcal{E}_{\varphi}(h,X,m),$

where $\mathcal{E}_{\varphi}(h,X,m)$ is defined by (3). And we call
$q_{\alpha,\varphi}(X)$ the robust generalized quantiles of $X$ if one has

$q_{\alpha,\varphi}(X)\in\underset{m\in\mathbb{R}}{\operatorname{argmin}}\hskip
3.00003pt\mathcal{E}_{\varphi}(h,X,m).$

Now, we give a sufficient condition for the existence of the robust
generalized quantiles.

######  Proposition 2.4.

Let $l_{1},l_{2}:[0,+\infty)\rightarrow[0,+\infty)$ be two convex and
increasing loss functions. For each $\alpha\in(0,1)$, a random variable $X$
and a penalization function $\varphi$, then it follows that

  * (a)

$\mathcal{E}_{\varphi}(h,X,m)$ is convex with respect to $m$, and

$\lim_{m\rightarrow-\infty}\mathcal{E}_{\varphi}(h,X,m)=\lim_{m\rightarrow+\infty}\mathcal{E}_{\varphi}(h,X,m)=+\infty.$

  * (b)

Suppose $\mathcal{E}_{\varphi}(h,X,0)<+\infty$, then there exists a closed
interval $[m_{1},m_{2}]$, such that

$[m_{1},m_{2}]=\underset{m\in\mathbb{R}}{\operatorname{argmin}}\hskip
3.00003pt\mathcal{E}_{\varphi}(h,X,m).$

Compared with generalized quantiles investigated by Bellini et al. (2014),
robust generalized quantiles consider the uncertainty distributions of future
losses, which also leads to an infinite dimension problem of calculation.
Fortunately, we can use the dual formula obtained by Bartl, Drapeau and Tangpi
(2020) to transform it into a finite dimensional convex function to solve the
extremum problem. More specifically, since $h(\cdot)$, defined in (4), is a
loss function, and its $\lambda c$-transform can be written as follows

$h^{\lambda c}(x)=\sup\\{h(y)-\lambda|x-y|^{p}:y\in\mathbb{R}\text{ such that
}h(y)<\infty\\},~{}~{}x\in\mathbb{R}.$

Using the dual formula (Lemma 4.1), we can obtain

$\displaystyle\mathcal{E}_{\varphi}(h,X,m)=\inf_{\lambda\geq
0}\left\\{E_{P}[h^{\lambda c}(X-m)]+\varphi^{*}(\lambda)\right\\}.$ (5)

To avoid of $h^{\lambda c}(\cdot)\equiv+\infty$, the following lemma provides
a sufficient condition by controlling the growth rate of the loss function.

######  Lemma 2.1.

For any loss function $h:\mathbb{R}\to\mathbb{R}$, suppose that there exists a
constant $C\geq 0$ such that for all $x\in\mathbb{R}$, $h(x)\leq
C(1+|x|^{p})$, where $p\geq 1$ is the power order for the cost function
$c(\cdot,\cdot)$. Then, there exists a constant $\lambda^{*}>C$, such that
$h^{\lambda^{*}c}(x)<+\infty$ for all $x\in\mathbb{R}$.

In particular, when $l_{1}(x)=l_{2}(x)=x$, then the corresponding robust
generalized quantiles is called robust VaR. Based on Lemma 2.1, we should
choose the cost function with $p\geq 1$. We find that the robust VaR
degenerates into VaR for any penalization function $\varphi$, when the cost
function $c(x,y)=|x-y|$, which means that the VaR itself has robustness.

######  Proposition 2.5.

Suppose the cost function is $c(x,y)=|x-y|$, the loss functions
$l_{1}(x)=l_{2}(x)=x$ , $h$ is defined by (4) for each $\alpha\in(0,1)$, and
$X\in L^{1}(\Omega,\mathcal{F},P)$. Then, for any penalization function
$\varphi$ and for any $\alpha\in(0,1)$, the robust VaR can be degenerated into
the classical VaR, i.e., $q_{\alpha,\varphi}(X)=VaR_{\alpha}(X)$.

##  3 Robust expectiles

Suppose $X\in L^{2}(\Omega,\mathcal{F},P)$. Choosing
$l_{1}(x)=l_{2}(x)=x^{2}$, for each $\alpha\in(0,1)$, then

$\displaystyle h(x)=\alpha(x^{+})^{2}+(1-\alpha)(x^{-})^{2},~{}~{}\forall
x\in\mathbb{R}.$ (6)

$h$ is a convex loss function. Newey and Powell (1987) have considered the
expectiles for random variables. Bellini et al. (2014) establish the
expectiles with the relationship for the risk measures.

This section, we will propose two kinds of robust generalized expectiles for
random variables with uncertainty distributions by introducing the following
two specific penalization functions $\varphi_{1}$ and $\varphi_{2}$:

  * 1.

$\varphi_{1}(x)=\delta_{1}x$ with $\delta_{1}>max\\{\alpha,1-\alpha\\}$;

  * 2.

$\varphi_{2}(x)=\infty I_{(\delta_{2},+\infty)}(x)$ with $\delta_{2}>0$.

By using dual formula, we transform the problem of solving for the robust
generalized quantiles into the minimum problem of finite dimensions, so as to
further study their properties as risk measures and the impact of penalization
parameters on them.

###  3.1 Robust expectiles with $\varphi_{1}$

This subsection considers the first specific penalization function
$\varphi_{1}(x)=\delta_{1}x$, $x\geq 0$, with
$\delta_{1}>max\\{\alpha,1-\alpha\\}$, which grows linearly with respect to
the distance, and it is named by the robust expectiles with $\varphi_{1}$.

######  Definition 3.1.

Suppose $X\in L^{2}(\Omega,\mathcal{F},P)$. For any $\alpha\in(0,1)$, let $h$
be the loss function defined by (6). The cost function $c(x,y)=|x-y|^{2}$.
Then the robust expectiles with penalization function $\varphi_{1}$ are
defined by:

$\displaystyle
e_{\alpha,\varphi_{1}}(X):=\underset{m\in\mathbb{R}}{\operatorname{argmin}}~{}\mathcal{E}_{\varphi_{1}}(h,X,m).$

######  Remark 3.1.

Lemma 2.1 explains why we choose $c(x,y)=|x-y|^{2}$ when we define the robust
expectiles with $\varphi_{1}$. Using the dual formula, we can easily obtain if
the cost function $c(x,y)=|x-y|$, and we find that $h^{\lambda
c}(x)\equiv+\infty$, which leads to
$\mathcal{E}_{\varphi_{1}}(h,X,m)\equiv+\infty$ with $m\in\mathbb{R}$. It is a
meaningless question. Therefore, for the loss function with $h$, we should
choose the cost function with $p\geq 2$.

The following Proposition 3.1 gives a representation of the robust expectiles
with $\varphi_{1}$. Compared to its definition, this characterization is very
straightforward. By means of the duality theorem, the influence of its
uncertain distribution is described by the penalty parameter $\delta_{1}$ and
its prior distribution under $P$.

######  Proposition 3.1.

Let $\alpha\in(0,1)$, and $e_{\alpha,\varphi_{1}}(X)$ be the robust expectiles
with $\varphi_{1}$ of $X$, Then

$\displaystyle
e_{\alpha,\varphi_{1}}(X)=\underset{m\in\mathbb{R}}{\operatorname{argmin}}~{}g_{1}(X,m,\delta_{1},\alpha),$

where

$g_{1}(X,m,\delta_{1},\alpha)=\frac{\alpha\delta_{1}}{\delta_{1}-\alpha}E_{P}\left[((X-m)^{+})^{2}\right]+\frac{(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}E_{P}\left[((X-m)^{-})^{2}\right].$

Similar to the expectiles in the classical situation, we can also establish
the relationship between robust expectiles with penalization function
$\varphi_{1}$ and risk measures. The following two theorems prove the robust
expectile with $\varphi_{1}$ is a coherent risk measure and give its
representation theorem.

######  Theorem 3.1.

Suppose that $\alpha\in(\frac{1}{2},1)$. Then, the robust expectile
$e_{\alpha,\varphi_{1}}(\cdot)$ is a coherent risk measure on
$L^{2}(\Omega,\mathcal{F},P)$.

In the proof procedure of Theorem 3.1, for any $X$ in
$L^{2}(\Omega,\mathcal{F},P)$, $\alpha\in(0,1)$ and
$\delta_{1}>max\\{\alpha,1-\alpha\\}$, we know that
$g_{1}^{\prime}(X,e_{\alpha,\varphi_{1}}(X),\delta_{1},\alpha)=0$. Let
$\psi:\mathbb{R}\to\mathbb{R}$ be defined by

$\psi(x):=\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}x^{+}-\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}x^{-},~{}~{}x\in\mathbb{R},$

which is increasing and convex with respect to $x$ when $\alpha$ in
$(\frac{1}{2},1)$. Then, in this situation, $e_{\alpha,\varphi_{1}}(X)$ also
satisfies

$E_{P}\left[\psi(X-e_{\alpha,\varphi_{1}}(X))\right]=0.$

Hence, when $\alpha\in(\frac{1}{2},1)$, the robust expectile can be regarded
as a special case of shortfall risk measure which is given by

$\rho(X)=\inf\\{m\in\mathbb{R}\hskip 1.99997pt|\hskip
1.99997ptE_{P}[l(X-m)]\leq x_{0}\\},$

where $l$ is a loss function and $x_{0}$ is a ceiling for expected loss. Then
by the representation theorem for expected shortfall risk measure (see,
Föllmer and Schied (2002) or Föllmer and Schied (2016)), we can obtain the
representation theorem for the robust expectile with $\varphi_{1}$.

######  Theorem 3.2.

For any $X\in L^{\infty}(\Omega,\mathcal{F},P)$, $\alpha\in(0,1)$ and
$\delta_{1}>max\\{\alpha,1-\alpha\\}$, the robust expectile
$e_{\alpha,\varphi_{1}}(X)$ has the following dual representation

$e_{\alpha,\varphi_{1}}(X)=\begin{cases}\underset{\mu\in\mathcal{M}_{1}(P)}{\operatorname{max}}E_{Q}[X],&\text{
if }\alpha\in(\frac{1}{2},1),\\\
\underset{\mu\in\mathcal{M}_{2}(P)}{\operatorname{min}}E_{Q}[X],&\text{ if
}\alpha\in(0,\frac{1}{2}),\\\ \end{cases}$

where

$\mathcal{M}_{1}(P)=\left\\{Q:\begin{array}[]{l}Q\hskip 3.00003ptis\hskip
3.00003ptabsolutely\hskip 3.00003ptcontinuous\hskip 3.00003ptwith\hskip
3.00003ptrespect\hskip 3.00003ptto\hskip 3.00003ptP,\\\ and\hskip
3.00003pt\exists\hskip 1.99997ptt_{0}>0,\hskip 3.00003ptsuch\hskip
3.00003ptthat\hskip
3.00003pt\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}\leq
t_{0}\frac{\mathrm{d}Q}{\mathrm{d}P}\leq\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}.\end{array}\right\\},$

and

$\mathcal{M}_{2}(P)=\left\\{Q:\begin{array}[]{l}Q\hskip 3.00003ptis\hskip
3.00003ptabsolutely\hskip 3.00003ptcontinuous\hskip 3.00003ptwith\hskip
3.00003ptrespect\hskip 3.00003ptto\hskip 3.00003ptP,\\\ and\hskip
3.00003pt\exists\hskip 1.99997ptt_{0}>0,\hskip 3.00003ptsuch\hskip
3.00003ptthat\hskip 3.00003pt\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}\leq
t_{0}\frac{\mathrm{d}Q}{\mathrm{d}P}\leq\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}\end{array}\right\\}.$

Compared with Proposition 8 in Bellini et al. (2014), they also established
the representation theorem for the generalized expectile when the random
variable has no uncertainty distribution. Our Theorem 3.2 provides a
representation of the robust expectile for the random variable with
uncertainty distributions. We can see the penalization parameter $\delta_{1}$
appears explicitly in the set of probability measures $\mathcal{M}_{1}(P)$ or
$\mathcal{M}_{2}(P)$.

Next, we will consider the impact of the penalization function on the robust
expectiles with $\varphi_{1}$ under a specific baseline distribution. For
$X\in L^{2}(\Omega,\mathcal{F},P)$, $\alpha\in(0,1)$ and
$\delta_{1}>max\\{\alpha,1-\alpha\\}$, we know there exists a unique
$m_{1}^{*}$ such that $g_{1}^{\prime}(X,m_{1}^{*},\delta_{1},\alpha)=0,$ i.e.,

$\displaystyle-\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}E_{P}\left[(X-m_{1}^{*})^{+}\right]+\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}E_{P}\left[(X-m_{1}^{*})^{-}\right]=0,$

where $e_{\alpha,\varphi_{1}}(X)=m_{1}^{*}$.

Firstly, it is obvious $m_{1}^{*}\equiv E_{P}[X]$ when $\alpha=\frac{1}{2}$.
We can only consider the situation $\alpha\neq\frac{1}{2}$, because we’re
going to focus on the effect of the penalization parameter $\delta_{1}$. Given
two random variables $X$ and $Y$, suppose the prior distributions of $X$ and
$Y$ are normal distribution and exponential distribution respectively. Now,
for each $\alpha\in(0,1)$, we consider the robust expectiles
$e_{\alpha,\varphi_{1}}(X)$ and $e_{\alpha,\varphi_{1}}(Y)$ for any
penalization parameter $\delta_{1}>\max\\{\alpha,1-\alpha\\}$.

Figure 1 depicts the $e_{\alpha,\varphi_{1}}(\cdot)$ with different
distributions and different distribution parameters. Observing that no matter
under normal distribution or exponential distribution,
$e_{\alpha,\varphi_{1}}(X)$ and $e_{\alpha,\varphi_{1}}(Y)$ have the similar
variation tendency. When $\alpha\in(\frac{1}{2},1)$, then
$e_{\alpha,\varphi_{1}}(X)$ and $e_{\alpha,\varphi_{1}}(Y)$ decrease gradually
with the increase of the penalization parameter $\delta_{1}$, but it is always
greater than the mean value of the random losses $X$ or $Y$ respectively. When
$\alpha\in(0,\frac{1}{2})$, $e_{\alpha,\varphi_{1}}(X)$ and
$e_{\alpha,\varphi_{1}}(Y)$ are increasing gradually with the increase of
penalization parameter $\delta_{1}$, but are always lower than the mean value
of uncertainty losses $X$ or $Y$ respectively. Moreover, for example, for
random loss $X$, it is not difficult to find that with the increase of
$\delta_{1}$, the change of $e_{\alpha,\varphi_{1}}(X)$ gradually slows down
and tends to be near the mean value of $X$. This means that those
distributions deviated far away from the baseline distribution have less
impact on the results. After all, the baseline distribution is the
distribution that the perceptions of financial agents are closer to the true
distribution, so the above results are reasonable.

(a) $e_{\alpha,\varphi_{1}}(X)$ with Normal prior distribution

(b) $e_{\alpha,\varphi_{1}}(Y)$ with Exponential prior distribution

Figure 1: The effect of penalization function $\varphi_{1}$ on
$e_{\alpha,\varphi_{1}}$ under different distributions

###  3.2 Robust expectiles with $\varphi_{2}$

This subsection considers the second specific penalization function
$\varphi_{2}(x)=\infty I_{(\delta_{2},+\infty)}(x)$, $x\geq 0$, with
$\delta_{2}>0$. It is named by the robust expectiles with $\varphi_{2}$. For
any $\mu$ in $\mathcal{M}(\mathbb{R})$,
$\varphi_{2}(d_{c}\left(\mu_{X},\mu\right))=\infty$ if
$d_{c}\left(\mu_{X},\mu\right)>\delta_{2}$, which means the potential
uncertainty distributions $\mu$ of $X$ should satisfy
$d_{c}\left(\mu_{X},\mu\right)\leq\delta_{2}$.

######  Definition 3.2.

Suppose $X\in L^{2}(\Omega,\mathcal{F},P)$. For any $\alpha\in(0,1)$, let $h$
be the loss function defined by (6). The cost function $c(x,y)=|x-y|^{2}$.
Then the robust expectiles with penalization function $\varphi_{2}$ are
defined by:

$\displaystyle
e_{\alpha,\varphi_{2}}(X):=\underset{m\in\mathbb{R}}{\operatorname{argmin}}~{}\mathcal{E}_{\varphi_{2}}(h,X,m).$

######  Proposition 3.2.

Let $\alpha\in(0,1)$, and $e_{\alpha,\varphi_{2}}(X)$ be the robust expectiles
of $X$, Then

$\displaystyle
e_{\alpha,\varphi_{2}}(X)=\underset{m\in\mathbb{R}}{\operatorname{argmin}}\left\\{\inf_{\lambda>max\\{\alpha,1-\alpha\\}}g_{2}(X,m,\lambda,\alpha)\right\\},$

where

$g_{2}(X,m,\lambda,\alpha)=\frac{\alpha\lambda}{\lambda-\alpha}E_{P}\left[((X-m)^{+})^{2}\right]+\frac{(1-\alpha)\lambda}{\lambda-(1-\alpha)}E_{P}\left[((X-m)^{-})^{2}\right]+\delta_{2}\lambda.$

Obviously, for the given random variable $X$ and $\alpha$,
$g_{2}(X,m,\lambda,\alpha)$ is a binary convex function with respect to
$(m,\lambda)$. To obtain the optimal solution when it reaches its extreme
value, we give the partial derivatives of $g_{2}(X,m,\lambda,\alpha)$ with
respect to $m$ and $\lambda$ respectively,

$\displaystyle\frac{\partial
g_{2}(X,m,\lambda,\alpha)}{\partial\lambda}=-\frac{\alpha^{2}}{(\lambda-\alpha)^{2}}E_{P}\left[((X-m)^{+})^{2}\right]-\frac{(1-\alpha)^{2}}{(\lambda-1+\alpha)^{2}}E_{P}\left[((X-m)^{-})^{2}\right]+\delta_{2},$
$\displaystyle\frac{\partial g_{2}(X,m,\lambda,\alpha)}{\partial
m}=-\frac{2\alpha\lambda}{\lambda-\alpha}E_{P}\left[(X-m)^{+}\right]+\frac{2(1-\alpha)\lambda}{\lambda-1+\alpha}E_{P}\left[(X-m)^{-}\right].\hskip
60.00009pt$

It is noting that when $\delta_{2}=0$, one has that

$\frac{\partial g_{2}(X,m,\lambda,\alpha)}{\partial\lambda}\leq 0,$

which implies that $g_{2}(X,m,\lambda,\alpha)$ is decreasing about $\lambda$.
Hence, $g_{2}(X,m,\lambda,\alpha)$ reaches the minimum value at
$\lambda\rightarrow+\infty$, it follows that

$\displaystyle\inf_{\lambda>max\\{\alpha,1-\alpha\\}}~{}g_{2}(X,m,\lambda,\alpha)~{}$
$\displaystyle=\lim_{\lambda\to+\infty}g_{2}(X,m,\lambda,\alpha)$
$\displaystyle=\alpha
E_{P}\left[((X-m)^{+})^{2}\right]+(1-\alpha)E_{P}\left[((X-m)^{-})^{2}\right].$

Hence, the minimization problem reduces to

$\displaystyle\underset{m\in\mathbb{R}}{\operatorname{argmin}}\Big{\\{}\alpha
E_{P}\left[((X-m)^{+})^{2}\right]+(1-\alpha)E_{P}\left[((X-m)^{-})^{2}\right]\Big{\\}},$

which is exactly the expectiles.

######  Remark 3.2.

For $\varphi_{2}(x)=\infty I_{(\delta_{2},+\infty)}(x)$, it means that we
consider only those distributions that satisfy
$d_{c}\left(\mu_{X},\mu\right)\leq\delta_{2}$. If we take $\delta_{2}=0$, it
implies that we only consider the distribution of uncertain future losses to
be deterministic and be $\mu_{X}$. Hence, the robust expectiles lead to the
classical expectiles without uncertainty distributions.

Now, let’s think about the case where $\delta_{2}>0$, one has that

$\lim_{\lambda\rightarrow max\\{\alpha,1-\alpha\\}}\frac{\partial
g_{2}(X,m,\lambda,\alpha)}{\partial\lambda}=-\infty;\hskip
3.00003pt\lim_{\lambda\rightarrow+\infty}\frac{\partial
g_{2}(X,m,\lambda,\alpha)}{\partial\lambda}=\delta_{2}>0.$

Hence, we know that there exist constants $m_{2}^{*}\in\mathbb{R}$ and
$\lambda^{*}>max\\{\alpha,1-\alpha\\}$, such that

$\left\\{\begin{aligned} \frac{\partial
g_{2}(X,m,\lambda,\alpha)}{\partial\lambda}\bigg{|}_{m=m_{2}^{*},~{}\lambda=\lambda^{*}}=0,\\\
\frac{\partial g_{2}(X,m,\lambda,\alpha)}{\partial
m}\bigg{|}_{m=m_{2}^{*},~{}\lambda=\lambda^{*}}=0,\\\ \end{aligned}\right.$

and $m_{2}^{*}$ is the robust expectile $e_{\alpha,\varphi_{2}}(X)$.

In the following, we will investigate the impact of the penalization function
on $e_{\alpha,\varphi_{2}}$ for the specific prior distribution.

Figure 2 depicts the $e_{\alpha,\varphi_{2}}$ with different prior
distributions and different parameters of prior distributions. Contrary to
$e_{\alpha,\varphi_{1}}$, when $\alpha\in(\frac{1}{2},1)$,
$e_{\alpha,\varphi_{2}}$ is increasing with respect to $\delta_{2}$, while if
$\alpha\in(0,\frac{1}{2})$, $e_{\alpha,\varphi_{2}}$ is decreasing with
respect to $\delta_{2}$. Similarly, the degree of changes becomes slower with
the increase of $\delta_{2}$. This is because no matter whether we use the
penalization functions $\varphi_{1}$ or $\varphi_{2}$, it tends to be that
those distributions far away from the baseline distribution should not have a
major impacts on our results.

(a) $e_{\alpha,\varphi_{2}}(X)$ with Normal prior distribution

(b) $e_{\alpha,\varphi_{2}}(Y)$ with Exponential prior distribution

Figure 2: The effect of penalization function $\varphi_{2}$ on
$e_{\alpha,\varphi_{2}}$ under different prior distributions

###  3.3 Comparisons with expectiles

According to Figure 1 and Figure 2, we have analyzed the influence of the
coefficient of the penalization functions on robust expectiles under normal
distribution and exponential distribution. In fact, the change trends of
robust expectiles with respect to $\delta_{1}$ or $\delta_{2}$ under the $T$
distribution are similarly consistent with that under the normal distribution.
This subsection considers the differences between the classical expectiles and
the robust expectiles under $T$ distribution.

Figure 3 indicates the impact on expectiles after introducing nonlinear
expectation with penalty. It reveals that the trends of
$e_{\alpha,\varphi_{2}}$, $e_{\alpha,\varphi_{1}}$ and expectiles with respect
to $\alpha$ are the similar. For $e_{\alpha,\varphi_{1}}$, it is always
smaller than the expectiles when $\alpha\in(0,\frac{1}{2}]$, and it has
already known that $e_{\alpha,\varphi_{1}}$ is increasing with respect
$\delta_{1}$, which implies that the larger $\delta_{1}$ is, the closer
$e_{\alpha,\varphi_{1}}$ is to the expectiles. While $e_{\alpha,\varphi_{1}}$
is always bigger than the expectiles when $\alpha\in[\frac{1}{2},1)$, and
$e_{\alpha,\varphi_{1}}$ is decreasing with respect $\delta_{1}$, which can
also implies that the larger $\delta_{1}$ is, the closer
$e_{\alpha,\varphi_{1}}$ is to the expectiles.

However, it is different for $e_{\alpha,\varphi_{2}}$. We find that the larger
$\delta_{2}$ is, the farther away $e_{\alpha,\varphi_{2}}$ is from the
expectiles. This is because for $e_{\alpha,\varphi_{2}}$, the larger
$\delta_{2}$ means the more distributions are considered, and the more
deviations from the expectiles should be expected.

Figure 3: The comparisons between robust expectiles and expectiles under $T$
prior distribution.

##  4 The Proofs

This section provides the proofs of the previous propositions and theorems. In
the following, we will use the duality theorem obtained by Bartl, Drapeau and
Tangpi (2020).

######  Lemma 4.1.

(Bartl, Drapeau and Tangpi (2020), Theorem 2.7) The following duality theorem
holds, i.e., for any random variable $X$,

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)=\inf_{\lambda\geq
0}\left\\{OCE_{l^{\lambda c}}(X)+\varphi^{*}(\lambda)\right\\},$ (7)

with

$\displaystyle OCE_{l^{\lambda
c}}(X)=\inf_{m\in\mathbb{R}}\left\\{m+\int_{\mathbb{R}}l^{\lambda
c}(x-m)\mu_{X}(dx)\right\\}=\inf_{m\in\mathbb{R}}\left\\{m+E_{P}[l^{\lambda
c}(X-m)]\right\\},$ (8)

where $l$ is a loss function, $\varphi^{*}$ is the convex conjugate of a
penalization function $\varphi$ and $l^{\lambda c}$ is $\lambda c$-transform
for $l$, which is defined by

$l^{\lambda
c}(x):=\sup_{y\in\mathbb{R}}\\{l(y)-\lambda|x-y|^{p}\\},~{}~{}x\in\mathbb{R}.$

######  Lemma 4.2.

Let a loss function $l(\cdot):\mathbb{R}\to\mathbb{R}$ be convex and
increasing. Then for any $\lambda\geq 0$, the $\lambda c$-transform for the
loss function $l$ has the following properties:

  * (a)

$l^{\lambda c}(x)\geq l(x)$ for any $x\in\mathbb{R}$;

  * (b)

$l^{\lambda c}(x)$ is increasing with respect to $x$;

  * (c)

$l^{\lambda c}(x)$ is convex with respect to $(x,\lambda).$

Proof. By substitution of variables, for any $\lambda\geq 0$, the $\lambda
c$-transform can be expressed as

$l^{\lambda
c}(x)=\sup_{y\in\mathbb{R}}\left\\{l(x-y)-\lambda|y|^{p}\right\\},~{}~{}\forall
x\in\mathbb{R}.$

Hence, $(a)$ obviously holds. Since the loss function $l$ is increasing, it
implies that $(b)$ is true.

To the part $(c)$, for any $t\in(0,1)$, $x_{1},x_{2}\in\mathbb{R}$ and
$\lambda_{1},\lambda_{2}\geq 0$, since the loss function $l(\cdot)$ is convex,
it follows that

$\displaystyle l^{(t\lambda_{1}+(1-t)\lambda_{2})c}(tx_{1}+(1-t)x_{2})$
$\displaystyle=~{}$
$\displaystyle\sup_{y\in\mathbb{R}}\left\\{l(tx_{1}+(1-t)x_{2}-y)-(t\lambda_{1}+(1-t)\lambda_{2})|y|^{p}\right\\}$
$\displaystyle\leq~{}$
$\displaystyle\sup_{y\in\mathbb{R}}\\{tl(x_{1}-y)+(1-t)l(x_{2}-y)-(t\lambda_{1}+(1-t)\lambda_{2})|y|^{p}\\}$
$\displaystyle\leq~{}$ $\displaystyle
t\sup_{y\in\mathbb{R}}\\{l(x_{1}-y)-\lambda_{1}|y|^{p}\\}+(1-t)\sup_{y\in\mathbb{R}}\\{l(x_{2}-y)-\lambda_{2}|y|^{p}\\}$
$\displaystyle=~{}$ $\displaystyle
tl^{\lambda_{1}c}(x_{1})+(1-t)l^{\lambda_{2}c}(x_{2}).$

$\Box$

_Proof of Proposition_ 2.1. The prior distribution invariance property $(a)$
is obvious derived from the definition of $\mathcal{OCE}_{l,\varphi}(\cdot)$.

For any random variable $X$, by Lemma 4.1, we obtain that

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)=\inf_{\lambda\geq
0}\left\\{OCE_{l^{\lambda c}}(X)+\varphi^{*}(\lambda)\right\\},$

with $OCE_{l^{\lambda c}}(X)=\inf_{m\in\mathbb{R}}\left\\{m+E_{P}[l^{\lambda
c}(X-m)]\right\\}.$ Therefore, for any $C\in\mathbb{R}$,

$\displaystyle OCE_{l^{\lambda c}}(X+C)$
$\displaystyle=\inf_{m\in\mathbb{R}}\left\\{m+E_{P}[l^{\lambda
c}(X+C-m)]\right\\}$
$\displaystyle=\inf_{m_{1}\in\mathbb{R}}\left\\{m_{1}+C+E_{P}[l^{\lambda
c}(X-m_{1})]\right\\}$ $\displaystyle=OCE_{l^{\lambda c}}(X)+C.$

It means that $(b)$ holds.

If $X\leq Y$,$P$-a.s., since $l$ is increasing, by Lemma 4.2, we know that the
$\lambda c$-transform $l^{\lambda c}(\cdot)$ is increasing, then it implies
that

$OCE_{l^{\lambda c}}(X)\leq OCE_{l^{\lambda c}}(Y).$

Hence, $\mathcal{OCE}_{l,\varphi}(X)\leq\mathcal{OCE}_{l,\varphi}(Y),$ i.e.,
$(c)$ is true.

Since $l(\cdot)$ is convex, Lemma 4.2 leads to $l^{\lambda c}(x)$ is convex
with respect to $(x,\lambda)$. Therefore, for any $t\in(0,1)$, random
variables $X,Y$ and $\lambda_{1},\lambda_{2}\geq 0$,

$\displaystyle\hskip
20.00003ptOCE_{l^{(t\lambda_{1}+(1-t)\lambda_{2})c}}(tX+(1-t)Y)$
$\displaystyle=\inf_{m_{1},m_{2}\in\mathbb{R}}\left\\{tm_{1}+(1-t)m_{2}+E_{P}[l^{(t\lambda_{1}+(1-t)\lambda_{2})c}(tX+(1-t)Y-(tm_{1}+(1-t)m_{2}))]\right\\}$
$\displaystyle\leq
t\inf_{m_{1}\in\mathbb{R}}\left\\{m_{1}+E_{P}[l^{\lambda_{1}c}(X-m_{1})]\right\\}+(1-t)\inf_{m_{2}\in\mathbb{R}}\left\\{m_{2}+E_{P}[l^{\lambda_{1}c}(Y-m_{2})]\right\\}$
$\displaystyle=tOCE_{l^{\lambda_{1}c}}(X)+(1-t)OCE_{l^{\lambda_{2}c}}(Y).$ (9)

On the other hand, since $\varphi^{*}$ is the convex conjugate of the
penalization function $\varphi$, then $\varphi^{*}(\cdot)$ is convex. Combing
the the convexity of OCE in (4), it implies that

$\displaystyle\mathcal{OCE}_{l,\varphi}(tX+(1-t)Y)$
$\displaystyle=\inf_{\lambda\geq 0}\left\\{OCE_{l^{\lambda
c}}(tX+(1-t)Y)+\varphi^{*}(\lambda)\right\\}$
$\displaystyle=\inf_{\lambda_{1},\lambda_{2}\geq
0}\left\\{OCE_{l^{(t\lambda_{1}+(1-t)\lambda_{2})c}}(tX+(1-t)Y)+\varphi^{*}(t\lambda_{1}+(1-t)\lambda_{2})\right\\}$
$\displaystyle\leq t\inf_{\lambda_{1}\geq
0}\left\\{OCE_{l^{\lambda_{1}c}}(X)+\varphi^{*}(\lambda_{1})\right\\}+(1-t)\inf_{\lambda_{2}\geq
0}\left\\{OCE_{l^{\lambda_{2}c}}(Y)+\varphi^{*}(\lambda_{2})\right\\}$
$\displaystyle=t\mathcal{OCE}_{l,\varphi}(X)+(1-t)\mathcal{OCE}_{l,\varphi}(Y).$

Thus, $\mathcal{OCE}_{l,\varphi}(\cdot)$ is convex. The proof is complete.
$\Box$

_Proof of Proposition_ 2.3

By Lemma 4.1, we obtain that

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)=\inf_{\lambda\geq
0}\left\\{OCE_{l^{\lambda c}}(X)+\varphi^{*}(\lambda)\right\\},$

with $OCE_{l^{\lambda c}}(X)=\inf_{m\in\mathbb{R}}\left\\{m+E_{P}[l^{\lambda
c}(X-m)]\right\\}.$

Since $\Lambda_{l}(X,\varphi)\neq\emptyset$, one has that

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)$ $\displaystyle=\inf_{\lambda\geq
0}\left\\{OCE_{l^{\lambda c}}(X)+\varphi^{*}(\lambda)\right\\}$
$\displaystyle=\inf_{\lambda\in\Lambda_{l}(X,\varphi)}\left\\{OCE_{l^{\lambda
c}}(X)+\varphi^{*}(\lambda)\right\\}.$ (10)

For any $\lambda\in\Lambda_{l}(X,\varphi)$, let $F(m)=m+E_{P}[l^{\lambda
c}(X-m)]$. Then we only need to consider those $m$ that make $F(m)<+\infty$.
In this case, since $suppX=[x_{min},x_{max}]$ with $-\infty<x_{min}\leq
x_{max}<+\infty$, by Lebesgue’s dominated convergence theorem, we can freely
interchange integration with one-sided derivation. Hence, one has that

$\displaystyle F^{{}^{\prime}}_{+}(m)=1-E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{+}(X-m)]\textrm{~{}~{}and~{}~{}}F^{{}^{\prime}}_{-}(m)=1-E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{-}(X-m)].$

Since $l$ is convex, which leads to $F(m)$ is convex in $m$. If $m^{*}$ is the
optimal solution of $OCE_{l^{\lambda c}}(X)$, then it should be

$\displaystyle F^{{}^{\prime}}_{-}(m^{*})\leq 0\leq
F^{{}^{\prime}}_{+}(m^{*}).$

Then, one has that

$\displaystyle E_{P}[(l^{\lambda c})^{{}^{\prime}}_{-}(X-m^{*})]\leq 1\leq
E_{P}[(l^{\lambda c})^{{}^{\prime}}_{+}(X-m^{*})].$

Next, we prove there exists $m^{*}\in suppX$. If $m^{*}<x_{min}$, then
$X-m^{*}>X-x_{min}$. Since $l\in L(X,\varphi)$ and $l^{\lambda c}(\cdot)$ is
convex, which leads to $(l^{\lambda c})^{{}^{\prime}}_{+}$ and $(l^{\lambda
c})^{{}^{\prime}}_{-}$ are nondecreasing, we can obtain that

$\displaystyle 1\geq E_{P}[(l^{\lambda c})^{{}^{\prime}}_{-}(X-m^{*})]\geq
E_{P}[(l^{\lambda c})^{{}^{\prime}}_{+}(X-x_{min})]\geq E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{-}(X-x_{min})]\geq(l^{\lambda c})^{{}^{\prime}}_{+}(0).$

Similarly, if $m^{*}>x_{max}$, we can obtain

$\displaystyle(l^{\lambda c})^{{}^{\prime}}_{-}(0)\geq E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{+}(X-x_{max})]\geq E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{-}(X-x_{max})]\geq E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{+}(X-m^{*})]\geq 1.$

Since $l\in L(X,\varphi)$, then for any $\lambda\in\Lambda_{l}(X,\varphi)$,
$l^{\lambda c}(x)\geq l^{\lambda c}(0)+x$ for all $x\in\mathbb{R}$, and
$l^{\lambda c}(\cdot)$ is convex, one has that

$\displaystyle(l^{\lambda c})^{{}^{\prime}}_{+}(0)\geq 1\geq(l^{\lambda
c})^{{}^{\prime}}_{-}(0),$

which leads to that

$\displaystyle E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{+}(X-x_{min})]=E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{-}(X-x_{min})]=1,$

or

$\displaystyle E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{+}(X-x_{max})]=E_{P}[(l^{\lambda
c})^{{}^{\prime}}_{-}(X-x_{max})]=1.$

Hence, $x_{min}$ or $x_{max}$ is the optimal solution of $OCE_{l^{\lambda
c}}(X)$. This is a contradiction.

Therefore, for any $\lambda\in\Lambda_{l}(X,\varphi)$, we have

$\displaystyle OCE_{l^{\lambda c}}(X)=\min_{m\in
suppX}\left\\{m+E_{P}[l^{\lambda c}(X-m)]\right\\}.$

Then, according to (4), one has that

$\displaystyle\mathcal{OCE}_{l,\varphi}(X)$
$\displaystyle=\inf_{\lambda\in\Lambda_{l}(X,\varphi)}\left\\{\min_{m\in
suppX}\left\\{m+E_{P}[l^{\lambda
c}(X-m)]\right\\}+\varphi^{*}(\lambda)\right\\}$
$\displaystyle=\inf_{\lambda\geq 0}\left\\{\min_{m\in
suppX}\left\\{m+E_{P}[l^{\lambda
c}(X-m)]\right\\}+\varphi^{*}(\lambda)\right\\}$ $\displaystyle=\min_{m\in
suppX}\left\\{m+\inf_{\lambda\geq
0}\left\\{\varphi^{*}(\lambda)+E_{P}[l^{\lambda c}(X-m)]\right\\}\right\\}$
$\displaystyle=\min_{m\in
suppX}\left\\{m+\mathcal{E}_{\varphi}(l,X,m)\right\\}.$

$\Box$

_Proof of Proposition_ 2.4

$(a)$ Since $l_{1}$ and $l_{2}$ are convex, then for each $x$, $h(x-m)$ is
convex in $m$. Hence, for each $t\in(0,1)$, for all
$m_{1},m_{2}\in\mathbb{R}$,

$\displaystyle\hskip 20.00003pt\mathcal{E}_{\varphi}(h,X,tm_{1}+(1-t)m_{2}))$
$\displaystyle=\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-(tm_{1}+(1-t)m_{2}))\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)$
$\displaystyle\leq\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(t\int_{\mathbb{R}}h(x-m_{1})\mu(dx)+(1-t)\int_{\mathbb{R}}h(x-m_{2})d\mu-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)$
$\displaystyle\leq
t\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-m_{1})\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)$
$\displaystyle+(1-t)\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-m_{2})\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)$
$\displaystyle=t\mathcal{E}_{\varphi}(h,X,m_{1})+(1-t)\mathcal{E}_{\varphi}(h,X,m_{2}),$

which implies $\mathcal{E}_{\varphi}(h,X,m)$ is convex about $m$.

On the other hand, from the definition of $\mathcal{E}_{\varphi}(h,X,m)$, it
follows that

$\displaystyle\mathcal{E}_{\varphi}(h,X,m)$
$\displaystyle=\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-m)\mu(dx)-\varphi\left(d_{c}\left(\mu_{X},\mu\right)\right)\right)$
$\displaystyle\geq E_{P}[\alpha l_{1}((X-m)^{+})+(1-\alpha)l_{2}((X-m)^{-})].$

Since the monotonicity and convexity properties of the loss functions $l_{1}$
and $l_{2}$, then by the monotone convergence theorem, it derives

$\displaystyle\lim_{m\rightarrow-\infty}E_{P}\left[\alpha
l_{1}((X-m)^{+})+(1-\alpha)l_{2}((X-m)^{-})\right]=+\infty,$

and

$\displaystyle\lim_{m\rightarrow+\infty}E_{P}\left[\alpha
l_{1}((X-m)^{+})+(1-\alpha)l_{2}((X-m)^{-})\right]=+\infty.$

Hence, one has that

$\lim_{m\rightarrow-\infty}\mathcal{E}_{\varphi}(h,X,m)=\lim_{m\rightarrow+\infty}\mathcal{E}_{\varphi}(h,X,m)=+\infty.$

$(b)$ Since $\mathcal{E}_{\varphi}(h,X,0)<+\infty$, then it implies
$\underset{m\in\mathbb{R}}{\operatorname{inf}}~{}\mathcal{E}_{\varphi}(h,X,m)<+\infty$.
Due to the facts that $\mathcal{E}_{\varphi}(h,X,m)$ is convex about $m$ and
$\lim_{m\rightarrow\pm\infty}\mathcal{E}_{\varphi}(h,X,m)=+\infty$, then it is
easy to find that there exists a closed interval $[m_{1},m_{2}]$, such that

$[m_{1},m_{2}]=\underset{m\in\mathbb{R}}{\operatorname{argmin}}~{}\hskip
1.00006pt\mathcal{E}_{\varphi}(h,X,m).$

$\Box$

_Proof of Lemma_ 2.1

Since $h$ has polynomial growth, it means there exists a constant $C\geq 0$
such that for all $x\in\mathbb{R}$, $h(x)\leq C(1+|x|^{p})$, then it implies
that, for each $\lambda\geq 0$ and $x\in\mathbb{R}$,

$\displaystyle\sup_{y\in\mathbb{R}}\\{h(y)-\lambda|x-y|^{p}\\}$
$\displaystyle\leq\sup_{y\in\mathbb{R}}\\{C(1+|y|^{p})-\lambda|x-y|^{p}\\}.$

Obviously, there exists a $\lambda^{*}>C$, such that
$h^{\lambda^{*}c}(x)<+\infty$ for all $x\in\mathbb{R}$. $\Box$

_Proof of Proposition_ 2.5

For any $\alpha\in(0,1)$, $m\in\mathbb{R}$, then

$h(x-m)=\alpha(x-m)^{+}+(1-\alpha)(x-m)^{-},~{}~{}\forall x\in\mathbb{R}.$

It is obvious that $h(\cdot-m)$ is bounded from below. Based on Lemma 4.1, we
can obtain the $\lambda c$-transform of $h(\cdot-m)$. Directly calculations,
it derives that

$h^{\lambda c}(x-m)=\begin{cases}\alpha(x-m)^{+}+(1-\alpha)(x-m)^{-},&\text{
if }\lambda\geq max\\{\alpha,1-\alpha\\},\\\ +\infty,&\text{ if
}\lambda<max\\{\alpha,1-\alpha\\},\end{cases}$

where $\varphi^{*}$ is the convex conjugate for the penalization function
$\phi$. Then,

$\mathcal{E}_{\varphi}(h,X,m)=\inf_{\lambda\geq
max\\{\alpha,1-\alpha\\}}\\{E_{P}\left[\alpha(X-m)^{+}+(1-\alpha)(X-m)^{-}\right]+\varphi^{*}(\lambda)\\}.$

Since $\varphi^{*}(\lambda)$ is increasing with respect to $\lambda$, then

$\displaystyle\mathcal{E}_{\varphi}(h,X,m)=E_{P}\left[\alpha(X-m)^{+}+(1-\alpha)(X-m)^{-}\right]+\varphi^{*}(max\\{\alpha,1-\alpha\\}).$

Hence, it obvious that

$\underset{m\in\mathbb{R}}{\operatorname{argmin}}\left\\{\mathcal{E}_{\varphi}(h,X,m)\right\\}=\underset{m\in\mathbb{R}}{\operatorname{argmin}}\\{\alpha
E_{P}\left[(X-m)^{+}\right]+(1-\alpha)E_{P}\left[(X-m)^{-}\right]\\}.$

By Exercise 4.4.1 in Föllmer and Schied (2016), its optimal solution
$m^{*}\in\underset{m\in\mathbb{R}}{\operatorname{argmin}}\mathcal{E}_{\varphi}(h,X,m)$
satisfies

$P(X<m^{*})\leq\alpha\leq P(X\leq m^{*}).$

It is exactly the $VaR_{\alpha}(X)$ in the classical situation, which means
$q_{\alpha,\varphi}(X)=VaR_{\alpha}(X)$. $\Box$

_Proof of Proposition_ 3.1

For each $\alpha\in(0,1)$ and $m\in\mathbb{R}$, denote

$h(x-m)=\alpha((x-m)^{+})^{2}+(1-\alpha))((x-m)^{-})^{2},\forall
x\in\mathbb{R}.$

Then, the $\lambda c$-transform of loss function $h(\cdot-m)$ can be
calculated as follows and it can be divided into three cases.

Case (i): When $\alpha\in(0,\frac{1}{2})$, then

$h^{\lambda c}(x-m)=\begin{cases}+\infty,&\text{ if }\lambda<1-\alpha,\\\ \\\
\frac{\alpha(1-\alpha)}{1-2\alpha}((x-m)^{+})^{2}+\infty
I_{(-\infty,m)}(x),&\text{ if }\lambda=1-\alpha,\\\ \\\
\frac{\alpha\lambda}{\lambda-\alpha}((x-m)^{+})^{2}+\frac{(1-\alpha)\lambda}{\lambda-(1-\alpha)}((x-m)^{-})^{2},&\text{
if }\lambda>1-\alpha.\end{cases}$

Case (ii): When $\alpha=\frac{1}{2}$, then

$h^{\lambda c}(x-m)=\begin{cases}+\infty,&\text{ if
}\lambda\leq\frac{1}{2},\\\ \\\
\frac{\alpha\lambda}{\lambda-\alpha}((x-m)^{+})^{2}+\frac{(1-\alpha)\lambda}{\lambda-(1-\alpha)}((x-m)^{-})^{2},&\text{
if }\lambda>\frac{1}{2}.\end{cases}$

Case (iii): When $\alpha\in(\frac{1}{2},1)$, then

$h^{\lambda c}(x-m)=\begin{cases}+\infty,&\text{ if }\lambda<\alpha,\\\ \\\
\infty
I_{(m,+\infty)}(x)+\frac{(1-\alpha)\alpha}{2\alpha-1}((x-m)^{-})^{2},&\text{
if }\lambda=\alpha,\\\ \\\
\frac{\alpha\lambda}{\lambda-\alpha}((x-m)^{+})^{2}+\frac{(1-\alpha)\lambda}{\lambda-(1-\alpha)}((x-m)^{-})^{2},&\text{
if }\lambda>\alpha.\end{cases}$

Since $\varphi_{1}(x)=\delta_{1}x$ , one has that
$\varphi_{1}^{*}(\lambda)=\infty I_{(\delta_{1},+\infty)}(\lambda)$.
Therefore, by Lemma 4.1, it follows that

$\displaystyle\mathcal{E}_{\varphi_{1}}(h,X,m)$
$\displaystyle=\sup_{\mu\in\mathcal{M}(\mathbb{R})}\left(\int_{\mathbb{R}}h(x-m)\mu(dx)-\delta_{1}d_{c}\left(\mu_{X},\mu\right)\right)$
$\displaystyle=\inf_{\lambda\geq 0}\left\\{E_{P}[h^{\lambda c}(X-m)]+\infty
I_{(\delta_{1},+\infty)}(\lambda)\right\\}$
$\displaystyle=\inf_{0\leq\lambda\leq\delta_{1}}~{}E_{P}[h^{\lambda c}(X-m)]$
$\displaystyle=\inf_{max\\{\alpha,1-\alpha\\}<\lambda\leq\delta_{1}}g_{1}(X,m,\lambda,\alpha),$

where

$g_{1}(X,m,\lambda,\alpha)=\frac{\alpha\lambda}{\lambda-\alpha}E_{P}\left[((X-m)^{+})^{2}\right]+\frac{(1-\alpha)\lambda}{\lambda-(1-\alpha)}E_{P}\left[((X-m)^{-})^{2}\right].$

It is clear that $g_{1}(X,m,\lambda,\alpha)$ is decreasing with respect to
$\lambda$. Thus, one has that

$\inf_{max\\{\alpha,1-\alpha\\}<\lambda\leq\delta_{1}}g_{1}(X,m,\lambda,\alpha)=g_{1}(X,m,\delta_{1},\alpha).$

$\Box$

_Proof of Theorem_ 3.1. For any $\alpha\in(\frac{1}{2},1)$ and
$\delta_{1}>\alpha$, we verify that $e_{\alpha,\varphi_{1}}(\cdot)$ satisfies
the axioms of the coherent risk measures.

_(i)_ Translation invariance. For any constants $C$ and $m$ in $\mathbb{R}$,
since

$g_{1}(X+C,m,\delta_{1},\alpha)=g_{1}(X,m-C,\delta_{1},\alpha).$

Hence, by the definition of robust expectile, then it implies that

$e_{\alpha,\varphi_{1}}(X+C)=e_{\alpha,\varphi_{1}}(X)+C.$

_(ii)_ Monotonicity. Since $X\in L^{2}(\Omega,\mathcal{F},P)$, then it can be
verified that $g_{1}(X,m,\delta_{1},\alpha)$ is differentiable with respect to
$m$, and

$\displaystyle
g_{1}^{\prime}(X,m,\delta_{1},\alpha)=-\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}E_{P}\left[(X-m)^{+}\right]+\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}E_{P}\left[(X-m)^{-}\right],~{}\forall
m\in\mathbb{R}.$

Since
$e_{\alpha,\varphi_{1}}(X)=\underset{m\in\mathbb{R}}{\operatorname{argmin}}~{}g_{1}(X,m,\delta_{1},\alpha)$
and $g_{1}(X,\cdot,\delta_{1},\alpha)$ is convex, then its optimal value for
$m$ should satisfy

$g_{1}^{\prime}(X,m,\delta_{1},\alpha)=0.$

Since $\delta_{1}>max\\{\alpha,1-\alpha\\}$, one has that

$-\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}<0~{}~{}\textrm{and}~{}~{}\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}>0.$

Then, for the given $\delta_{1},m$ and $\alpha$,
$g_{1}^{\prime}(X,m,\delta_{1},\alpha)$ is decreasing with respect to $X$,
i.e., if $X\leq Y$, $P$-a.s., then $g_{1}^{\prime}(X,m,\delta_{1},\alpha)\geq
g_{1}^{\prime}(Y,m,\delta_{1},\alpha)$.

Therefore, when $X\leq Y$, $P$-a.s., then

$g_{1}^{\prime}(Y,e_{\alpha,\varphi_{1}}(X),\delta_{1},\alpha)\leq
g_{1}^{\prime}(X,e_{\alpha,\varphi_{1}}(X),\delta_{1},\alpha)=0.$

Since $g_{1}^{\prime}(Y,m,\delta_{1},\alpha)$ is increasing with respect to
$m$, and $g_{1}^{\prime}(Y,e_{\alpha,\varphi_{1}}(Y),\delta_{1},\alpha)=0$,
then it implies that

$e_{\alpha,\varphi_{1}}(X)\leq e_{\alpha,\varphi_{1}}(Y).$

_(iii)_ Convexity. Since $\alpha\in(\frac{1}{2},1)$ and
$\delta_{1}>max\\{\alpha,1-\alpha\\}$, then it can derive that

$\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha}\geq\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)},$

which inplies that $g_{1}^{\prime}(X,m,\delta_{1},\alpha)$ is concave with
respect to $(X,m)$. Recall that

$g_{1}^{\prime}(X,e_{\alpha,\varphi_{1}}(X),\delta_{1},\alpha)=0\textrm{ and
}g_{1}^{\prime}(Y,e_{\alpha,\varphi_{1}}(Y),\delta_{1},\alpha)=0.$

For any $t\in(0,1)$, it then implies

$\displaystyle
g_{1}^{\prime}\left(tX+(1-t)Y,te_{\alpha,\varphi_{1}}(X)+(1-t)e_{\alpha,\varphi_{1}}(Y),\delta_{1},\alpha\right)$
$\displaystyle\geq$
$\displaystyle~{}tg_{1}^{\prime}\left(X,e_{\alpha,\varphi_{1}}(X),\delta_{1},\alpha\right)+(1-t)g_{1}^{\prime}\left(Y,e_{\alpha,\varphi_{1}}(Y),\delta_{1},\alpha\right)$
$\displaystyle=$ $\displaystyle 0.$

Since $g_{1}^{\prime}(X,m,\delta_{1},\alpha)$ is increasing with respect to
$m$, and

$g_{1}^{\prime}(tX+(1-t)Y,e_{\alpha,\varphi_{1}}(tX+(1-t)Y),\delta_{1},\alpha)=0.$

Therefore, for the given random variables $X$ and $Y$, for each $t\in(0,1)$,
$\alpha\in(\frac{1}{2},1)$, and $\delta_{1}>max\\{\alpha,1-\alpha\\}$, then

$e_{\alpha,\varphi_{1}}(tX+(1-t)Y)\leq
te_{\alpha,\varphi_{1}}(X)+(1-t)e_{\alpha,\varphi_{1}}(Y).$

_(iv)_ Positive homogeneity. When $t=0$, it is obvious that
$e_{\alpha,\varphi_{1}}(0)=0$. For any $t>0$, one has that

$g_{1}^{\prime}(tX,m,\delta_{1},\alpha)=tg_{1}^{\prime}(X,\frac{m}{t},\delta_{1},\alpha).$

Hence, for any $t\geq 0$, $e_{\alpha,\varphi_{1}}(tX)=t\hskip
1.99997pte_{\alpha,\varphi_{1}}(X)$. $\Box$

_Proof of Theorem_ 3.2

Suppose $\alpha\in(\frac{1}{2},1)$. By Theorem 3.1,
$e_{\alpha,\varphi_{1}}(X)$ is also a convex risk measure. Based on
Proposition 4.113 and Theorem 4.115 in Föllmer and Schied (2016),
$e_{\alpha,\varphi_{1}}(X)$ can be represented as

$e_{\alpha,\varphi_{1}}(X)=\underset{Q\in\mathcal{M}(P_{X})}{\operatorname{max}}\left\\{E_{Q}[X]-\inf_{t>0}\frac{1}{t}E_{P}\left[\psi^{*}(t\frac{\mathrm{d}Q}{\mathrm{d}P})\right]\right\\},$

where

$\mathcal{M}(P)=\left\\{Q:Q\textrm{ is absolutely continuous with respect to
}P\right\\},$

and $\psi^{*}$ is the dual conjugate function of $\psi$ with

$\psi^{*}(x)=\begin{cases}0&\text{
}\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}\leq
x\leq\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha},\\\ +\infty&\text{ }else.\\\
\end{cases}$

Hence, it derives that

$\inf_{t>0}\frac{1}{t}E_{P}[\psi^{*}(t\frac{\mathrm{d}Q}{\mathrm{d}P})]=\begin{cases}0&\exists
t_{0}>0,s.t.,\frac{2(1-\alpha)\delta_{1}}{\delta_{1}-(1-\alpha)}\leq
t_{0}\frac{\mathrm{d}Q}{\mathrm{d}P}\leq\frac{2\alpha\delta_{1}}{\delta_{1}-\alpha},\\\
+\infty&\text{ }else.\\\ \end{cases}$

Thus,
$e_{\alpha,\varphi_{1}}(X)=\underset{\mu\in\mathcal{M}_{1}(P)}{\operatorname{max}}E_{Q}[X]$.

Since $e_{\alpha,\varphi_{1}}(X)=-e_{1-\alpha,\varphi_{1}}(-X)$, it is easy to
obtain the dual representation of $e_{\alpha,\varphi_{1}}(X)$ when
$\alpha\in(0,\frac{1}{2})$. $\Box$

_Proof of Proposition_ 3.2

Since $\varphi_{2}(x)=\infty I_{(\delta_{2},+\infty)}(x)$, then the convex
conjugate function of $\varphi_{2}$ is

$\varphi_{2}^{*}(y)=\delta_{2}y^{+},~{}~{}y\in\mathbb{R}.$

By Lemma 4.1, it implies that

$\displaystyle\mathcal{E}_{2}(h,X,m)$
$\displaystyle=\sup_{\mu\in\mathcal{M}(\mathbb{R}),d_{c}(\mu_{X},\mu)\leq\delta_{2}}\int_{\mathbb{R}}h(x-m)\mu(dx)$
$\displaystyle=\inf_{\lambda\geq 0}\left\\{E_{P}[h^{\lambda
c}(X-m)]+\delta_{2}\lambda\right\\}$
$\displaystyle=\inf_{\lambda>max\\{\alpha,1-\alpha\\}}g_{2}(X,m,\lambda),$

where $h^{\lambda c}$ can be found in Proposition 3.1. We complete the proof.
$\Box$

##  5 The Conclusions

Inspired by Bartl, Drapeau and Tangpi (2020), the paper analyzes the relevant
properties of the robust optimized certainty equivalents as risk measures for
loss positions with distribution uncertainty. Based on the robust optimized
certainty equivalents, we propose the robust generalized quantiles, which is a
natural generalization for the quantiles. Furthermore, we focus on two kinds
of specific robust expectiles corresponding to two penalization functions
$\varphi_{1}$ and $\varphi_{2}$. The robust expectiles with $\varphi_{1}$ are
proved to be coherent risk measures, and the dual representation theorems are
established. The results are a development and complement to Bellini et al.
(2014) and Bartl, Drapeau and Tangpi (2020). Besides, we also study the
influences of penalization functions on the robust expectiles and compare them
with expectiles for some specific prior distributions by numerical
simulations.

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