# Partially Law-Invariant Risk Measures

Yi Shen Zachary Van Oosten Ruodu Wang Department of Statistics and Actuarial
Science, University of Waterloo, Canada. ✉<EMAIL_ADDRESS>author. Department of Statistics and Actuarial Science, University of
Waterloo, Canada. ✉<EMAIL_ADDRESS>of Statistics and
Actuarial Science, University of Waterloo, Canada. ✉<EMAIL_ADDRESS>
###### Abstract

We introduce the concept of partial law invariance, generalizing the concept
of law-invariant risk measures widely used in statistical and financial
applications. This new concept is motivated by practical considerations of
decision making under uncertainty, and thus connecting the literature on
decision theory and that on financial risk management. We fully characterize
partially law-invariant coherent risk measures via a novel representation
formula, which, surprisingly, has little resemblance to the classical formula
for law-invariant coherent risk measures. A notion of strong partial law
invariance is introduced, allowing for a representation formula akin to the
classical one. We propose a few classes of new risk measures, including
partially law-invariant versions of the Expected Shortfall and the entropic
risk measures, and illustrate their applications in risk assessment under
model uncertainty.

Keywords: law invariance, model uncertainty, coherent risk measures, source
theory, Expected Shortfall

##  1 Introduction

Risk measures are popular tools in financial regulation and decision making,
and they have been an active field of study since the seminal paper of Artzner
et al. (1999). Risk measures such as the Value-at-Risk and the Expected
Shortfall (ES, also known as CVaR) are implemented in the regulatory framework
of BCBS (2019), and they are also useful tools in optimization; see Föllmer
and Schied (2016) and McNeil et al. (2015) for general treatments on risk
measures in risk management and finance, and Shapiro et al. (2021) in the
context of optimization.

The most fundamental property that makes risk measures applicable to practice
is law invariance. Law-invariant risk measures evaluate the riskiness of
financial positions using their distributions. This connects the theory of
risk measures to statistics, thus allowing for a description of the
probabilistic structure of financial positions, arguably a basic step in any
quantitative financial analysis. Although law invariance is such a fundamental
concept, it may not be universally desirable in all contexts. Model
misspecification, or model uncertainty, occurs when the assumed probabilistic
model for various financial risks does not accurately represent the true
distribution of these risks. There are many avenues of research that account
for model uncertainty in risk assessment in different ways. One area is
distributionally robust optimization, in which optimization is performed
against a worst-case scenario; see e.g., Ghaoui et al. (2003) and Zhu and
Fukushima (2009). Another area is to study risk measures formulated from
multiple scenarios, see e.g. Delage et al. (2019) and Wang and Ziegel (2021).
Uncertainty is deeply rooted in decision theory (often called ambiguity
there), with influential work pioneered by Schmeidler (1989) and Gilboa and
Schmeidler (1989), followed by e.g., Klibanoff et al. (2005) and Maccheroni et
al. (2006).

The vast majority of the literature, including the pieces of work mentioned
above, either starts from law-invariant decision models and incorporate model
uncertainty on top, or starts from a general decision model without
specifically imposing law invariance. We take a different route in this paper,
and seek a direct way to weaken law invariance to address model uncertainty.
Our idea is motivated by the following observation. When estimating the
distributions of various loss factors, it is natural that different factors
will have different levels of uncertainty, which may be quantified
statistically by, e.g., confidence regions. To account for this discrepancy, a
manager would like to be robust to the factors they are more uncertain of.
With this guiding principle in mind, this paper introduces a new property
called partial law invariance. Fix a source of information, mathematically
modelled by a sub-$\sigma$-algebra $\mathcal{G}$. A risk measure is said to be
partially law invariant if it equates the risks of positions that (a) both
belong to this source (i.e., $\mathcal{G}$-measurable) and (b) share identical
distributions. The idea is that losses from this source have no uncertainty
(or an acceptable level of uncertainty) associated with them. Therefore, when
a loss belongs to this source, the risk assessment model, a risk measure in
our context, only takes into account the distribution of this loss.

This property was inspired by, and closely related to, the recent paper of
Baillon et al. (2023), where the authors formalize the idea of sources. In
that paper, sources are defined as algebras of events, and the main idea is
that uncertainty attitudes may differ between sources. Our paper takes one
fixed source with the argument that no (or little) uncertainty exists on this
source, and this allows us to obtain mathematically tractable representation
results. A similar idea to have a $\sigma$-algebra with certainty in decision
theory appears in the expected uncertainty utility model of Gul and
Pesendorfer (2014).

We summarize the main contributions of the paper below. As the first
contribution, we introduce the concept of partial law invariance as a new and
relevant property for risk measures, with many examples in Section 2 and 3. As
mentioned earlier, the new property is mainly motivated by the need for risk
measures to take into account model uncertainty, but it also naturally appears
in several other contexts. This approach of modeling risk measures can address
the situation where some sources of risks are more uncertain than the others,
a setting that cannot be handled by traditional approaches such as
distributionally robust optimization with Wasserstein uncertainty or
likelihood uncertainty, as these models of uncertainty cannot directly
incorporate $\sigma$-algebras. A main theoretical result, Theorem 1 in Section
4, is a representation for coherent risk measures satisfying partial law
invariance. Although having a clear resemblance of the classic representation
of coherent risk measures of Artzner et al. (1999), Theorem 1 does not pave a
clear path to a representation of the type of Kusuoka (2001) and Frittelli and
Rosazza Gianin (2005), which is the most popular type of representation
results for law-invariant coherent or convex risk measures (see also Shapiro
(2013)). To bridge this gap, through introducing the technical tool called
coherent adjustment, we obtain another alternative representation in Theorem
2, which eventually gives rise to a new representation in Theorem 3 that
generalizes Kusuoka’s. It turns out that such representation requires a
property of strong partial law invariance, another natural generalization of
law invariance, introduced and studied in Section 5. With no surprise to
specialists in risk measures, most results can be smoothly generalized to the
case of convex risk measures, and we present some of them in Section 6. We
will focus on an atomless probability space in the main part of the paper, and
it is further assumed to be standard Borel in Section 5. The analysis in the
case of a finite probability space is presented in the appendices, along with
omitted proofs of some results.

We should distinguish our work from conditional and dynamic risk measures,
which is another area of research where it is common to fix a smaller source
of information; see e.g., Riedel (2004), Detlefsen and Scandolo (2005) and
Filipović et al. (2012). Although we also fix a smaller source of information,
we are in the static setting (i.e., our risk measures are real-valued). It
will be clear from the next sections that our motivation and analysis are
fundamentally different from this area of research, and our results have no
overlap with the results in this literature.

##  2 Partial law invariance

In this section, we define and motivate the concept of partial law invariance.
We first introduce some notation that is used throughout the paper.

For a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a
sub-$\sigma$-algebra $\mathcal{G}\subseteq\mathcal{F}$, we denote by
$L^{\infty}(\mathcal{G})$ (resp. $L^{1}(\mathcal{G})$) the set of essentially
bounded (resp. integrable) random variables on
$(\Omega,\mathcal{G},\mathbb{P}|_{\mathcal{G}})$, and by
$\mathcal{M}(\mathcal{G})$ (resp. $\mathcal{M}_{1}(\mathcal{G})$) the set of
finite signed measures (resp. probability measures) on $(\Omega,\mathcal{G})$
that are absolutely continuous with respect to $\mathbb{P}|_{\mathcal{G}}$. We
write $L^{\infty}=L^{\infty}(\mathcal{F})$, $L^{1}=L^{1}(\mathcal{F})$,
$\mathcal{M}=\mathcal{M}(\mathcal{F})$, and
$\mathcal{M}_{1}=\mathcal{M}_{1}(\mathcal{F})$ for simplicity. For
$\mu\in\mathcal{M}(\mathcal{G})$, let
$\eta_{\mu}=\mathrm{d}\mu/\mathrm{d}(\mathbb{P}|_{\mathcal{G}})\in
L^{1}(\mathcal{G})$, that is, its Radon–Nikodym derivative with respect to
$\mathbb{P}|_{\mathcal{G}}$. Given any $X,Y\in L^{1}$ and
$\mu\in\mathcal{M}_{1}$, denote by $X\overset{\mathrm{d}}{=}_{\mathbb{\mu}}Y$
if $X$ and $Y$ have the same distribution under $\mu$. We write
$\sigma(\cdot)$ for the $\sigma$-algebra generated by either a random
variable, a collection of random variables, or a collection of sets in
$\mathcal{F}$.

A common risk measure that we will encounter is the Expected Shortfall (ES,
also called CVaR), which is the standard risk measure for market risk in the
banking regulatory framework of BCBS (2019). For a given probability measure
$\mu\in\mathcal{M}_{1}$, the ES at level $\alpha\in[0,1)$ is given by

$\mathrm{ES}^{\mu}_{\alpha}(X)=\min_{x\in\mathbb{R}}\left\\{x+\frac{1}{1-\alpha}\mathbb{E}^{\mu}[(X-x)_{+}]\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty}.$

ES can be defined via other ways (e.g., Rockafellar and Uryasev (2002)) and on
larger spaces such as $L^{1}(\Omega,\mathcal{F},\mu)$; see e.g., McNeil et al.
(2015, Chapter 8). For simplicity, we write
$\mathrm{ES}_{\alpha}=\mathrm{ES}_{\alpha}^{\mathbb{P}}$.

###  2.1 Definition and motivating examples

We now present the main property in this paper. Let $\mathcal{X}$ a generic
set of random variables on the $(\Omega,\mathcal{F})$. For a risk measure
$\rho:\mathcal{X}\to\mathbb{R}$, we introduce the following property.

  1. $\mathcal{G}$-law invariance: $\rho(X)=\rho(Y)$ for all $X,Y$ that are $\mathcal{G}$-measurable such that $X\buildrel\mathrm{d}\over{=}_{\mathbb{P}}Y$.

The property $\mathcal{F}$-law invariance is widely studied in the risk
measure literature, which is simply called law invariance. As far as we are
aware, $\mathcal{G}$-law invariance has not been studied in the literature
when $\mathcal{G}\neq\mathcal{F}$. It is easy to see that $\mathcal{F}$-law
invariance is stronger than $\mathcal{G}$-law invariance. For example,
$\mathrm{ES}_{\alpha}$ for $\alpha\in[0,1)$ is $\mathcal{F}$-law invariant,
and hence also $\mathcal{G}$-law invariant. When making a contrast, we will
refer to $\mathcal{G}$-law invariance as _partial law invariance_ and to
$\mathcal{F}$-law invariance as _full law invariance_ , where the
$\sigma$-algebras should be clear from the context.

It is also clear that $\mathcal{G}$-law invariance is strictly weaker than
$\mathcal{F}$-law invariance. For instance, for any $\mu\in\mathcal{M}_{1}$
satisfying $\mu|_{\mathcal{G}}=\mathbb{P}|_{\mathcal{G}}$, the mapping
$X\mapsto\mathbb{E}^{\mu}[X]$ is $\mathcal{G}$-law invariant, but not
$\mathcal{F}$-law invariant unless $\mathcal{G}=\mathcal{F}$. We also say that
a set $\mathcal{X}\subseteq L^{1}$ is $\mathcal{G}$-law invariant if for all
$X\in\mathcal{X}\cap L^{1}(\mathcal{G})$ and
$X\overset{\mathrm{d}}{=}_{\mathbb{P}}Y\in L^{1}(\mathcal{G})$, we have
$Y\in\mathcal{X}$.

The main motivation for partial law invariance, in the absence of full law
invariance, is model uncertainty in risk evaluation, illustrated in Example 1,
as well as ambiguity in decision making, illustrated in Example 2. A second
motivation comes from the analysis of conditional expectations, which appears
naturally in the context of a complete financial market. On these points, we
will elaborate with a few more concrete examples in Sections 2.2–2.4.

######  Example 1 (Model uncertainty).

Suppose a firm’s portfolio manager is tasked with making investment decisions
involving random losses/payoffs in a set $\mathcal{X}$. To assess the risk
associated with investment choices, an analyst provides a probability measure
$\mathbb{P}$ for $(\Omega,\mathcal{F})$, via estimation and modeling, and the
manager uses the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Among
random variables in $\mathcal{X}$, for some $Z$ the manager knows that the
distribution $F_{\mathbb{P},Z}$ is accurate, i.e., there is no (or little, in
practice) uncertainty; for instance, $Z$ may be the outcome of a simulation
with known parameters, a coin toss, or some other losses that the manager is
the most familiar with. For the other random variables $X$, their distribution
$F_{\mathbb{P},X}$ may be the output of an estimation or approximation
procedure, which is subject to a considerable amount of uncertainty; for
instance, the return of a traded stock. The manager needs to decide a risk
measure to use. In this context, law invariance naturally applies when dealing
with positions $g(Z)$ that are functions of $Z$, because the manager is
certain about its distribution. However, for random variables $X$ that are not
$\sigma(Z)$-measurable, even if $X\buildrel\mathrm{d}\over{=}g(Z)$, it is
unreasonable to insist that $X$ and $g(Z)$ are equally risky due to
uncertainty associated with $X$; the same can be said for two general random
variables $X$ and $Y$. In other words, it is natural to require
$\sigma(Z)$-law invariance but not (full) law invariance. This example will be
continued in Section 2.2 below with a concrete risk measure.

######  Example 2 (Matching probability).

The idea of matching probability is used to analyze decisions under ambiguity;
see e.g., Dimmock et al. (2016) and Baillon et al. (2018). We explain a
connection between using matching probability and partial law invariance.
Suppose that an agent has preferences over a set of functions on
$(\Omega,\mathcal{F},\mathbb{P})$, including the indicator functions. The
agent does not have access to $\mathbb{P}$, and therefore events in
$\mathcal{F}$ are associated with ambiguity to this agent. There are some
events, such as coin toss or simulation of a random number, that are not
ambiguous to this agent, and we denote by $\mathcal{G}$ the $\sigma$-algebra
generated by these certain events. For a decision scientist to understand the
ambiguity attitude of the agent, the approach of Dimmock et al. (2016) is to
associate each event $A\in\mathcal{F}$ with a number $p_{A}\in[0,1]$ that
corresponds to the known probability of an equally preferred event. A
questionnaire for the agent would look like “would you prefer getting $\$1$ if
event $A$ happens, or getting $\$1$ with probability $p$”. Assuming certain
continuity of the preference, there exists a unique $p_{A}\in[0,1]$ such that
the agent is indifferent between $\mathds{1}_{A}$ and a payoff with
probability $p_{A}$. This $p_{A}$ is called the matching probability of $A$.
In order for this procedure to be sensible, an implicit assumption is that all
events with the same known probability $p$ are equivalent to the agent. Note
that $p_{A}$ is a numerical representation of the preference of the agent on
$\\{\mathds{1}_{A}:A\in\mathcal{F}\\}$, and this numerical representation
satisfies $\mathcal{G}$-law invariance, but not full law invariance.

######  Remark 1.

There is a large literature on conditional and dynamic risk measures, which
also involve a fixed sub-$\sigma$-algebra $\mathcal{G}$; see Riedel (2004),
Detlefsen and Scandolo (2005), and Filipović et al. (2012). A conditional risk
measures is a mapping $\rho:L^{\infty}\to L^{\infty}(\mathcal{G})$ satisfying
some specific properties. We emphasize that this paper deals with the static
setting, in the sense that our risk measures take real values. As such, our
mathematical results have no overlap with the results in the above literature.
Moreover, the motivation for partial law invariance is completely different
from that for dynamic risk measures, although both involving
sub-$\sigma$-algebras.

Partial law invariance naturally appears in several different concrete
settings, which we discuss below.

###  2.2 Robust risk measures under model uncertainty

Let us continue to make the setting of Example 1 more explicit. Due to the
presence of model uncertainty, and to be robust to model misspecification, the
manager has chosen a collection of probability measures
$\mathcal{Q}\subseteq\mathcal{M}_{1}$ with the property that for all
$\mu\in\mathcal{Q}$, $F_{\mu,Z}=F_{\mathbb{P},Z}$, where $Z$ is the loss the
manager is certain about, and $F_{\mu,Z}$ is cumulative distribution function
(cdf) of $Z$ under $\mu$. For example, the manager could take

$\displaystyle\mathcal{Q}=\left\\{\mu\in\mathcal{M}_{1}\mid
F_{\mu,Z}=F_{\mathbb{P},Z}\text{ and }d(\mu,\mathbb{P})<\epsilon\right\\},$
(1)

where $\epsilon>0$ and $d:\mathcal{M}_{1}\times\mathcal{M}_{1}\to[0,\infty)$
is some measure of divergence, such as a Wasserstein distance or the
Kullback–Leibler divergence.

For $X\in L^{\infty}$, the manager plans to find a way to evaluate the risk of
$X$ under each $\mu\in\mathcal{Q}$. Afterward, the manager can use the worst-
case approach of Gilboa and Schmeidler (1989) to assess the overall risk of
$X$. Following the guidelines of BCBS (2019), for each $\mu\in\mathcal{Q}$,
she uses $\mathrm{ES}_{\alpha}^{\mu}$ at level $\alpha=0.975$ to evaluate the
risk of $X\in L^{\infty}$ under $\mu$. The resulting risk measure is the
worst-case ES (see Zhu and Fukushima (2009) and Wang and Ziegel (2021)), given
by

$\overline{\mathrm{ES}}^{\mathcal{Q}}_{\alpha}(X)=\sup_{\mu\in\mathcal{Q}}\mathrm{ES}_{\alpha}^{\mu}(X),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty}.$

It can easily be verified that the risk measure
$\overline{\mathrm{ES}}^{\mathcal{Q}}_{\alpha}$ with $\mathcal{Q}$ in (1) is
$\sigma(Z)$-law invariant. In fact,

$\displaystyle\overline{\mathrm{ES}}^{\mathcal{Q}}_{\alpha}(X)=\mathrm{ES}_{\alpha}(X)\leavevmode\nobreak\
\mbox{for\leavevmode\nobreak\ }X\in L^{\infty}(\sigma(Z)),$

by noting that for any real function $g$, the distribution of $X=g(Z)$ under
$\mu\in\mathcal{Q}$ is the same as that under $\mathbb{P}$, as for any Borel
$A\subseteq\mathbb{R}$,

$\mu(X\in A)=\mu(g(Z)\in A)=\mu(Z\in g^{-1}(A))=\mathbb{P}(Z\in
g^{-1}(A))=\mathbb{P}(X\in A).$

Therefore, the consideration of robust risk optimization will naturally lead
to partial law invariance.

###  2.3 Scenario-based risk measures

Another common way to assess model uncertainty is through the use of multiple
probability measures, in decision theory this has rooted in the study of
ambiguity since Gilboa and Schmeidler (1989). The concept of law invariant
risk measures under multiple probability measures has been studied by Wang and
Ziegel (2021) under the name of scenario-based risk measures. It turns out
that partial law invariance is closely related to scenario-based risk
meausres.

Take a set $\mathcal{Q}\subseteq\mathcal{M}_{1}$. Wang and Ziegel (2021)
defined the following property.

  1. $\mathcal{Q}$-basedness: $\rho(X)=\rho(Y)$ for all $X,Y\in L^{\infty}$ satisfying $X\buildrel\mathrm{d}\over{=}_{Q}Y$ for all $Q\in\mathcal{Q}$.

It is obvious that law invariance is a special case of both $\mathcal{G}$-law
invariance, by choosing $\mathcal{G}=\mathcal{F}$, and
$\mathcal{Q}$-basedness, by choosing $\mathcal{Q}=\\{\mathbb{P}\\}$. Moreover,
law invariance is stronger in general than both $\mathcal{G}$-law invariance
and $\mathcal{Q}$-basedness (if $\mathbb{P}\in\mathcal{Q}$). What is not so
obvious is how $\mathcal{G}$-law invariance and $\mathcal{Q}$-basedness are
connected, and we will first address this question.

######  Proposition 1.

Suppose that $\rho:L^{\infty}\to\mathbb{R}$ is $\mathcal{Q}$-based. Then,
$\rho$ is $\mathcal{G}$-law invariant for any $\mathcal{G}$ independent of
$\sigma\left(\eta_{\mu}:\mu\in\mathcal{Q}\right)$.

###### Proof.

Let $X,Y\in L^{\infty}(\mathcal{G})$ and
$X\buildrel\mathrm{d}\over{=}_{\mathbb{P}}Y$. For $\mu\in\mathcal{Q}$ and
$x\in\mathbb{R}$ we have

$\displaystyle\mu(X\leq x)=\mathbb{E}\left[\eta_{\mu}\mathds{1}_{\\{X\leq
x\\}}\right]=\mathbb{E}\left[\eta_{\mu}\right]\mathbb{E}\left[\mathds{1}_{\\{X\leq
x\\}}\right]=\mathbb{P}(X\leq x)$

and similarly $\mu(Y\leq x)=\mathbb{P}(Y\leq x)$. Hence, $X$ and $Y$ are
identically distributed under each element of $\mathcal{Q}$. Since $\rho$ is
$\mathcal{Q}$-based, we have $\rho(X)=\rho(Y)$. This implies that $\rho$ is
$\mathcal{G}$-law invariant. ∎

The condition of independence between $\mathcal{G}$ and
$\sigma\left(\eta_{\mu}:\mu\in\mathcal{Q}\right)$ in Proposition 1 can be
safely weakened to $\mathbb{E}[\eta_{\mu}|\mathcal{G}]=1$ for
$\mu\in\mathcal{Q}$, following the same proof.

######  Remark 2.

The notion of conditional non-atomicity proposed by Shen et al. (2019) is
related to Proposition 1. Let $\mathcal{Q}$ be a collection of probability
measures on $(\Omega,\mathcal{F})$ and $\mathcal{M}_{1}(\mathcal{Q})$ be the
set of probability measures dominating each element of $\mathcal{Q}$, which is
assumed to be non-empty (it is always non-empty if $\mathcal{Q}$ is finite or
$\mathcal{Q}\subseteq\mathcal{M}_{1}$). The set $\mathcal{Q}$ is said to be
_conditionally atomless_ if there exist $\nu\in\mathcal{M}_{1}(\mathcal{Q})$
and random variable $X$ such that under $\nu$, $X$ is continuously distributed
and independent of $\sigma(\eta_{\mu}:\mu\in\mathcal{Q})$. As discussed by
Delbaen (2021), an equivalent statement for conditional non-atomicity of
$\mathcal{Q}$ is that there exists an atomless $\sigma$-algebra $\mathcal{G}$
independent of $\sigma(\eta_{\mu}:\mu\in\mathcal{Q})$ for some
$\nu\in\mathcal{M}_{1}(\mathcal{Q})$. Clearly, when
$(\Omega,\mathcal{G},\mathbb{P}|_{\mathcal{G}})$ is atomless, the condition
that $\mathcal{G}$ is independent of $\sigma(\eta_{\mu}:\mu\in\mathcal{Q})$ in
Proposition 1 is sufficient for conditional non-atomicity of $\mathcal{Q}$.

###  2.4 Risk measures for conditional expectations

Next, we discuss another concrete example of partial law invariance. Different
from the presence of model uncertainty discussed above, partial law invariance
appears in this context because of the analysis of conditional expectations on
the coarser $\sigma$-algebra that is built in the model setting.

Consider an arbitrage-free and complete financial market with $0$ interest
rate, time horizon $\mathbb{T}=[0,2]$, information filtration
$(\mathcal{F}_{t})_{t\in\mathbb{T}}$, and a unique martingale measure
$\mathbb{P}$. For $X\in L^{\infty}(\mathcal{F}_{2})$, let
$(X_{t})_{t\in\mathbb{T}}$ be the price of a self-financing portfolio with
$X_{2}=X$, thus replicating $X$. Since the price process of such a portfolio
is a martingale, $X_{t}$ is determined by $X$ via
$X_{t}=\mathbb{E}[X|\mathcal{F}_{t}]$ for $t\in\mathbb{T}$. Given a Borel
function $g:\mathbb{R}\to\mathbb{R}$, consider a European style option that
pays $g(X_{1})$ at time $t=1$. For example, the function $g(x)=(x-K)^{+}$
could be used, which represents a call option with strike price $K$. Via risk-
neutral evaluation under the martingale measure $\mathbb{P}$, the price
$\pi_{g}(X)$ of this option is given by

$\displaystyle\pi_{g}(X)=\mathbb{E}\left[g\left(\mathbb{E}\left[X|\mathcal{F}_{1}\right]\right)\right].$

It is clear that $\pi_{g}$ is $\mathcal{F}_{1}$-law invariant, but not
necessarily $\mathcal{F}_{2}$-law invariant; thus partial law invariance holds
but not full law invariance.

Generalizing the above idea, we consider risk measures $\rho$ of the form

$\displaystyle{\rho}(X)=\tilde{\rho}(\mathbb{E}[X|\mathcal{G}]),\hskip
3.0ptX\in L^{\infty}.$ (2)

where $\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ is a law-invariant
mapping; in the above example, it is $\tilde{\rho}:Y\mapsto\mathbb{E}[g(Y)]$.
Clearly, $\rho$ defined in (2) is $\mathcal{G}$-law invariant because
$\rho(X)=\tilde{\rho}(X)=\tilde{\rho}(Y)=\rho(Y)$ for any $X,Y$ that are
$\mathcal{G}$-measurable and equally distributed. Therefore, (2) yields a
simple way to construct partially law-invariant risk measures. The risk
measures in (2) will be further studied in Propositions 5–6.

##  3 Preliminaries and review of risk measures

In this section, we review coherent risk measures and analyze the simplest
type of partially law-invariant coherent risk measures: those induced by
conditional expectation as in (2).

###  3.1 Notation

From now on, fix the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and
sub-$\sigma$-algebra $\mathcal{G}$. The restriction of $\mathbb{P}$ to
$\mathcal{G}$ will also be denoted by $\mathbb{P}$ for simplicity. Assume that
$(\Omega,\mathcal{G},\mathbb{P})$ is atomless, that is,
$\\{\mathbb{P}(A):A\subseteq B,\leavevmode\nobreak\
A\in\mathcal{G}\\}=[0,\mathbb{P}(B)]$ for all $B\in\mathcal{G}$. For $X,Y\in
L^{1}$, the inequality $X\leq Y$ means $X\leq Y$ $\mathbb{P}$-a.s. For
notational simplicity, when $\mu\in\mathcal{M}_{1}(\mathcal{G})$ we define
$\mathbb{E}^{\mu}$ on $L^{\infty}$ by
$\mathbb{E}^{\mu}[X]=\mathbb{E}[\eta_{\mu}X]$. For
$\mathcal{S}\subseteq\mathcal{M}(\mathcal{G})$, we write
$\widehat{\mathcal{S}}=\\{\eta_{\mu}:\mu\in\mathcal{S}\\}\subseteq
L^{1}(\mathcal{G})$. A set $\mathcal{S}\subseteq\mathcal{M}(\mathcal{G})$ is
_$\mathcal{G}$ -law invariant_ if the set $\widehat{\mathcal{S}}$ is
$\mathcal{G}$-law invariant.

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be two vector spaces. Given a dual
pairing $\left(\mathfrak{X},\mathfrak{Y},\langle\;,\;\rangle\right)$ let
$w(\mathfrak{X},\mathfrak{Y})$ denote the weak topology on $\mathfrak{X}$.
This makes $\left(\mathfrak{X},w(\mathfrak{X},\mathfrak{Y})\right)$ a locally
convex (Hausdorff) topological vector space. Given some
$\mathcal{R}\subseteq\mathfrak{X}$, let $\operatorname*{conv}(\mathcal{R})$
denote the convex hull of $\mathcal{R}$, and we say that $\mathcal{R}$ is
$w(\mathfrak{X},\mathfrak{Y})$-precompact if the
$w(\mathfrak{X},\mathfrak{Y})$-closure of $\mathcal{R}$ is
$w(\mathfrak{X},\mathfrak{Y})$-compact.

It is well known that
$\left(L^{\infty},\mathcal{M},\langle\;,\;\rangle\right)$ is a dual pairing
under $\langle
X,\mu\rangle=\mathbb{E}^{\mu}[X]=\mathbb{E}\left[\eta_{\mu}X\right].$ It is
also well known that $\left(L^{\infty},L^{1},\langle\;,\;\rangle\right)$ is a
dual pairing under $\langle X,Y\rangle=\mathbb{E}[XY].$ Indeed,
$w\left(L^{\infty},\mathcal{M}\right)$ and $w\left(L^{\infty},L^{1}\right)$
coincide since
$\left(\mathcal{M},w\left(\mathcal{M},L^{\infty}\right)\right)\cong\left(L^{1},w\left(L^{1},L^{\infty}\right)\right)$
through the mapping $\mu\mapsto\eta_{\mu}$.

###  3.2 Review of coherent risk measures

We now review coherent risk measures. The interpretation taken in this paper
is that $X\in L^{\infty}$ represents a loss/profit of some financial position
for various states of the world $\omega\in\Omega$. A _coherent risk measure_
(Artzner et al. (1999)) is a mapping $\rho:L^{\infty}\to\mathbb{R}$ satisfying
the following properties.

  * Cash invariance: $\rho(X+c)=\rho(X)+c$ for all $X\in L^{\infty}$ and $c\in\mathbb{R}$;

  * Monotonicity: $\rho(X)\leq\rho(Y)$ for all $X,Y\in L^{\infty}$ with $X\leq Y$;

  * Convexity: $\rho(\lambda X+(1-\lambda)Y)\leq\lambda\rho(X)+(1-\lambda)\rho(Y)$ for all $X,Y\in L^{\infty}$ and $\lambda\in[0,1]$;

  * Positive homogeneity: $\rho(\lambda X)=\lambda\rho(X)$ for all $X\in L^{\infty}$ and $\lambda\geq 0$.

Moreover, a _convex risk measure_ is a mapping that satisfies the first three
properties in the above list, and a _monetary risk measure_ is a mapping that
satisfies the first two properties in the above list; see Frittelli and
Rosazza Gianin (2002) and Föllmer and Schied (2002).

For $\rho:L^{\infty}\to\mathbb{R}$, we say that $\rho$ is _Fatou (continuous)_
if for all uniformly bounded sequences $(X_{n})_{n\in\mathbb{N}}$ in
$L^{\infty}$ that converge ($\mathbb{P}$-a.s.) to some $X\in L^{\infty}$, it
holds that

$\displaystyle\rho(X)\leq\liminf_{n\to\infty}\rho(X_{n}).$

An example of a Fatou coherent risk measure is $\mathrm{ES}_{\alpha}^{\mu}$
for $\mu\in\mathcal{M}_{1}$ and $\alpha\in[0,1)$.

Some well-known facts are collected below. A coherent risk measure $\rho$ is
Fatou if and only if $\rho$ is $w(L^{\infty},\mathcal{M})$-lower
semicontinuous, a proof of which can be found in Föllmer and Schied (2016,
Theorem 4.33). A coherent risk measure has a dual representation in the
following form; see, e.g., Föllmer and Schied (2016, Theorem 4.33).

######  Proposition 2 (Delbaen (2002)).

For $\rho:L^{\infty}\to\mathbb{R}$, the following are equivalent.

  1. (i)

The mapping $\rho$ is a Fatou coherent risk measure;

  2. (ii)

there exists a $w(\mathcal{M},L^{\infty})$-closed and convex set
$\mathcal{S}_{\rho}\subseteq\mathcal{M}_{1}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}^{\mu}[X],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$ (3)

In case (ii) holds, $\mathcal{S}_{\rho}$ is uniquely given by
$\left\\{\mu\in\mathcal{M}_{1}:\mathbb{E}^{\mu}[X]\leq\rho(X)\text{ for all
}X\in L^{\infty}\right\\}$. Moreover, $\mathcal{R}\subseteq\mathcal{M}_{1}$
gives representation (3) if and only if $\operatorname*{conv}(\mathcal{R})$ is
$w(\mathcal{M},L^{\infty})$-dense in $\mathcal{S}_{\rho}$.

For a Fatou coherent risk measure $\rho$, the set

$\mathcal{S}_{\rho}=\left\\{\mu\in\mathcal{M}_{1}:\mathbb{E}^{\mu}[X]\leq\rho(X)\text{
for all }X\in L^{\infty}\right\\}$

will be called the _supporting set_ of $\rho$. If $\rho$ is defined on
$L^{\infty}(\mathcal{G})$ instead of $L^{\infty}$, then its support set
$\mathcal{S}_{\rho}$ is understood as a subset of
$\mathcal{M}_{1}(\mathcal{G})$. For $\mu\in\mathcal{M}_{1}$,
$\mathrm{ES}^{\mu}$ admits the dual representation

$\displaystyle\mathrm{ES}^{\mu}_{\alpha}(X)=\sup_{\nu\in\mathcal{S}_{\alpha,\mu}}\mathbb{E}^{\nu}[X],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty},$

where
$\mathcal{S}_{\alpha,\mu}=\\{\nu\in\mathcal{M}_{1}\mid({1-\alpha})\eta_{\nu}\leq\eta_{\mu}\\}$;
see e.g., Föllmer and Schied (2016, Theorem 4.52). Since
$\mathcal{S}_{\alpha,\mu}$ is $w\left(\mathcal{M},L^{\infty}\right)$-closed
and convex,
$\mathcal{S}_{\mathrm{ES}^{\mu}_{\alpha}}=\mathcal{S}_{\alpha,\mu}$ holds.

Law-invariant coherent risk measures also admit a representation of Kusuoka
(2001), and Fatou continuity is automatic for law-invariant convex risk
measures, as shown by Jouini et al. (2006) and Svindland (2010). Let
$\mathcal{M}_{B}$ denote the probability measures on $[0,1)$ equipped with the
restricted Borel $\sigma$-algebra.

######  Proposition 3.

For $\rho:L^{\infty}\to\mathbb{R}$, the following are equivalent..

  1. (i)

The mapping $\rho$ is a Fatou law-invariant coherent risk measure;

  2. (ii)

the mapping $\rho$ is a law-invariant coherent risk measure;

  3. (iii)

there exists some $\mathcal{D}\subseteq\mathcal{M}_{B}$ such that

$\displaystyle\rho(X)=\sup_{Q\in\mathcal{D}}\int_{[0,1)}\mathrm{ES}_{\alpha}(X)Q(\mathrm{d}\alpha),\hskip
3.0ptX\in L^{\infty}.$ (4)

Given some monetary risk measure $\rho:L^{\infty}\to\mathbb{R}$, we call
$\mathcal{A}_{\rho}=\\{X\in L^{\infty}\mid\rho(X)\leq 0\\}$ the _acceptance
set_ of $\rho$. It is well known that the monetary risk measure
$\rho:L^{\infty}\to\mathbb{R}$ is convex if and only if $\mathcal{A}_{\rho}$
is convex, and coherent if and only if $\mathcal{A}_{\rho}$ is a convex cone.

######  Proposition 4.

A monetary risk measure is $\mathcal{G}$-law invariant if and only if
$\mathcal{A}_{\rho}$ is $\mathcal{G}$-law invariant.

###### Proof.

The forward is trivial. For the converse note that $\\{X\in
L^{\infty}\mid\rho(X)\leq c\\}=\mathcal{A}_{\rho}+c$. Therefore, the level
sets of $\rho$ are $\mathcal{G}$-law invariant; the claim is straightforward
to prove after this observation. ∎

###  3.3 Examples induced by conditional expectations

This section focuses on coherent risk measures of the form
${\rho}(X)=\tilde{\rho}(\mathbb{E}[X|\mathcal{G}])$ for $X\in L^{\infty}$,
which are the simplest type of $\mathcal{G}$-law invariant coherent risk
measures, and presented earlier in (2).

######  Proposition 5.

Let $\tilde{\rho}$ in (2) be coherent and Fatou. Then the resulting risk
measure $\rho$ in (2) is also coherent and Fatou.

###### Proof.

Coherence of $\rho$ is clear, but Fatou continuity needs some care. We claim
that $\rho$ is $w(L^{\infty},\mathcal{M})$-lower semicontinuous. By
Proposition 25 in Appendix B, $\mathbb{E}[\cdot|\mathcal{G}]$ is
$w(L^{\infty},\mathcal{M})/w(L^{\infty}(\mathcal{G})/\mathcal{M}(\mathcal{G}))$-continuous.
Since $\tilde{\rho}$ is coherent and Fatou, $\tilde{\rho}$ is
$w(L^{\infty}(\mathcal{G}),\mathcal{M}(\mathcal{G}))$-lower semicontinuous.
Therefore, $\rho$ is $w(L^{\infty},\mathcal{M})$-lower semicontinuous and thus
Fatou. ∎

Coherent risk measures of this form are characterized in the following result.
Recall that for a Fatou coherent risk measure $\rho$, $\mathcal{S}_{\rho}$ is
the support set of $\rho$ and
$\widehat{\mathcal{S}}_{\rho}=\\{\eta_{\mu}:\mu\in\mathcal{S}_{\rho}\\}$.

######  Proposition 6.

For a Fatou coherent risk measure $\rho:L^{\infty}\to\mathbb{R}$, equivalent
are:

  1. (i)

There exists a Fatou coherent risk measure
$\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ such that (2) holds;

  2. (ii)

there exists $\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ such that (2)
holds;

  3. (iii)

$\rho(X)=\rho(\mathbb{E}[X|\mathcal{G}])$ for all $X\in L^{\infty}$;

  4. (iv)

$\widehat{\mathcal{S}}_{\rho}\subseteq L^{1}(\mathcal{G})$.

Moreover, in the case that any of the above conditions are satisfied, $\rho$
is $\mathcal{G}$-law invariant if and only if $\tilde{\rho}$ is law invariant.

###### Proof.

$(\mathrm{i})\Rightarrow(\mathrm{ii})$ and
$(\mathrm{ii})\Rightarrow(\mathrm{iii})$: These are clear.
$(\mathrm{iii})\Rightarrow(\mathrm{iv})$: We have

$\displaystyle\rho(X)=\rho(\mathbb{E}[X|\mathcal{G}])=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}\mathbb{E}[X|\mathcal{G}]]=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\mathbb{E}[\eta_{\mu}|\mathcal{G}]X].$

Therefore the set
$\\{\mathbb{E}[\eta_{\mu}|\mathcal{G}]:\mu\in\mathcal{S}_{\rho}\\}\subseteq\widehat{\mathcal{S}}_{\rho}$
is $w(L^{1},L^{\infty})$-dense by Proposition 2. Since $L^{1}(\mathcal{G})$ is
$w(L^{1},L^{\infty})$-closed (see Corollary 3 in Appendix B), we have
$\widehat{\mathcal{S}}_{\rho}\subseteq L^{1}(\mathcal{G})$.
$(\mathrm{iv})\Rightarrow(\mathrm{i})$: Define the map

$\displaystyle\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}:X\mapsto\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}X].$

The mapping $\tilde{\rho}$ is clearly coherent and Fatou. By the tower
property of conditional expectation, we have

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}X]=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}\mathbb{E}[X|\mathcal{G}]]=\tilde{\rho}(\mathbb{E}[X|\mathcal{G}]).$

The final statement of the proposition follows by definition. ∎

The risk measure in (2) further satisfies a stronger form of partial law
invariance, which will be studied in Section 5.

The idea of using conditional expectations can be generalized further to
incorporate different sub-$\sigma$-fields. Let $\mathcal{H}$ be a
sub-$\sigma$-algebra independent of $\mathcal{G}$. Let
$\rho_{1}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ and
$\rho_{2}:L^{\infty}(\mathcal{H})\to\mathbb{R}$ both be coherent risk
measures, with the former being law invariant. Define the following risk
measure

$\displaystyle\rho(X)=\rho_{1}(\mathbb{E}[X|\mathcal{G}])\vee\rho_{2}(\mathbb{E}[X|\mathcal{H}]),\hskip
3.0ptX\in L^{\infty},$

where $a\vee b$ denotes the maximum between two real numbers $a$ and $b$. To
show that $\rho$ above is $\mathcal{G}$-law invariant, we first notice that
for $X$ that is $\mathcal{G}$-measurable, we have
$\rho(X)=\rho_{1}(X)\vee\mathbb{E}[X].$ Coherence and law invariance of
$\rho_{1}$ implies $\rho_{1}(X)\geq\mathbb{E}[X]$ for $X\in
L^{\infty}(\mathcal{G})$; for this result, see Theorem 4.3 of Bäuerle and
Müller (2006). Therefore $\rho(X)=\rho_{1}(X)$ for all $X\in
L^{\infty}(\mathcal{G})$, which implies that $\rho$ is $\mathcal{G}$-law
invariant.

##  4 Coherent risk measures and partial law invariance

Given the importance of coherent risk measures in finance and portfolio
optimization (e.g., McNeil et al. (2015) and Föllmer and Schied (2016)), our
next task is to understand the structure of all $\mathcal{G}$-law invariant
coherent risk measures, ideally in parallel to the characterization of law-
invariant coherent risk measures of Kusuoka (2001). We first present a general
representation using supporting sets. Then, we introduce the technical tool,
called coherent adjustments, which themselves admit interesting mathematical
properties. The coherent adjustments will lead us to an alternative
representation, which will be very helpful for establishing a representation
of the type of Kusuoka (2001) in Section 5.

###  4.1 Main representation

We first present a representation of partially law-invariant coherent risk
measures, in a form that is closest to the well-known representation in
Proposition 2.

######  Theorem 1.

A mapping $\rho:L^{\infty}\to\mathbb{R}$ is a Fatou and $\mathcal{G}$-law-
invariant coherent risk measure if and only if

$\rho(X)=\sup_{\mu\in\mathcal{S}}\mathbb{E}^{\mu}[X],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty}$ (5)

holds for some convex $\mathcal{S}\subseteq\mathcal{M}_{1}$ such that the
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-closure of
$\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{S}\\}$ is $\mathcal{G}$-law invariant.

For later use, we call the property of $\mathcal{S}$ in Theorem 1 almost
$\mathcal{G}$-law invariance.

To interpret the representation (5) and the almost $\mathcal{G}$-law
invariance condition, we first consider a simple example. For
$\alpha\in(0,1)$, let
$\mathcal{S}=\\{\mu\in\mathcal{M}_{1}:\mu|_{\mathcal{G}}=\mathbb{P}|_{\mathcal{G}};\leavevmode\nobreak\
\eta_{\mu}\leq 1/(1-\alpha)\\},$ which is almost $\mathcal{G}$-law invariant,
since
$\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{S}\\}=\\{\mathbb{P}|_{\mathcal{G}}\\}$.
The corresponding risk measure $\rho$ in (5) has interesting properties in (6)
below. For convenience, Take any $\sigma$-algebra $\mathcal{H}$ independent of
$\mathcal{G}$, such that $(\Omega,\mathcal{H},\mathbb{P})$ is atomless
(assumed to exist; this corresponds to a notion of conditional atomless
property; see Remark 2). We claim that

$\displaystyle\rho(X)=\mathbb{E}[X]\leavevmode\nobreak\ \mbox{if $X\in
L^{\infty}(\mathcal{G})$\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}\rho(X)=\mathrm{ES}_{\alpha}(X)\leavevmode\nobreak\ \mbox{ $X\in
L^{\infty}(\mathcal{H})$.}$ (6)

The first equality is straightforward. To show the second equality, first,
from the dual representation of ES, we get $\rho(X)\leq\sup_{\eta_{\mu}\leq
1/(1-\alpha)}\mathbb{E}^{\mu}[X]=\mathrm{ES}_{\alpha}(X).$ For the other
direction, let $A\in\mathcal{F}$ be an event satisfying
$\mathbb{E}[X|A]=\mathrm{ES}_{\alpha}(X)$ and independent of $\mathcal{G}$.
The existence of such an event (called an $\alpha$-tail event) is justified by
Wang and Zitikis (2021, Lemmas A.4 and A.7). Let $\nu$ be given by
$\eta_{\nu}=\mathds{1}_{A}/(1-\alpha)$. Since $\mathds{1}_{A}$ is independent
of $\mathcal{G}$, we have $\nu|_{\mathcal{G}}=\mathbb{P}|_{\mathcal{G}}$, and
therefore $\nu\in\mathcal{S}$, and
$\rho(X)\geq\mathbb{E}^{\nu}[X]=\mathbb{E}[X|A]=\mathrm{ES}_{\alpha}(X)$.

By allowing $\mathcal{S}$ to vary but maintaining the condition that
$\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{S}\\}=\\{\mathbb{P}|_{\mathcal{G}}\\}$,
we allow the risk measure to behave “nicely” on $L^{\infty}(\mathcal{G})$,
while leaving flexibility outside this $\sigma$-field. Certainly, it is not
necessary to require $\mu|_{\mathcal{G}}=\mathbb{P}|_{\mathcal{G}}$ for
$\mu\in\mathcal{S}$; according to Theorem 1, it is necessary and sufficient to
take $\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{S}\\}$ to be almost law invariant.
Indeed, generalizing (6), we will be able let $\rho$ behave like a law-
invariant risk measure $\tilde{\rho}$ on $\mathcal{G}$ and like a tail risk
measure generated by $\tilde{\rho}$ on $\mathcal{H}$ in the sense of Liu and
Wang (2021), through the coherent adjustments that we study below.

Theorem 1 also gives a feasible way to construct $\mathcal{G}$-law-invariant
risk measures. We can take any law-invariant coherent risk measure
$\tilde{\rho}$ on $L^{\infty}(\mathcal{G})$ (e.g., $\mathrm{ES}_{\alpha}$),
and for each
$\mu\in\mathcal{S}_{\tilde{\rho}}\subseteq\mathcal{M}_{1}(\mathcal{G})$ we
take any set $\mathcal{S}(\mu)\subseteq\mathcal{M}_{1}$ satisfying
$\nu|_{\mathcal{G}}=\mu$ for all $\nu\in\mathcal{S}(\mu)$. Setting

$\displaystyle\rho(X)=\sup_{\nu\in\mathcal{S}}\mathbb{E}^{\nu}[X],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty},\mbox{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ where
}\mathcal{S}=\bigcup_{\mu\in\mathcal{S}_{\tilde{\rho}}}\mathcal{S}(\mu),$ (7)

yields a $\mathcal{G}$-law-invariant risk measure. Moreover, $\rho$ coincides
with $\tilde{\rho}$ on $L^{\infty}(\mathcal{G})$. For instance, the risk
measure $\overline{\mathrm{ES}}^{\mathcal{Q}}_{\alpha}$ in Section 2.2
coincides with $\mathrm{ES}_{\alpha}$ on $L^{\infty}(\mathcal{G})$.

######  Example 3.

Let $\rho$ be the mapping $X\mapsto\mathbb{E}^{\nu}[X]$ where
$\nu|_{\mathcal{G}}=\mathbb{P}|_{\mathcal{G}}$. Clearly, $\rho$ is
$\mathcal{G}$-law invariant, and the only possible set $\mathcal{S}$ in (5) is
given by $\\{\nu\\}$. The set $\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{S}\\}$ is
equal to $\\{\mathbb{P}|_{\mathcal{G}}\\}$, which is $\mathcal{G}$-law
invariant.

It is well-known that a Fatou coherent risk measure $\rho$ is law invariant if
and only if its supporting set $\mathcal{S}_{\rho}\subseteq\mathcal{M}_{1}$ is
law invariant; see Föllmer and Schied (2016, Theorem 4.59). Theorem 1 is
reminiscent of this result, but with some subtle differences; in particular,
one needs to take the restriction of elements of $\mathcal{S}$, as well as a
closure of the resulting set. The first operation is clearly needed, and the
second one seems to be also essential (see Remark 3).

A proof of Theorem 1 follows from an analysis on the supporting set of $\rho$,
which will be presented in a few separate results.

For $\mathcal{R}\subseteq\mathcal{M}$ let
$\mathcal{R}^{\mathcal{G}}=\\{\mu|_{\mathcal{G}}:\mu\in\mathcal{R}\\}\subseteq\mathcal{M}(\mathcal{G})$.
Given $\mathcal{B}\subseteq L^{1}$, let
$\mathcal{B}^{\mathcal{G}}=\\{\mathbb{E}[X|\mathcal{G}]:X\in\mathcal{B}\\}\subseteq
L^{1}(\mathcal{G})$. There is no ambiguity when we use the notation
$\widehat{\mathcal{R}}^{\mathcal{G}}$ because
$(\widehat{\mathcal{R}})^{\mathcal{G}}=\widehat{(\mathcal{R}^{\mathcal{G}})}$.

######  Proposition 7.

Let $\rho:L^{\infty}\to\mathbb{R}$ be a Fatou coherent risk measure and
$\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$. Then, $\tilde{\rho}$ is a
Fatou coherent risk measure. Furthermore,
$\mathcal{S}_{\rho}^{\mathcal{G}}\subseteq\mathcal{S}_{\tilde{\rho}}$ and
$\mathcal{S}_{\rho}^{\mathcal{G}}$ is
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-dense in
$\mathcal{S}_{\tilde{\rho}}$.

###### Proof.

The fact that $\tilde{\rho}$ is a Fatou coherent risk measure is clear. Since
$\mathcal{S}_{\rho}^{\mathcal{G}}$ is convex and

$\displaystyle\tilde{\rho}(X)=\rho(X)=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}X]=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\mathbb{E}[\eta_{\mu}|\mathcal{G}]X]=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\mathbb{E}[\eta_{\mu}X],$

for $X\in L^{\infty}(\mathcal{G})$, we have that
$\mathcal{S}_{\rho}^{\mathcal{G}}$ is
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-dense in
$\mathcal{S}_{\tilde{\rho}}$ by Proposition 2. ∎

######  Proposition 8.

Let $\rho:L^{\infty}\to\mathbb{R}$ be a Fatou coherent risk measure and
$\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$. The following are sufficient
conditions for $\mathcal{S}_{\rho}^{\mathcal{G}}=\mathcal{S}_{\tilde{\rho}}$.

  1. (i)

The risk measure $\rho$ satisfies $\rho(\mathbb{E}[X|\mathcal{G}])\leq\rho(X)$
for all $X\in L^{\infty}$ (equivalently,
$\widehat{\mathcal{S}}^{\mathcal{G}}_{\rho}\subseteq\widehat{\mathcal{S}}_{\rho}$).

  2. (ii)

The risk measure $\rho$ is Lebesgue continuous. That is, if
$(X_{n})_{n=1}^{\infty}$ is a sequence in $L^{\infty}$ of uniformly bounded
random variables that converge $\mathbb{P}$-a.s. to $X\in L^{\infty}$, then
$\rho(X_{n})\to\rho(X)$.

###### Proof.

$(\mathrm{i})$ For $\mu\in\mathcal{S}_{\tilde{\rho}}$, we have

$\displaystyle\mathbb{E}[\eta_{\mu}X]=\mathbb{E}[\eta_{\mu}\mathbb{E}[X|\mathcal{G}]]\leq\rho(\mathbb{E}[X|\mathcal{G}])\leq\rho(X),$

for all $X\in L^{\infty}$. Therefore, the extension of $\mu$ to $\mathcal{F}$,
i.e., the measure $\hat{\mu}:\mathcal{F}\to[0,1]$ given by
$A\mapsto\mathbb{E}^{\mu}[\mathds{1}_{A}]$, is in $\mathcal{S}_{\rho}$. Since
$\hat{\mu}|_{\mathcal{G}}=\mu$, we have
$\mathcal{S}_{\tilde{\rho}}\subseteq\mathcal{S}_{\rho}^{\mathcal{G}}$.
$(\mathrm{ii})$ Since the restriction map $\mu\mapsto\mu|_{\mathcal{G}}$ is
$w(\mathcal{M},L^{\infty})/w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-continuous
(see Proposition 21 in Appendix C) and $\mathcal{S}_{\rho}$ is
$w(\mathcal{M},L^{\infty})$-compact (see Föllmer and Schied (2016, Corollary
4.38)) we have that $\mathcal{S}^{\mathcal{G}}_{\rho}$ is
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-compact. Therefore, it
is closed and $\mathcal{S}_{\rho}^{\mathcal{G}}=\mathcal{S}_{\tilde{\rho}}$. ∎

######  Remark 3.

Although Proposition 8 only provides sufficient conditions for
$\mathcal{S}_{\rho}^{\mathcal{G}}=\mathcal{S}_{\tilde{\rho}}$, we did not find
any counter-example in which
$\mathcal{S}_{\rho}^{\mathcal{G}}\neq\mathcal{S}_{\tilde{\rho}}$, which seems
to be a challenging question. This relates to the question of whether in
Theorem 1 we can omit the operation of taking a closure. If
$\mathcal{S}_{\rho}^{\mathcal{G}}=\mathcal{S}_{\tilde{\rho}}$ always holds
true, then that operation can be omitted.

Now we are ready to prove Theorem 1.

###### Proof of Theorem 1.

Assume that $\rho$ is a Fatou and $\mathcal{G}$-law-invariant coherent risk
measure. The claim follows if $\mathcal{S}^{\mathcal{G}}_{\rho}$ is almost
$\mathcal{G}$-law invariant. Since
$\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$ is law-invaraint. We have that
$\mathcal{S}_{\tilde{\rho}}$ is $\mathcal{G}$-law invariant. By Proposition 7,
$\mathcal{S}^{\mathcal{G}}_{\rho}$ is almost $\mathcal{G}$-law invariant. For
the converse, it is clear that $\rho$ is a Fatou coherent risk measure. It is
also clear that $\mathcal{S}^{\mathcal{G}}$ is convex and supports
$\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$. Since
$\mathcal{S}^{\mathcal{\mathcal{G}}}$ is almost $\mathcal{G}$-law invariant,
$\tilde{\rho}$ is law invariant. This implies that $\rho$ is $\mathcal{G}$-law
invariant. ∎

###  4.2 Coherent adjustments and an alternative representation

Next, we take a different route to introduce the concept of coherent
adjustments, a useful technical tool that helps us further understand
$\mathcal{G}$-law-invariant coherent risk measures. In particular, coherent
adjustments make precise how the risk measures with supporting sets
$\mathcal{S}(\mu)$ are built in (7).

######  Definition 1.

Let $\mu\in\mathcal{M}_{1}(\mathcal{G})$, a _$\mu$ -coherent adjustment_ is a
mapping $\tau:L^{\infty}\to\mathbb{R}$ following the properties

  * $\mathcal{G}$-invariance: $\tau(X+Y)=\tau(X)$ for all $X\in L^{\infty}$ and $Y\in L^{\infty}(\mathcal{G})$;

  * $\mu$-monotonicity: $\mathbb{E}^{\mu}[X]+\tau(X)\leq\mathbb{E}^{\mu}[Y]+\tau(Y)$ for all $X,Y\in L^{\infty}$ with $X\leq Y$;

  * Convexity: $\tau(\lambda X+(1-\lambda)Y)\leq\lambda\tau(X)+(1-\lambda)\tau(Y)$ for all $X,Y\in L^{\infty}$ and $\lambda\in[0,1]$;

  * Positive homogeneity: $\tau(\lambda X)=\lambda\tau(X)$ for all $X\in L^{\infty}$ and $\lambda\geq 0$.

Moreover, a $\mu$-_convex adjustment_ is a mapping that satisfies the first
three properties in the above list, or $\tau=-\infty$ on $L^{\infty}$. Given
$\mu\in\mathcal{M}_{1}(\mathcal{G})$ we denote the set of all Fatou
$\mu$-coherent adjustments by $\mathrm{CA}(\mu)$. Let
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$, we say that
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ is a _CA-assignment_ if
$\tau_{\mu}\in\mathrm{CA}(\mu)$ for all $\mu\in\mathcal{R}$.

From the above properties, it is straightforward that the supremum of coherent
adjustments is again a $\mu$-coherent adjustment. A simple example of a Fatou
$\mu$-coherent adjustment is $\tau=\mathbb{E}^{\nu}-\mathbb{E}^{\mu}$ for any
$\nu\in\mathcal{M}_{1}$ with $\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$,
which is easy to check. Later, in Proposition 10, we will show that indeed,
the supremum of the form $\tau=\mathbb{E}^{\nu}-\mathbb{E}^{\mu}$ contains all
possible Fatou $\mu$-coherent adjustments.

We also note that $\mathcal{G}$-invariance is equivalent to $\tau(X)=0$ for
all $X\in L^{\infty}(\mathcal{G})$ in the presence of subadditivity.

######  Remark 4.

The properties of $\mu$-coherent adjustment $\tau$ appear similar to a
generalized deviation measure $D$ of Rockafellar et al. (2006), which is
defined by four properties: location invariance (i.e., $D(X+c)=D(X)$ for
$c\in\mathbb{R}$), subadditivity, positive homogeneity and $D(X)\geq 0$ for
all $X$ with strict $D(X)>0$ when $X$ is not constant. By choosing
$\mathcal{G}=\\{\varnothing,\Omega\\}$ (in which case
$\mathcal{M}_{1}(\mathcal{G})=\\{\mu\\}$ and
$\eta_{\mu}=1=\mathbb{E}[\eta_{\nu}|\mathcal{G}]$ for all
$\nu\in\mathcal{M}_{1}$), $\mathcal{G}$-invariance is the same as location
invariance; in this case, $\tau$ and $D$ share three properties. However,
there is an important difference between $\tau$ and $D$, as $\tau$ may take
negative values. For instance, $\tau=\mathbb{E}^{\nu}-\mathbb{E}^{\mu}$
defines a $\mu$-coherent adjustment, but not a generalized deviation measure.

The name “$\mu$-coherent adjustment” is explained by the following
proposition.

######  Proposition 9.

Let $\rho:L^{\infty}\to\mathbb{R}$ and $\mu\in\mathcal{M}_{1}(\mathcal{G})$.
The following are equivalent.

  1. (i)

The mapping $\rho$ is a Fatou coherent risk measure with
$\rho|_{L^{\infty}(\mathcal{G})}=\mathbb{E}^{\mu}|_{L^{\infty}(\mathcal{G})}$;

  2. (ii)

there exists $\tau\in\mathrm{CA}(\mu)$ such that $\rho=\mathbb{E}^{\mu}+\tau$.

###### Proof.

(i)$\Rightarrow$(ii): For $X\in L^{\infty}$ define
$\tau(X)=\rho(X)-\mathbb{E}\left[\eta_{\mu}X\right]$. The fact that $\tau$ is
a $\mu$-coherent adjustment is easily verified. To check that $\tau$ is Fatou,
let $(X_{n})_{n=1}^{\infty}$ be a uniformly bounded sequence in $L^{\infty}$
which converges $\mathbb{P}$-a.s. to some $X\in L^{\infty}$. By the Dominated
Convergence Theorem (DCT) we have
$\lim_{n\to\infty}\mathbb{E}\left[\eta_{\mu}X_{n}\right]=\mathbb{E}\left[\eta_{\mu}X\right]$.
Therefore,

$\displaystyle\liminf_{n\to\infty}\tau(X_{n})=\liminf_{n\to\infty}(\rho(X_{n})-\mathbb{E}\left[\eta_{\mu}X_{n}\right])$
$\displaystyle\geq\liminf_{n\to\infty}\rho(X_{n})-\lim_{n\to\infty}\mathbb{E}\left[\eta_{\mu}X_{n}\right]$
$\displaystyle\geq\rho(X)-\mathbb{E}\left[\eta_{\mu}X\right]=\tau(X).$

The claim follows since $\rho(X)=\mathbb{E}\left[\eta_{\mu}X\right]+\tau(X)$
for all $X\in L^{\infty}$. (ii)$\Rightarrow$(i): The facts that $\rho$ is a
coherent risk measure and that its restriction is $\mathbb{E}^{\mu}$ are
easily verified. Fatou continuity follows a similar argument as the one given
above. ∎

For some $\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and CA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{R}}$, by Proposition 9, the following risk
measure

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}^{\mu}[X]+\tau_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty},$ (8)

is coherent, as it is the supremum of coherent risk measures. A converse
statement to this requires some further conditions, which will be studied
below.

To explain why $\mu$-coherent adjustments help us extend a coherent risk
measure on $L^{\infty}(\mathcal{G})$ to $L^{\infty}$, let
$\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ be a law-invariant
coherent risk measure. For a convex
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ that is
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-dense in
$\mathcal{S}_{\tilde{\rho}}$ and a CA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{R}}$, let $\rho$ be given by (8). It is easy to
see that $\rho|_{L^{\infty}(\mathcal{G})}=\tilde{\rho}$, meaning that it is
$\mathcal{G}$-law invariant. In the special case that $\tau_{\mu}=0$ for all
$\mu\in\mathcal{R}$, this extension technique precisely gives the class
studied in Proposition 6, because
$\widehat{\mathcal{S}}_{\rho}=\widehat{\mathcal{S}}_{\tilde{\rho}}\subseteq
L^{1}(\mathcal{G})$.

We proceed by proving some facts regarding the representation of
$\mu$-coherent adjustments. For the proofs of the following results, see
Appendix C.

######  Proposition 10.

Let $\tau:L^{\infty}\to\mathbb{R}$ and $\mu\in\mathcal{M}_{1}(\mathcal{G})$.
The following are equivalent.

  1. (i)

The mapping $\tau\in\mathrm{CA}(\mu)$;

  2. (ii)

there exists a $w(\mathcal{M},L^{\infty})$-closed and convex set
$\mathcal{S}_{\tau}\subseteq\mathcal{M}_{1}$ such that
$\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$ for all
$\nu\in\mathcal{S}_{\tau}$ and
$\tau(X)=\sup_{\nu\in\mathcal{S}_{\tau}}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\right\\}$,
$X\in L^{\infty}$;

  3. (iii)

there exists $\mathcal{R}\subseteq\mathcal{M}_{1}$ such that
$\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$ for all $\nu\in\mathcal{R}$
and
$\tau(X)=\sup_{\nu\in\mathcal{R}}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\right\\}$,
$X\in L^{\infty}$.

In case (ii) holds, the set $\mathcal{S}_{\tau}$ is uniquely given by
$\left\\{\nu\in\mathcal{M}_{1}:\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\leq\tau(X)\text{
for all }X\in L^{\infty}\right\\}$.

######  Corollary 1.

Let $\mu\in\mathcal{M}_{1}(\mathcal{G})$ and $\tau:L^{\infty}\to\mathbb{R}$ be
a $\mu$-coherent adjustment. The following are equivalent.

  1. (i)

The mapping $\tau\in\mathrm{CA}(\mu)$;

  2. (ii)

the mapping $\tau$ is $w(L^{\infty},\mathcal{M})$-lower semicontinuous;

  3. (iii)

the mapping $\mathbb{E}^{\mu}+\tau$ is Fatou;

  4. (iv)

the mapping $\mathbb{E}^{\mu}+\tau$ is $w(L^{\infty},\mathcal{M})$-lower
semicontinuous.

Based on the results on supporting sets and coherent adjustments, we are ready
to provide an alternative representation result for coherent risk measures
with partial law invariance.

######  Theorem 2.

The mapping $\rho:L^{\infty}\to\mathbb{R}$ is a Fatou coherent risk measure if
and only if there exists a convex set
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and a CA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}^{\mu}[X]+\tau_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$ (9)

Moreover, in (9), $\rho$ is $\mathcal{G}$-law invariant if and only if
$\mathcal{R}$ is almost $\mathcal{G}$-law invariant.

###### Proof.

Let $\rho$ be a Fatou coherent risk measure. For
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$ we define
$\mathcal{S}(\mu)=\\{\nu\in\mathcal{S}_{\rho}\hskip 2.0pt:\hskip
2.0pt\nu|_{\mathcal{G}}=\mu\\}$, and note that
$\mathcal{S}_{\rho}=\bigcup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\mathcal{S}(\mu)$.
We have

$\displaystyle\rho(X)=\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\eta_{\mu}X\right]$
$\displaystyle=\sup_{\nu\in\mathcal{S}_{\rho}}\left\\{\mathbb{E}\left[\mathbb{E}[\eta_{\nu}|\mathcal{G}]X\right]+\mathbb{E}[(\eta_{\nu}-\mathbb{E}[\eta_{\nu}|\mathcal{G}])X]\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\sup_{\nu\in\mathcal{S}(\mu)}\left\\{\mathbb{E}\left[\eta_{\mu}X\right]+\mathbb{E}\left[(\eta_{\nu}-\eta_{\mu})X\right]\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\left\\{\mathbb{E}[\eta_{\mu}X]+\tau_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty},$

where
$\tau_{\mu}(X)=\sup_{\nu\in\mathcal{S}(\mu)}\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\\}$,
$X\in L^{\infty}$. The fact that $\tau_{\mu}\in\mathrm{CA}(\mu)$ follows from
Proposition 10. It is straightforward to see that
$\mathcal{S}_{\rho}^{\mathcal{G}}$ is convex. For the converse, note that (9)
is the supremum of coherent risk measures (by Proposition 9; see (8)) and thus
a coherent risk measure. By Corollary 1, $\mathbb{E}^{\mu}+\tau_{\mu}$ is
$w(L^{\infty},\mathcal{M})$-lower semicontinuous for all $\mu\in\mathcal{R}$.
As the supremum of $w(L^{\infty},\mathcal{M})$-lower semicontinuous and
coherent risk measures, $\rho$ is also one, and hence it is Fatou. Let
$\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$. We have that $\mathcal{R}$ in
(9) is dense in $\mathcal{S}_{\tilde{\rho}}$, and the last claim now follows
from the same argument in the proof of Theorem 1. ∎

######  Remark 5.

There is a straightforward way to change between representations (5) and (9)
for a Fatou and $\mathcal{G}$-law-invariant coherent risk measure. Let a
convex set $\mathcal{S}\subseteq\mathcal{M}_{1}(\mathcal{G})$ that represents
$\rho$ in Theorem 1 be given, and define
$\mathcal{R}=\mathcal{S}^{\mathcal{G}}$, which is almost $\mathcal{G}$-law
invariant. For each $\mu\in\mathcal{R}$ define
$\mathcal{S}(\mu)=\\{\nu\in\mathcal{S}\mid\nu|_{\mathcal{G}}=\mu\\}$ and
$\tau_{\mu}(X)=\sup_{\nu\in\mathcal{S}(\mu)}\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\\}$.
We can use these in representation (9). To go the other direction, let an
almost $\mathcal{G}$-law invariant convex set
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and a CA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ representing $\rho$ in Theorem 2 be given.
For $\mu\in\mathcal{R}$, let $\mathcal{S}_{\tau_{\mu}}$ be the
$w(\mathcal{M},L^{\infty})$-closed and convex set in Proposition 10 that
represents $\tau_{\mu}$. It is easy to show that
$\mathcal{S}=\bigcup_{\mu\in\mathcal{R}}$ is convex and
$S^{\mathcal{G}}=\mathcal{R}$. We can use these in representation (5). The
above connection clarifies the construction in (7).

For $\rho:L^{\infty}\to\mathbb{R}$, we say that $\rho$ is _$\mathcal{G}$
-additive_ if $\rho(X+Y)=\rho(X)+\rho(Y)$ for all $X,Y\in
L^{\infty}(\mathcal{G})$. Additivity has the interpretation as risk neutrality
under a particular probability, an idea that goes back to at least de Finetti
(1931); see Wakker (2010, p. 23) for discussions on this. Hence,
$\mathcal{G}$-additivity reflects that the risk measure is risk neutral on the
subset $L^{\infty}(\mathcal{G})$ of $L^{\infty}$. In other words, such a risk
measure is neutral for risks without uncertainty; recall that in the context
of Example 1, random losses in $L^{\infty}(\mathcal{G})$ do not have
uncertainty in their distributions.

For $X\in L^{\infty}$, we say that $\rho$ is _linear along_ $X$ if
$\rho(\lambda X)=\lambda\rho(X)$ for all $\lambda\in\mathbb{R}$; see Bellini
et al. (2021) and Liebrich and Munari (2022) for more discussions on this
technical property. The following proposition gives necessary and sufficient
conditions for dropping the supremum in representation (9) in the case of
$\mathcal{G}$-law invariance.

######  Proposition 11.

Let $\rho:L^{\infty}\to\mathbb{R}$ be a Fatou coherent risk measure. The
following are equivalent.

  1. (i)

The mapping $\rho$ is $\mathcal{G}$-additive and $\mathcal{G}$-law invariant;

  2. (ii)

the mapping $\rho$ is linear along a nonconstant $X\in
L^{\infty}(\mathcal{G})$ and $\mathcal{G}$-law invariant;

  3. (iii)

there exists $\tau\in\mathrm{CA}(\mathbb{P})$ such that

$\displaystyle\rho(X)=\mathbb{E}[X]+\tau(X),\hskip 3.0ptX\in L^{\infty}.$

###### Proof.

(i)$\Rightarrow$(ii): Since $\rho$ is $\mathcal{G}$-additive and coherent, it
is linear on $L^{\infty}(\mathcal{G})$. Since $L^{\infty}(\mathcal{G})$
contains non-constant random variables, (ii) holds true.
(ii)$\Rightarrow$(iii): The fact that
$\rho|_{L^{\infty}(\mathcal{G})}=\mathbb{E}|_{L^{\infty}(\mathcal{G})}$
follows from Theorem 4.5 and Theorem 4.7 of Bellini et al. (2021). Using
Proposition 9, we prove the claim. (iii)$\Rightarrow$(i): This is clear by
definition. ∎

A natural question would be, under what conditions is representation (9)
unique? When the coherent risk measure is Lebesgue continuous, there is a
definite answer to this. Let
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ be
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-closed and convex, and
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ be a CA-assignment. We say that
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ is _Lebesgue continuous_ if $\tau_{\mu}$ is
Lebesgue continuous for all $\mu\in\mathcal{R}$. We say that
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ is _regular_ if

  1. (a)

For any fixed $X\in L^{\infty}$, concavity of $\mu\mapsto\tau_{\mu}(X)$ holds;
that is, for all $\mu_{1},\mu_{2}\in\mathcal{R}$ and $\lambda\in(0,1)$ we have

$\displaystyle\lambda\tau_{\mu_{1}}(X)+(1-\lambda)\tau_{\mu_{2}}(X)\leq\tau_{\lambda\mu_{1}+(1-\lambda)\mu_{2}}(X).$

See e.g., Wang et al. (2020) for concavity for distributional mixtures in the
context of Choquet integrals.

  2. (b)

A weak form of semicontinuity holds: If $(\mu_{\lambda})_{\lambda\in\Lambda}$
is a net in $\mathcal{R}$ that converges to $\mu\in\mathcal{R}$ in
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$, then for all $X\in
L^{\infty}$,

$\displaystyle\tau_{\mu}(X)\geq\liminf_{\lambda}\tau_{\mu_{\lambda}}(X).$

See the appendix for background information on nets, subnets, and their
convergence.

######  Proposition 12.

Let $\rho:L^{\infty}\to\mathbb{R}$ be a Lebesgue-continuous coherent risk
measure. There exists a unique
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-compact and convex set
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and a unique regular
Lebesgue-continuous CA-assignment $(\tau_{\mu})_{\mu\in\mathcal{R}}$ such that
$\rho$ can be represented by equation (9).

For a proof of Proposition 12, see Appendix C.

We note that representation (9) has little resemblance to the classical
representation result of Kusuoka (2001) for law-invariant coherent risk
measures, as we see in Proposition 3. The next section addresses this gap by
further imposing a condition that turns out to be necessary and sufficient for
a similar representation along that direction.

##  5 Kusuoka-type representation

This section defines a new property called strong partial law invariance. This
new property implies partial law invariance and is necessary and sufficient to
reduce (9) in Section 4 to a Kusuoka-type representation, under a slight
restriction of our probability space. Later in the section, we give explicit
examples of coherent risk measures that are partially law invariant but not
strongly partially law invariant.

###  5.1 Strong partial law invariance and the main result

From now on, we further assume that $(\Omega,\mathcal{G},\mathbb{P})$ is
standard, i.e., isomorphic to the standard Borel space. Define the set

$\displaystyle\mathfrak{T}(\mathcal{G})=\\{T:\Omega\to\Omega\mid T\text{ is
bijective, bi-measurable with repsect to }\mathcal{G}\text{, and measure
preserving}\\}.$

Note that for $X,Y\in L^{\infty}(\mathcal{G})$ we have
$X\overset{\mathrm{d}}{=}_{\mathbb{P}}Y$ if and only if $X=Y\circ T$ for some
$T\in\mathfrak{T}(\mathcal{G})$ (this property was extensively used in Shapiro
(2013) and a proof of this claim can be found in Jouini et al. (2006)). For
$T\in\mathfrak{T}(\mathcal{G})$, define the mapping

$\displaystyle\Psi_{T}:L^{1}\to L^{1}:X\mapsto\mathbb{E}[X|\mathcal{G}]\circ
T^{-1}+X-\mathbb{E}[X|\mathcal{G}].$

The mapping $\Psi_{T}$ needs some explanation. Since the conditional
expectation with respect to $\mathcal{G}$ is a projection, we can decompose
$L^{\infty}$ into the direct sum of the image and the kernel. That is, for all
$X\in L^{\infty}$, to write

$\displaystyle X=\mathbb{E}[X|\mathcal{G}]+X-\mathbb{E}[X|\mathcal{G}].$

Let $T\in\mathfrak{T}(\mathcal{G})$, the mapping $\Psi_{T}$ applied to $X\in
L^{\infty}$ decomposes $X$ like above and then transforms the image component
by precomposing it with $T^{-1}$.

Noting that $\Psi_{T}(X)=X\circ T^{-1}$ for $X\in L^{\infty}(\mathcal{G})$, a
mapping $\rho$ is $\mathcal{G}$-law invariant if and only if
$\rho(\Psi_{T}(X))=\rho(X)$ for all $T\in\mathfrak{T}(\mathcal{G})$ and $X\in
L^{\infty}(\mathcal{G})$. Here, the “if” direction uses the fact that
$(\Omega,\mathcal{G},\mathbb{P})$ is standard. This motivates the following
property for a risk measure $\rho:L^{\infty}\to\mathbb{R}$.

  1. Strong $\mathcal{G}$-law invariance: $\rho\left(\Psi_{T}(X)\right)=\rho(X)$ for all $X\in L^{\infty}$ with $T\in\mathfrak{T}(\mathcal{G})$.

As discussed above, strong $\mathcal{G}$-law invariance is indeed stronger
than $\mathcal{G}$-law invariance, thus the name. When
$(\Omega,\mathcal{F},\mathbb{P})$ is standard, $\rho$ is law invariant if and
only if $\rho$ is strongly $\mathcal{F}$-law invariant. Therefore, strong
$\mathcal{G}$-law invariance and $\mathcal{G}$-law invariance are both a
generalization of law invariance, in subtly different ways.

An example of strongly $\mathcal{G}$-law invariant risk measure $\rho$ is
$\rho(X)=\tilde{\rho}(\mathbb{E}[X|\mathcal{G}])$ given in (2), where
$\tilde{\rho}:L^{\infty}(\mathcal{G})\to\mathbb{R}$ is law invariant. Examples
of functionals satisfying $\mathcal{G}$-law invariance but not strongly
$\mathcal{G}$-law invariance will be provided in Section 5.3.

Before we give the main result of the section, we need some technical details.
Given $X\in L^{1}$, define $q_{X}(\alpha)=\inf\\{x\in\mathbb{R}\mid
F_{X}(x)>\alpha\\}$ for $\alpha\in[0,1)$. Given
$\mu\in\mathcal{M}_{1}(\mathcal{G})$, we can define $Q\in\mathcal{M}_{B}$ (see
Föllmer and Schied (2016, Theorem 4.62)) by

$\displaystyle
Q([0,t))=\int_{[0,t)}(1-s)\mathrm{d}q_{\eta_{\mu}}(s),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{for all $t\in[0,1)$.}$ (10)

For $Q\in\mathcal{M}_{B}$, define
$\mathcal{M}_{Q}=\\{\mu\in\mathcal{M}_{1}(\mathcal{G}):\eqref{eq:Q-def}\mbox{\leavevmode\nobreak\
holds}\\},$ which is non-empty because $(\Omega,\mathcal{G},\mathbb{P})$ is
atomless, and

$\displaystyle\sup_{\mu\in\mathcal{M}_{Q}}\mathbb{E}[\eta_{\mu}X]=\int_{[0,1)}\mathrm{ES}_{\alpha}(\mathbb{E}[X|\mathcal{G}])Q(\mathrm{d}\alpha);$

see Föllmer and Schied (2016, Theorem 4.62). Let

$\displaystyle\mathrm{CA}(Q)=\bigcap_{\mu\in\mathcal{M}_{Q}}\mathrm{CA}(\mu).$

This set is clearly non-empty (it contains the zero map). We now present the
following result generalizing the representation of Kusuoka (2001).

######  Theorem 3.

Let $\rho:L^{\infty}\to\mathbb{R}$, the following are equivalent.

  1. (i)

The mapping $\rho$ is a Fatou and strongly $\mathcal{G}$-law-invariant
coherent risk measure;

  2. (ii)

there exist $\mathcal{D}\subseteq\mathcal{M}_{B}$ and
$\tau_{Q}\in\mathrm{CA}(Q)$ for each $Q\in\mathcal{D}$ such that

$\displaystyle\rho(X)=\sup_{Q\in\mathcal{D}}\left\\{\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}[X|\mathcal{G}]\right)Q(\mathrm{d}\alpha)+\tau_{Q}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$ (11)

In the trivial setting $\mathcal{G}=\mathcal{F}$, the representation (11)
reduces to the Kusuoka representation,

$\displaystyle\rho(X)=\sup_{Q\in\mathcal{D}}\int_{[0,1)}\mathrm{ES}_{\alpha}(X)Q(\mathrm{d}\alpha),\hskip
3.0ptX\in L^{\infty},$

because in this case, $\mathbb{E}[X|\mathcal{G}]=X$, $\tau_{Q}(X)=0$, and
strong $\mathcal{G}$-law invariance is law invariance. Another example is to
choose $\tau_{Q}=0$ for all $Q\in\mathcal{D}$, which yields the risk measure
$\rho:X\mapsto\tilde{\rho}(\mathbb{E}[X|\mathcal{G}])$ studied in Proposition
6. The proof of Theorem 3 relies on several results that will be studied in
Section 5.2.

###  5.2 Results leading to the proof of Theorem 3

We next present several results, which will eventually lead to a proof of
Theorem 3. We first analyze some properties of the transform $\Psi_{T}$.
Below, let $\mathbb{I}_{A}$ be the identity mapping on a set $A$.

######  Proposition 13.

Let $X\in L^{\infty}$, $Y\in L^{1}$ and $T,S\in\mathfrak{T}(\mathcal{G})$. We
have

  1. (i)

$\Psi_{ST}=\Psi_{S}\circ\Psi_{T}$;

  2. (ii)

$\Psi_{\mathbb{I}_{\Omega}}=\mathbb{I}_{L^{1}}$;

  3. (iii)

$\Psi_{T}$ is bijective;

  4. (iv)

$\Psi_{T}(X)\in L^{\infty}$;

  5. (v)

$\mathbb{E}[Y\Psi_{T}(X)]=\mathbb{E}[X\Psi_{T^{-1}}(Y)]$.

###### Proof.

(i): Let $Y\in L^{1}$ and $S,T\in\mathfrak{T}(\mathcal{G})$, we have

$\displaystyle\Psi_{ST}(Y)$
$\displaystyle=\mathbb{E}[Y|\mathcal{G}]\circ(ST)^{-1}+Y-\mathbb{E}[Y|\mathcal{G}]$
$\displaystyle=\mathbb{E}[Y|\mathcal{G}]\circ T^{-1}\circ
S^{-1}+Y-\mathbb{E}[Y|\mathcal{G}]$
$\displaystyle=\left(\mathbb{E}[\mathbb{E}[Y|\mathcal{G}]\circ
T^{-1}+Y-\mathbb{E}[Y|\mathcal{G}]|\mathcal{G}]\right)\circ
S^{-1}+Y-\mathbb{E}[Y|\mathcal{G}]$
$\displaystyle=\mathbb{E}[\Psi_{T}(Y)|\mathcal{G}]\circ
S^{-1}+Y-\mathbb{E}[Y|\mathcal{G}]$
$\displaystyle=\mathbb{E}[\Psi_{T}(Y)|\mathcal{G}]\circ
S^{-1}+\Psi_{T}(Y)-\mathbb{E}[Y|\mathcal{G}]\circ T^{-1}$
$\displaystyle=\mathbb{E}[\Psi_{T}(Y)|\mathcal{G}]\circ
S^{-1}+\Psi_{T}(Y)-\mathbb{E}[\Psi_{T}(Y)|\mathcal{G}]=\Psi_{S}(\Psi_{T}(Y)).$

Statements (ii), (iii) and (iv) can be seen from the standard properties of a
projection. (v): Let $X\in L^{\infty},Y\in L^{1}$ and
$T\in\mathfrak{T}(\mathcal{G})$.

$\displaystyle\mathbb{E}[Y\Psi_{T}(X)]$
$\displaystyle=\mathbb{E}[Y(\mathbb{E}[X|\mathcal{G}]\circ
T^{-1}+X-\mathbb{E}[X|\mathcal{G}])]$
$\displaystyle=\mathbb{E}[\mathbb{E}[Y|\mathcal{G}]\mathbb{E}[X|\mathcal{G}]\circ
T^{-1}]+\mathbb{E}[Y(X-\mathbb{E}[X|\mathcal{G}])]$
$\displaystyle=\mathbb{E}[\mathbb{E}[X|\mathcal{G}]\mathbb{E}[Y|\mathcal{G}]\circ
T]+\mathbb{E}[(Y-\mathbb{E}[Y|\mathcal{G}])(X-\mathbb{E}[X|\mathcal{G}])]$
$\displaystyle=\mathbb{E}[X\mathbb{E}[Y|\mathcal{G}]\circ
T]+\mathbb{E}[X(Y-\mathbb{E}[Y|\mathcal{G}])]$
$\displaystyle=\mathbb{E}[X(\mathbb{E}[Y|\mathcal{G}]\circ
T+Y-\mathbb{E}[Y|\mathcal{G}])]=\mathbb{E}[X\Psi_{T^{-1}}(Y)].$

This completes the proof. ∎

Proposition 13 implies, in particular, that $(\Psi_{T},\circ)$ forms a group.

######  Definition 2.

We say that a set $\mathcal{B}\subseteq L^{1}$ is _strongly $\mathcal{G}$-law
invariant_ if $\Psi_{T}(\mathcal{B})=\mathcal{B}$ for all
$T\in\mathfrak{T}(\mathcal{G})$. We say $\mathcal{R}\subseteq\mathcal{M}$ is
strongly $\mathcal{G}$-law invariant if the set $\widehat{\mathcal{R}}$ is
strongly $\mathcal{G}$-law invariant.

Since $\Psi_{T}$ is bijective for all $T\in\mathfrak{T}(\mathcal{G})$, we have
that $\mathcal{B}\subseteq L^{1}$ is strongly $\mathcal{G}$-law invariant if
and only if $\Psi_{T}(\mathcal{B})\subseteq\mathcal{B}$ for all
$T\in\mathfrak{T}(\mathcal{G})$. This fact can be helpful in the proofs. For
$\mathcal{B}\subseteq L^{1}$, strong $\mathcal{G}$-law invariance implies
$\mathcal{G}$-law invariance.

######  Lemma 1.

Let $\mathcal{B}\subseteq L^{1}$ be strongly $\mathcal{G}$-law invariant. Then
$\mathcal{B}^{\mathcal{G}}$ is $\mathcal{G}$-law invariant.

###### Proof.

Let $Y\in\mathcal{B}$, $X\in L^{1}(\mathcal{G})$ and
$X\overset{\mathrm{d}}{=}_{\mathbb{P}}\mathbb{E}[Y|\mathcal{G}]$. There exists
$T\in\mathfrak{T}(\mathcal{G})$ such that $X=\mathbb{E}[Y|\mathcal{G}]\circ
T$. Since $\Psi_{T^{-1}}(Y)\in\mathcal{B}$ we have

$\displaystyle X=\mathbb{E}[Y|\mathcal{G}]\circ
T=\mathbb{E}[\Psi_{T^{-1}}(Y)|\mathcal{G}]\in\mathcal{B}^{\mathcal{G}}.$

Since $X\in L^{1}(\mathcal{G})$ is arbitrary, $\mathcal{B}^{\mathcal{G}}$ is
$\mathcal{G}$-law invariant. ∎

We have the following connection between the risk measure $\rho$ and the
supporting set $\mathcal{S}_{\rho}$ in the case of strong $\mathcal{G}$-law
invariance.

######  Proposition 14.

Let $\rho:L^{\infty}\to\mathbb{R}$ be a Fatou coherent risk measure. The risk
measure $\rho$ is strongly $\mathcal{G}$-law invariant if and only if
$\mathcal{S}_{\rho}$ is strongly $\mathcal{G}$-law invariant.

###### Proof.

Let $\mathcal{S}_{\rho}$ be strongly $\mathcal{G}$-law invariant. Then we have

$\displaystyle\rho\left(\Psi_{T}(X)\right)=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\eta_{\mu}\Psi_{T}(X)\right]=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}\left[X\Psi_{T^{-1}}(\eta_{\mu})\right]=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\eta_{\mu}X]=\rho(X)$

for all $T\in\mathfrak{T}(\mathcal{G})$. Now assume that $\mathcal{S}_{\rho}$
is not strongly $\mathcal{G}$-law invariant. Therefore there exists
$\mu\in\mathcal{S}_{\rho}$ and $T\in\mathfrak{T}(\mathcal{G})$ such that
$\Psi_{T}(\eta_{\mu})\notin\widehat{\mathcal{S}}_{\rho}$. By the Hahn-Banach
separation theorem, we have that there exists $X\in L^{\infty}$ such that

$\displaystyle\rho(X)=\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\eta_{\nu}X\right]<\mathbb{E}\left[\Psi_{T}(\eta_{\mu})X\right]=\mathbb{E}\left[\Psi_{T^{-1}}(X)\eta_{\mu}\right]\leq\rho\left(\Psi_{T^{-1}}(X)\right).$

Thus, $\rho$ is not strongly $\mathcal{G}$-law invariant. ∎

######  Corollary 2.

If $\rho$ is strongly $\mathcal{G}$-law invariant, then
$\mathcal{S}_{\rho}^{\mathcal{G}}$ is $\mathcal{G}$-law invariant.

The following lemma will be used in the proceeding proposition.

######  Lemma 2.

Let $\mu,\nu\in\mathcal{M}_{1}(\mathcal{G})$. The following are equivalent.

  1. (i)

$\eta_{\nu}\overset{\mathrm{d}}{=}_{\mathbb{P}}\eta_{\mu}$;

  2. (ii)

$\eta_{\nu}=\eta_{\mu}\circ T^{-1}$ for some $T\in\mathfrak{T}(\mathcal{G})$;

  3. (iii)

$\nu=\mu\circ T^{-1}$ for some $T\in\mathfrak{T}(\mathcal{G})$.

###### Proof.

$(\mathrm{i})\Leftrightarrow(\mathrm{ii})$: This is mentioned earlier in the
section. $(\mathrm{ii})\Rightarrow(\mathrm{iii})$: For $A\in\mathcal{G}$ and
$T\in\mathfrak{T}(\mathcal{G})$,

$\displaystyle\nu(A)=\mathbb{E}\left[\eta_{\nu}\mathds{1}_{A}\right]=\mathbb{E}\left[\eta_{\mu}\circ
T^{-1}\mathds{1}_{A}\right]=\mathbb{E}\left[\eta_{\mu}\circ
T^{-1}\mathds{1}_{T^{-1}(A)}\circ
T^{-1}\right]=\mathbb{E}\left[\eta_{\mu}\mathds{1}_{T^{-1}(A)}\right]=\mu\left(T^{-1}(A)\right).$

$(\mathrm{iii})\Rightarrow(\mathrm{ii})$: For $A\in\mathcal{G}$ and
$T\in\mathfrak{T}(\mathcal{G})$,

$\displaystyle\mathbb{E}\left[\eta_{\mu}\circ
T^{-1}\mathds{1}_{A}\right]=\mu\left(T^{-1}(A)\right)=\nu(A)=\mathbb{E}\left[\eta_{\nu}\mathds{1}_{A}\right].$

Since this is true for all $A\in\mathcal{G}$ and $\eta_{\nu}$ and
$\eta_{\mu}\circ T^{-1}$ are $\mathcal{G}$ measurable, we have
$\eta_{\nu}=\eta_{\mu}\circ T^{-1}$. ∎

Let $\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ be $\mathcal{G}$-law
invariant and $(\tau_{\mu})_{\mu\in\mathcal{R}}$ be a CA-assignment. We say
that $(\tau_{\mu})_{\mu\in\mathcal{R}}$ is $\mathcal{G}$-law invariant if
$\tau_{\mu}=\tau_{\nu}$ when
$\eta_{\mu}\overset{d}{=}_{\mathbb{P}}\eta_{\nu}$.

Since strong $\mathcal{G}$-law invariance implies $\mathcal{G}$-law
invariance, we know that $\rho$ has representation (9). The next result is an
essential step to prove Theorem 3.

######  Proposition 15.

The risk measure $\rho$ is a Fatou and strongly $\mathcal{G}$-law-invariant
coherent risk measure if and only if there exist a $\mathcal{G}$-law invariant
convex set $\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and a
$\mathcal{G}$-law-invariant CA-assignment $(\tau_{\mu})_{\mu\in\mathcal{R}}$
such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}^{\mu}[X]+\tau_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

###### Proof.

Assume the latter condition. By Theorem 2, $\rho$ is a Fatou coherent risk
measure. Let $X\in L^{\infty}$ and $T\in\mathfrak{T}(\mathcal{G})$, we have

$\displaystyle\rho\left(\Psi_{T}(X)\right)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}\left[\eta_{\mu}\Psi_{T}(X)\right]+\tau_{\mu}\left(\Psi_{T}(X)\right)\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}\left[\eta_{\mu}\mathbb{E}[X|\mathcal{G}]\circ
T^{-1}\right]+\tau_{\mu}(X)\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}\left[\mathbb{E}[X|\mathcal{G}]\eta_{\mu}\circ
T\right]+\tau_{\mu}(X)\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}\left[\eta_{\mu}\mathbb{E}[X|\mathcal{G}]\right]+\tau_{\mu\circ
T^{-1}}(X)\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}\left[\eta_{\mu}X\right]+\tau_{\mu}(X)\right\\}=\rho(X).$

Conversely, using the same construction as Theorem 2, we have that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{S}^{\mathcal{G}}_{\rho}}\\{\mathbb{E}^{\mu}[X]+\tau_{\mu}(X)\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty},$

where
$\tau_{\mu}(X)=\sup_{\nu\in\mathcal{S}(\mu)}\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\\}$,
$X\in L^{\infty}$. Notice that $\mathcal{S}^{\mathcal{G}}_{\rho}$ is
$\mathcal{G}$-law invariant by Corollary 2. To see the final statement, note
that for all $\mu_{1},\mu_{2}\in\mathcal{S}_{\rho}^{\mathcal{G}}$ with
$\eta_{\mu_{1}}\overset{\mathrm{d}}{=}_{\mathbb{P}}\eta_{\mu_{2}}$, we have
that $\eta_{\mu_{1}}\circ T^{-1}=\eta_{\mu_{2}}$ for some
$T\in\mathfrak{T}(\mathcal{G})$. Let $\nu_{1}\in\mathcal{S}(\mu_{1})$. Since
$\mathcal{S}_{\rho}$ is strong $\mathcal{G}$-law invariant, we have that there
exists $\upsilon\in\mathcal{S}_{\rho}$ with
$\eta_{\upsilon}=\Psi_{T}(\eta_{\nu_{1}})$. This implies that
$\upsilon|_{\mathcal{G}}=\mu_{2}$ and thus $\upsilon\in\mathcal{S}(\mu_{2})$.
Yet we have $\eta_{\upsilon}-\eta_{\mu_{2}}=\eta_{\nu_{1}}-\eta_{\mu_{1}}$,
implying that

$\displaystyle\tau_{\mu_{2}}(X)=\sup_{\upsilon\in\mathcal{S}(\mu_{2})}\mathbb{E}[(\eta_{\upsilon}-\eta_{\mu_{2}})X]\geq\sup_{\upsilon\in\mathcal{S}(\mu_{1})}\mathbb{E}[(\eta_{\upsilon}-\eta_{\mu_{1}})X]=\tau_{\mu_{1}}(X).$

By symmetry, we get the reverse inequality and this completes the proof. ∎

We are finally ready to prove Theorem 3.

###### Proof of Theorem 3.

$(\mathrm{i})\Rightarrow(\mathrm{ii})$: By Proposition 15, we have that there
exist a $\mathcal{G}$-law invariant convex
$\mathcal{R}\subseteq\mathcal{M}_{1}(\mathcal{G})$ and a law-invariant CA-
assignment $(\tau_{\mu})_{\mu\in\mathcal{R}}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}[\eta_{\mu}X]+\tau_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

By Föllmer and Schied (2016, Theorem 4.62), for $Q\in\mathcal{M}_{B}$ and
$\mu\in\mathcal{M}_{Q}$, we have

$\displaystyle\mathcal{M}_{Q}=\left\\{\nu\in\mathcal{M}_{1}(\mathcal{G})\mid\eta_{\nu}\overset{\mathrm{d}}{=}\eta_{\mu}\right\\}.$

Define
$\mathcal{D}=\\{Q\in\mathcal{M}_{\mathcal{B}}\mid\mathcal{M}_{Q}\cap\mathcal{R}\neq\varnothing\\}$,
and note that $\mathcal{R}=\bigcup_{Q\in\mathcal{D}}\mathcal{M}_{Q}$ since
$\mathcal{R}$ is $\mathcal{G}$-law invariant. Define $\tau_{Q}=\tau_{\mu}$ for
some $\mu\in\mathcal{M}_{Q}$ (thus all for $\mu\in\mathcal{M}_{Q}$ since
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ is law invariant), therefore

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left\\{\mathbb{E}[\eta_{\mu}X]+\tau_{\mu}(X)\right\\}$
$\displaystyle=\sup_{Q\in\mathcal{D}}\sup_{\mu\in\mathcal{M}_{Q}}\left\\{\mathbb{E}[\eta_{\mu}X]+\tau_{\mu}(X)\right\\}$
$\displaystyle=\sup_{Q\in\mathcal{D}}\left\\{\int_{[0,1)}\mathrm{ES}_{\alpha}(\mathbb{E}[X|\mathcal{G}])Q(\mathrm{d}\alpha)+\tau_{Q}(X)\right\\}.$

$(\mathrm{ii})\Rightarrow(\mathrm{i})$: The fact that $\rho$ is cash
invariant, subadditive, and positively homogenous is clear. Let $X,Y\in
L^{\infty}$ with $X\leq Y$ and $Q\in\mathcal{D}$. Then we have

$\displaystyle\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}[X|\mathcal{G}]\right)Q(\mathrm{d}\alpha)+\tau_{Q}(X)$
$\displaystyle=\sup_{\mu\in\mathcal{M}_{Q}}\left\\{\mathbb{E}\left[\eta_{\mu}X\right]+\tau_{Q}(X)\right\\}$
$\displaystyle\leq\sup_{\mu\in\mathcal{M}_{Q}}\left\\{\mathbb{E}\left[\eta_{\mu}Y\right]+\tau_{Q}(Y)\right\\}$
$\displaystyle=\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}[Y|\mathcal{G}]\right)Q(\mathrm{d}\alpha)+\tau_{Q}(Y).$

Therefore $\rho$ is monotonic as it is the supremum of monotonic functionals.
By Proposition 5 for all $Q\in\mathcal{M}_{B}$, we have

$\displaystyle
L^{\infty}\to\mathbb{R}:X\mapsto\int_{[0,1)}\mathrm{ES}_{\alpha}(\mathbb{E}[X|\mathcal{G}])Q(\mathrm{d}\alpha)$

is $w(L^{\infty},\mathcal{M})$-lower semicontinuous. For all $Q\in\mathcal{D}$
the mapping

$\displaystyle
L^{\infty}\to\mathbb{R}:X\mapsto\int_{[0,1)}\mathrm{ES}_{\alpha}(\mathbb{E}[X|\mathcal{G}])Q(\mathrm{d}\alpha)+\tau_{Q}(X)$

is $w(L^{\infty},\mathcal{M})$-lower semicontinuous by Corollary 1, therefore
$\rho$ is $w(L^{\infty},\mathcal{M})$-lower semicontinuous as it is the
supremum of $w(L^{\infty},\mathcal{M})$-lower semicontinuous functionals.
Since $\rho$ is coherent, it is Fatou. Let $T\in\mathfrak{T}(\mathcal{G})$, we
have

$\displaystyle\rho\left(\Psi_{T}(X)\right)$
$\displaystyle=\sup_{Q\in\mathcal{D}}\left\\{\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}\left[\Psi_{T}(X)|\mathcal{G}\right]\right)Q(\mathrm{d}\alpha)+\tau_{Q}\left(\Psi_{T}(X)\right)\right\\}$
$\displaystyle=\sup_{Q\in\mathcal{D}}\left\\{\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}\left[X|\mathcal{G}\right]\circ
T^{-1}\right)Q(\mathrm{d}\alpha)+\tau_{Q}\left(X\right)\right\\}$
$\displaystyle=\sup_{Q\in\mathcal{D}}\left\\{\int_{[0,1)}\mathrm{ES}_{\alpha}\left(\mathbb{E}\left[X|\mathcal{G}\right]\right)Q(\mathrm{d}\alpha)+\tau_{Q}\left(X\right)\right\\}=\rho(X),$

where the last equality comes from the fact that $\mathrm{ES}_{\alpha}$ is law
invariant for all $\alpha\in[0,1)$. ∎

###  5.3 Examples

This section will give examples of partially law-invariant and strongly
partially law-invariant risk measures. We use the following lemma from Delbaen
(2000) frequently in this section.

######  Lemma 3.

Let $\rho_{1}:L^{\infty}\to\mathbb{R}$ and $\rho_{2}:L^{\infty}\to\mathbb{R}$
be Fatou coherent risk measures. Then $\rho_{1}\vee\rho_{2}$ is a Fatou
coherent risk measure with supporting set equal to
$\operatorname*{conv}(\mathcal{S}_{\rho_{1}}\cup\mathcal{S}_{\rho_{2}})$.

Let
$(\Omega_{k},\mathcal{F}_{k},\mathbb{P}_{k})=([0,1],\mathcal{B}([0,1]),\lambda)$
for $k=1,2$. For the rest of the section let
$(\Omega,\mathcal{F},\mathbb{P})=(\Omega_{1}\times\Omega_{2},\mathcal{F}_{1}\otimes\mathcal{F}_{2},\mathbb{P}_{1}\times\mathbb{P}_{2})$
and $\mathcal{G}=\sigma(\pi_{1})$ where
$\pi_{1}(\omega,\tilde{\omega})=\omega$. Define the set
$\mathcal{S}_{1}=\\{\mu\in\mathcal{M}_{1}\mid\eta_{\mu}\in
L^{1}(\mathcal{G}),\eta_{\mu}\leq 2\\}$. This set is
$w(\mathcal{M},L^{\infty})$-closed and convex and corresponds to the Fatou
coherent risk measure $\rho_{1}=\mathrm{ES}_{1/2}(\mathbb{E}[X|\mathcal{G}])$.

The following example gives a Fatou $\mathcal{G}$-law-invariant coherent risk
measure that is not strongly $\mathcal{G}$-law invariant.

######  Example 4.

Define $\nu\in\mathcal{M}_{1}$ by
$\eta_{\nu}:\Omega\to\mathbb{R}:(\omega,\tilde{\omega})\mapsto
4\omega\tilde{\omega}$. We claim that the Fatou coherent risk measure
$\rho:L^{\infty}\to\mathbb{R}:X\mapsto\mathbb{E}[\eta_{\nu}X]\vee\rho_{1}(X)$,
which has supporting set
$\mathcal{S}_{\rho}=\operatorname*{conv}(\mathcal{S}_{1}\cup\\{\nu\\})$, is
$\mathcal{G}$-law invariant but not strongly $\mathcal{G}$-law invariant. To
see this, note
$\mathbb{E}[\eta_{\nu}|\mathcal{G}](\omega,\tilde{\omega})=2\omega$. So the
measure associated to $\mathbb{E}[\eta_{\nu}|\mathcal{G}]$ is in the set
$\mathcal{S}_{1}$. Therefore

$\displaystyle\operatorname*{conv}(\mathcal{S}_{1}\cup\\{\nu\\})^{\mathcal{G}}=\operatorname*{conv}(\mathcal{S}_{1}^{\mathcal{G}}\cup\\{\nu\\}^{\mathcal{G}})=\mathcal{S}_{1}$

which is $\mathcal{G}$-law invariant, meaning $\rho$ is $\mathcal{G}$-law
invariant by Theorem 1. Now define
$T:\Omega\to\Omega:(\omega,\tilde{\omega})\mapsto(1-\omega,\tilde{\omega})$,
and it is clear that $T\in\mathfrak{T}(\mathcal{G})$. We have

$\displaystyle\Phi_{T}(\eta_{\nu})(\omega,\tilde{\omega})=2(1-\omega)+4\omega\tilde{\omega}-2\omega=2+4\omega\tilde{\omega}-4\omega.$

Let $A=\\{(\omega,\tilde{\omega})\in\Omega\mid 2/3\leq\omega\leq 1\text{ and
}0\leq\tilde{\omega}\leq 1/4\\}\in\mathcal{F}$. We have

$\displaystyle\int_{A}\Phi_{T}(\eta_{\nu})d\lambda$
$\displaystyle=\int_{0}^{1/4}\left(\int_{2/3}^{1}2+4\omega\tilde{\omega}-4\omega\hskip
2.0ptd\omega\right)d\tilde{\omega}=-11/144.$

Therefore $\Phi_{T}(\eta_{\nu})$ is associated with a signed measure and not a
probability measure as it takes negative values, implying that it is not
contained in $\operatorname*{conv}(\mathcal{S}_{1}\cup\\{\nu\\})$ and $\rho$
is not strongly $\mathcal{G}$-law invariant by Proposition 14.

The following example gives a Fatou strongly $\mathcal{G}$-law invariant
coherent risk measure that is not in the form of (2).

######  Example 5.

Define $\nu\in\mathcal{M}_{1}$ by
$\eta_{\nu}:\Omega\to\mathbb{R}:(\omega,\tilde{\omega})\mapsto
2\tilde{\omega}$. We claim that the Fatou coherent risk measure
$\rho:L^{\infty}\to\mathbb{R}:X\mapsto\mathbb{E}[\eta_{\nu}X]\vee\rho_{1}(X)$
is strongly $\mathcal{G}$-law invariant. To see this let
$\mu\in\mathcal{S}_{1}$, $T\in\mathfrak{T}(\mathcal{G})$, and
$0\leq\lambda\leq 1$. We have

$\displaystyle\Psi_{T}(\lambda\eta_{\nu}+(1-\lambda)\eta_{\mu})=\lambda\eta_{\nu}+(1-\lambda)\eta_{\mu}\circ
T^{-1}.$

Therefore the measure associated to
$\Psi_{T}(\lambda\eta_{\nu}+(1-\lambda)\eta_{\mu})$ is in
$\operatorname*{conv}(\mathcal{S}_{1}\cup\\{\nu\\})$, proving that $\rho$ is
strongly $\mathcal{G}$-law invariant by Proposition 14. Since
$\widehat{\mathcal{S}}_{\rho}$ is not contained in $L^{1}(\mathcal{G})$,
$\rho$ is not in the form of (2).

##  6 Generalization to convex risk measures

In this section, we quickly generalize some results to the case of convex risk
measures, by providing representations akin to Proposition 10 and Theorem 2,
which will yield a representation in the form of Theorem 1. We only present
here the necessary ingredients. All proofs and related results are provided in
Appendix D.

For $\mu\in\mathcal{M}_{1}(\mathcal{G})$, define
$\mathcal{E}(\mu)=\\{\nu\in\mathcal{M}_{1}:\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}\\}$.
First, we obtain a representation of $\mu$-convex adjustments, similar to
Proposition 10.

######  Proposition 16.

Let $\tau:L^{\infty}\to\mathbb{R}$, $\mu\in\mathcal{M}_{1}(\mathcal{G})$, and
$\rho=\mathbb{E}^{\mu}+\tau$. The following are equivalent.

  1. (i)

The mapping $\tau$ is a Fatou $\mu$-convex adjustment;

  2. (ii)

the mapping $\rho$ is a $w(L^{\infty},L^{1})$-lower semicontinuous convex risk
measure with $\operatorname*{dom}(\rho^{*})\subseteq\mathcal{E}(\mu)$;

  3. (iii)

$\tau$ can be represented by

$\displaystyle\tau(X)=\sup_{\nu\in\mathcal{E}(\mu)}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]-\rho^{*}(\nu)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty};$

  4. (iv)

there exists $\alpha:\mathcal{E}(\mu)\to\mathbb{R}\cup\\{\infty\\}$ such that
$\tau$ can be represented by

$\displaystyle\tau(X)=\sup_{\nu\in\mathcal{E}(\mu)}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]-\alpha(\nu)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

Similarly to CA-assignments, we say that
$(\tau_{\mu})_{\mu\in\mathcal{M}_{1}(\mathcal{G})}$ is a _CoA-assignment_ if
for all $\mu\in\mathcal{M}_{1}(\mathcal{G})$, $\tau_{\mu}$ is a Fatou
$\mu$-convex adjustment and the mapping $\mu\mapsto\tau_{\mu}(0)$ is concave.
The following result is the convex version of Theorem 2.

######  Theorem 4.

The mapping $\rho:L^{\infty}\to\mathbb{R}$ is a Fatou convex risk measure if
and only if there exists a CoA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{M}_{1}(\mathcal{G})}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{M}_{1}(\mathcal{G})}\left\\{\mathbb{E}^{\mu}[X]+\tau_{\mu}(X)\right\\}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

Moreover, $\rho$ is $\mathcal{G}$-law invariant if and only if for all
$c>-\rho(0)$ the set
$\left\\{\mu\in\mathcal{M}_{1}(\mathcal{G})\mid-\tau_{\mu}(0)\leq c\right\\}$
is almost $\mathcal{G}$-law invariant.

As entropic risk measures (Föllmer and Schied (2002)) are the most common
examples for convex risk measures, we discuss the $\mathcal{G}$-partially law-
invariant version of this family in the following example.

######  Example 6 ($\mathcal{G}$-entropic risk measures).

We give examples of $\mathcal{G}$-partially law-invariant convex risk measures
built on the usual entropic risk measure (ER), defined by, for $\beta>0$,

$\mathrm{ER}_{\beta}(X)=\frac{1}{\beta}\log\mathbb{E}\left[e^{\beta
X}\right],\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ X\in
L^{\infty}.$

Moreover, we write the conditional version of ER (this is not a risk measure
in our definition) as

$\mathrm{ER}_{\beta}(X|\mathcal{G})=\frac{1}{\beta}\log\mathbb{E}\left[e^{\beta
X}|\mathcal{G}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ X\in L^{\infty}.$

There are three natural ways to generalize the family of ER to satisfy
$\mathcal{G}$-partial law-invariance. The first is to use the mean of the
conditional ER , that is,

${\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)=\mathbb{E}[\mathrm{ER}_{\beta}(X|\mathcal{G})]=\mathbb{E}\left[\frac{1}{\beta}\log\mathbb{E}\left[e^{\beta
X}|\mathcal{G}\right]\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ X\in L^{\infty}.$

The second is to compute the ER of the conditional mean, that is,

$\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)=\mathrm{ER}_{\beta}(\mathbb{E}[X|\mathcal{G}])=\frac{1}{\beta}\log\mathbb{E}\left[e^{\beta\mathbb{E}[X|\mathcal{G}]}\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

Finally, we can also compute ER of the conditional ER, but this gives back the
usual ER, that is

$\mathrm{ER}_{\beta}(\mathrm{ER}_{\beta}(X|\mathcal{G}))=\frac{1}{\beta}\log\mathbb{E}\left[\exp\left(\beta\frac{1}{\beta}\log\mathbb{E}\left[e^{\beta
X}|\mathcal{G}\right]\right)\right]=\mathrm{ER}_{\beta}(X),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

(This is time-consistency of ER as a dynamic risk measure; see Kupper and
Schachermayer (2009).) The last version is fully law invariant and not
interesting to us. The first two versions, both being $\mathcal{G}$-law
invariant, have intriguing and distinct properties. For instance, if $X$ is
$\mathcal{G}$-measurable, then

${\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)=\mathbb{E}[X]\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
}\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)=\mathrm{ER}_{\beta}(X);$

if $Y$ is independent of $\mathcal{G}$, then

${\mathrm{ER}}^{\mathcal{G}}_{\beta}(Y)=\mathrm{ER}_{\beta}(Y)\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
}\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}(Y)=\mathbb{E}[Y].$

The above formulas show that ${\mathrm{ER}}^{\mathcal{G}}_{\beta}$ and
$\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}$ are not fully law invariant,
and none of them dominates the other. Moreover, if $X$ and $Y$ above are
identically distributed and non-degenerate, then
${\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)<{\mathrm{ER}}^{\mathcal{G}}_{\beta}(Y)$
and
$\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}(X)>\widetilde{\mathrm{ER}}^{\mathcal{G}}_{\beta}(Y)$,
showing different emphases of the two risk measures when assessing risks from
different sources.

##  7 Conclusion

The main content of the paper is the propose and study the concept of partial
law invariance, as a generalization of law invariance. This new property is
both appealing from a decision-theoretic perspective and interesting from a
purely mathematical perspective. To summarize the motivation, different
economic sources have different levels of ambiguity, and a law-invariant
mapping does not take these differences into account, thus calling for the
concept of partial law invariance to rectify this discrepancy.

The economic relevance of the new framework is illustrated with several
realistic situations described in Section 2. This new concept gives raise to
different procedures in robust risk assessment applications, which will be
explored later. The mathematical novelty of the new framework lies in new
classes of risk measures and new representation results in forms that are not
present in the literature. In particular, the concept of coherent and convex
adjustments plays an important role in understanding the new property as well
as its representation in both the context of coherent risk measures that that
of convex ones.

There are several mathematical questions for future research, and we describe
two promising ones below. The first weakens the assumption of coherence, and
the second generalizes the property of law invariance.

The star-shaped risk measures, studied by Castagnoli et al. (2022), are a
natural generalization of convex risk measures, and they satisfy the following
property in addition to $\rho(0)=0$.

  1. Star-shapedness: $\rho(\lambda X)\leq\lambda\rho(X)$ for all $X\in L^{\infty}$ and $\lambda\in[0,1]$.

One area of further research could be to generalize the results in this paper
to this more general class, and study the corresponding representation.
Representations of law-invariant (quasi-)star-shaped risk measures are
recently studied by Han et al. (2021) and Laeven et al. (2023), which admit a
quite different structure than the Kusuoka-type.

This paper only considers one source (sub-$\sigma$-algebra) on which the risk
measure is law invariant. In a practical application, there may be multiple
sources of randomness on which the risk measure is law invariant. To formalize
this, let $\mathfrak{G}$ be a set of sub-$\sigma$-algebras. For a function
$\rho:L^{\infty}\to\mathbb{R}$, we say that $\rho$ is $\mathfrak{G}$-law
invariant if it is $\mathcal{G}$-law invariant for each
$\mathcal{G}\in\mathfrak{G}$. When $\mathfrak{G}=\\{\mathcal{G}\\}$,
$\mathfrak{G}$-law invariance is the same as $\mathcal{G}$-law invariance. It
may be tempting to consider the sub-$\sigma$-algebra
$\sigma_{\mathfrak{G}}=\sigma\left(\bigcup_{\mathcal{G}\in\mathfrak{G}}\mathcal{G}\right)$
generated by $\mathfrak{G}$, and one wonders whether $\mathfrak{G}$-law
invariance is simply equivalent to $\sigma_{\mathfrak{G}}$-law invariance. It
is however clear from (6) in Section 4 that this is not the case.
Characterizing risk measures satisfying $\mathfrak{G}$-law invariance seems to
be beyond the available techniques.

## References

  * Artzner et al. (1999) Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. _Mathematical Finance_ , 9(3), 203–228.
  * Baillon et al. (2023) Baillon, A., Bleichrodt, H., Li, C. and Wakker, P. P. (2023). Source theory: A tractable and positive ambiguity theory. Working paper available at https://personal.eur.nl/wakker/pdf/sourceth.pdf
  * Baillon et al. (2018) Baillon, A., Huang, Z., Selim, A. and Wakker, P. P. (2018). Measuring ambiguity attitudes for all (natural) events. _Econometrica_ , 86(5), 1839–1858.
  * Bäuerle and Müller (2006) Bäuerle, N. and Müller, A. (2006). Stochastic orders and risk measures: Consistency and bounds. _Insurance: Mathematics and Economics_ , 38(1), 132–148.
  * BCBS (2019) BCBS (2019). Minimum Capital Requirements for Market Risk. February 2019. Basel Committee on Banking Supervision. Basel: Bank for International Settlements. https://www.bis.org/bcbs/publ/d457.htm
  * Bellini et al. (2021) Bellini, F., Koch-Medina, P., Munari, C. and Svindland, G. (2021). Law-invariant functionals that collapse to the mean. _Insurance: Mathematics and Economics_ , 98, 83–91.
  * Bredon (1993) Bredon, G. (1993). _Topology and Geometry_. Springer, Berlin.
  * Castagnoli et al. (2022) Castagnoli, E., Cattelan, G., Maccheroni, F., Tebaldi, C. and Wang, R. (2022). Star-shaped risk measures. _Operations Research_ , 70(5), 2637–2654.
  * de Finetti (1931) de Finetti, B. (1931). Sul significato soggettivo della probabilitá. _Fundamenta Mathematicae_ , 17, 298–329. Translated into English by Khale, M. as _On the Subjective Meaning of Probability_. In Monari, P. & Cocchi, D. (1993, eds.). _Probabilitá e Induzione_ , 291–321. Clueb, Bologna.
  * Delage et al. (2019) Delage, E., Kuhn, D. and Wiesemann, W. (2019). “Dice”-sion-making under uncertainty: When can a random decision reduce risk? _Management Science_ , 65(7), 3282–3301.
  * Delbaen (2000) Delbaen, F. (2000). Coherent risk measures. _Lecture Notes_ , Pisa.
  * Delbaen (2002) Delbaen, F. (2002). Coherent risk measures on general probability spaces. In _Advances in Finance and Stochastics_ (pp. 1–37). Springer, Berlin, Heidelberg.
  * Delbaen (2021) Delbaen, F. (2021). Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions. _Finance and Stochastics_ , 25, 597–614.
  * Detlefsen and Scandolo (2005) Detlefsen, K. and Scandolo, G. (2005). Conditional and dynamic convex risk measures. _Finance and Stochastics_ , 9, 539–561.
  * Dimmock et al. (2016) Dimmock, S., Kouwenberg, R., Wakker, P. (2016). Ambiguity attitudes in a large representative sample. _Management Science_ , 62(5), 1363–-1380.
  * Filipović et al. (2012) Filipović, D., Kupper, M., Vogelpoth, N. (2012). Approaches to conditional risk. _SIAM Journal on Financial Mathematics_ , 3(1), 402–432.
  * Föllmer and Schied (2002) Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. _Finance and Stochastics_ , 6(4), 429–447.
  * Föllmer and Schied (2016) Föllmer, H. and Schied, A. (2016). _Stochastic Finance. An Introduction in Discrete Time_. Fourth Edition. Walter de Gruyter, Berlin.
  * Frittelli and Rosazza Gianin (2005) Frittelli, M. and Rosazza Gianin, E. (2005). Law invariant convex risk measures. _Advances in Mathematical Economics_ , 7, 33–46.
  * Frittelli and Rosazza Gianin (2002) Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. _Journal of Banking and Finance_ , 26,1473–1486.
  * Ghaoui et al. (2003) Ghaoui, L. M. E., Oks, M. and Oustry, F. (2003). Worst-case value-at-risk and robust portfolio optimization. _Operations Research_ , 51, 543–556.
  * Gilboa and Schmeidler (1989) Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. _Journal of Mathematical Economics_ , 18, 141–153.
  * Gul and Pesendorfer (2014) Gul, F. and Pesendorfer, W. (2014). Expected uncertain utility theory. _Econometrica_ , 82(1), 1–39.
  * Han et al. (2021) Han, X., Wang, Q., Wang, R. and Xia, J. (2021). Cash-subadditive risk measures without quasi-convexity. _arXiv_ : 2110.12198.
  * Jouini et al. (2006) Jouini, E., Schachermayer, W. and Touzi, N. (2006). Law invariant risk measures have the Fatou property. _Advances in Mathematical Economics_ , 9, 49–71.
  * Klibanoff et al. (2005) Klibanoff, P., Marinacci, M. and Mukerji, S. (2005). A smooth model of decision making under uncertainty. _Econometrica_ , 73(6), 1849–1892.
  * Kupper and Schachermayer (2009) Kupper, M. and Schachermayer, W. (2009). Representation results for law invariant time consistent functions. Mathematics and Financial Economics, 2(3), 189–210.
  * Kusuoka (2001) Kusuoka, S. (2001). On law invariant coherent risk measures. _Advances in Mathematical Economics_ , 3, 83–95.
  * Laeven et al. (2023) Laeven, R. J., Rosazza Gianin, E. and Zullino, M. (2023). Law-invariant return and star-shaped risk measures. _arXiv_ : 2310.19552.
  * Liebrich and Munari (2022) Liebrich, F. B. and Munari, C. (2022). Law-invariant functionals that collapse to the mean: Beyond convexity. _Mathematics and Financial Economics_ , 16(3), 447–480.
  * Liu and Wang (2021) Liu, F. and Wang, R. (2021). A theory for measures of tail risk. _Mathematics of Operations Research_ , 46(3), 1109–1128.
  * Maccheroni et al. (2006) Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. _Econometrica_ , 74(6), 1447–1498.
  * McNeil et al. (2015) McNeil, A. J., Frey, R. and Embrechts, P. (2015). _Quantitative Risk Management: Concepts, Techniques and Tools_. Revised Edition. Princeton, NJ: Princeton University Press.
  * Riedel (2004) Riedel, F. (2004). Dynamic coherent risk measures. _Stochastic Processes and their Applications_ , 112(2), 185–200.
  * Rockafellar and Uryasev (2002) Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. _Journal of Banking and Finance_ , 26(7), 1443–1471.
  * Rockafellar et al. (2006) Rockafellar, R. T., Uryasev, S. and Zabarankin, M. (2006). Generalized deviation in risk analysis. _Finance and Stochastics_ , 10, 51–74.
  * Schmeidler (1989) Schmeidler, D. (1989). Subjective probability and expected utility without additivity. _Econometrica_ , 57(3), 571–587.
  * Shapiro (2013) Shapiro, A. (2013). On Kusuoka representation of law invariant risk measures. _Mathematics of Operations Research_ , 38, 142–152.
  * Shapiro et al. (2021) Shapiro, A., Dentcheva, D. and Ruszczynski, A. (2021). _Lectures on Stochastic Programming: Modeling and Theory_. Society for Industrial and Applied Mathematics.
  * Shen et al. (2019) Shen, J., Shen, Y., Wang, B. and Wang, R. (2019). Distributional compatibility for change of measures. _Finance and Stochastics_ , 23(3), 761–794.
  * Svindland (2010) Svindland, G. (2010). Continuity properties of law-invariant (quasi-)convex risk functions on $L^{\infty}$. _Mathematics and Financial Economics_ , 3, 39–43.
  * Wakker (2010) Wakker, P. P. (2010). _Prospect Theory: For Risk and Ambiguity_. Cambridge University Press.
  * Wang et al. (2020) Wang, R., Wei, Y. and Willmot, G. E. (2020). Characterization, robustness and aggregation of signed Choquet integrals. _Mathematics of Operations Research_ , 45(3), 993–1015.
  * Wang and Ziegel (2021) Wang, R., and Ziegel, J. F. (2021). Scenario-based risk evaluation. _Finance and Stochastics_ , 25, 725–756.
  * Wang and Zitikis (2021) Wang, R. and Zitikis, R. (2021). An axiomatic foundation for the Expected Shortfall. _Management Science_ , 67, 1413–1429.
  * Zalinescu (2002) Zalinescu, C. (2002) _Convex Analysis in General Vector Spaces_. World Scientific, Singapore.
  * Zhu and Fukushima (2009) Zhu, S. and Fukushima, M. (2009). Worst-case conditional value-at-risk with application to robust portfolio management. _Operations Research_ , 57(5), 1155-1168.

##  Appendix A Analysis in a finite probability space

Will first show that law invariance is equivalent to permutation invariance in
a finite uniform probability space. Afterward, we will study partially law-
invariant coherent risk measures when our probability space has a natural
structure.

###  A.1 Law-invariant coherent risk measures

In the literature, when proving results regarding law-invariant risk measures,
one often assumes that the underlying probability space is atomless. Since
finite probability spaces always have atoms, we instead will assume that the
underlying probability measure is uniform.

Let $N\in\mathbb{N}$, denote by $[N]=\\{1,\dots,N\\}$. Assume that
$\Omega=[N]$ and $\mathcal{F}=2^{\Omega}$ and put the uniform probability
measure $\mathbb{P}$ on $(\Omega,\mathcal{F})$, that is
$\mathbb{P}(\\{k\\})=1/N$ for all $k\in[N]$. We can identify elements $X\in
L^{\infty}$ with $\mathbb{R}^{N}$. For the rest of the section, we will refer
only to $\mathbb{R}^{N}$, but this is to be regarded as $L^{\infty}$.

Let $\Delta^{N}=\left\\{\mu\in\mathbb{R}_{+}^{N}\hskip 2.0pt:\hskip
2.0pt\mathbb{E}[\mu]=1\right\\}$, it is easy to see that we can identify
$\mathcal{M}_{1}$ with $\Delta^{N}$. Since we are working in $\mathbb{R}^{N}$,
the $w\left(\mathbb{R}^{N},\mathbb{R}^{N}\right)$ topology
($w\left(L^{\infty},\mathcal{M}\right)$ and
$w\left(\mathcal{M},L^{\infty}\right)$) is the standard euclidean topology.
Therefore, there will be no mention in this section as to which topology we
are dealing with. Note that any closed set $\mathcal{R}\subseteq\Delta^{N}$ is
compact. Finally, Fatou continuity is implied by any coherent risk measure
$\rho:\mathbb{R}^{N}\to\mathbb{R}$. Thus, there will be no mention of
continuity in this section.

To describe law invariance in this space define
$S_{N}=\\{\sigma:[N]\to[N]\mid\sigma\text{ is a bijection}\\}$. For $\sigma\in
S_{N}$ define the map $T_{\sigma}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ where
$(T_{\sigma}(X))_{n}=X_{\sigma(n)}$.

######  Definition 3.

We say that a mapping $\rho:\mathbb{R}^{N}\to\mathbb{R}$ is _permutation
invariant_ if $\rho(T_{\sigma}(X))=\rho(X)$ for all $\sigma\in S_{N}$ and
$X\in\mathbb{R}^{N}$. We say a set $\mathcal{R}\subseteq\mathbb{R}^{N}$ is
_permutation invariant_ if $T_{\sigma}(\mathcal{R})\subseteq\mathcal{R}$ for
all $\sigma\in S_{N}$.

It is easy to see that a mapping is law invariant if and only if it is
permutation invariant. If $\mathcal{R}\subseteq\mathbb{R}^{N}$ is permutation
invariant then $T_{\sigma}(\mathcal{R})\supseteq
T_{\sigma}(T_{\sigma^{-1}}(\mathcal{R}))=\mathcal{R}$ so we actually have
equality.

######  Proposition 17.

Let $\rho:\mathbb{R}^{N}\to\mathbb{R}$ be a coherent risk measure. Then $\rho$
is law invariant if and only if $\mathcal{S}_{\rho}$ is permutation invariant.

###### Proof.

Let $\mathcal{S}_{\rho}$ be permutation invariant then we have

$\displaystyle\rho(T_{\sigma}(X))=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}[\mu
T_{\sigma}(X)]$
$\displaystyle=\frac{1}{N}\sup_{\mu\in\mathcal{S}_{\rho}}(\mu\cdot
T_{\sigma}(X))$
$\displaystyle=\frac{1}{N}\sup_{\mu\in\mathcal{S}_{\rho}}(T_{\sigma^{-1}}(\mu)\cdot
X)=\frac{1}{N}\sup_{\mu\in\mathcal{S}_{\rho}}(\mu\cdot X)=\rho(X),$

for all $\sigma\in S_{n}$. Conversely, assume that $\mathcal{S}_{\rho}$ is not
permutation invariant. Find $R\in\mathcal{S}_{\rho}$ and $\sigma\in S_{n}$
such that $T_{\sigma}(R)\notin\mathcal{S}_{\rho}$. Since $\mathcal{S}_{\rho}$
is compact and convex, by the Hahn Banach separation theorem, we can find
$X\in\mathbb{R}^{N}$ such that

$\displaystyle\rho(X)=\frac{1}{N}\sup_{\mu\in\mathcal{S}_{\rho}}(\mu\cdot
X)<\frac{1}{N}X\cdot T_{\sigma}(R).$

Therefore we have

$\displaystyle\rho(X)<\frac{1}{N}X\cdot
T_{\sigma}(R)=\frac{1}{N}T_{\sigma^{-1}}(X)\cdot
R\leq\rho(T_{\sigma^{-1}}(X)).$

Implying that $\rho$ is not permutation invariant and thus not law invariant.
∎

###  A.2 Partially law-invariant coherent risk measures

We write $A\times B=\\{(a,b):a\in A,\leavevmode\nobreak\ b\in B\\}$ for two
sets $A$ and $B$.

Let $\Omega=[M]\times[N]$ for some $M,N\in\mathbb{N}$,
$\mathcal{F}=2^{\Omega}$, $\mathcal{G}$ is generated by the projection map
onto the second coordinate, and $\mathbb{P}$ is positive (that is,
$\mathbb{P}>0$ on $\mathcal{F}\setminus\\{\varnothing\\}$) and uniform in the
second coordinate (that is, $\mathbb{P}([M]\times\\{n\\})=1/N$ for all
$n\in[N]$).

We can identify elements $X\in L^{\infty}$ with $\mathbb{R}^{M\times N}$. For
the rest of the section, we will refer only to $\mathbb{R}^{M\times N}$ but
this is to be regarded as $L^{\infty}$. Let $\Delta^{M\times
N}=\\{\mu\in\mathbb{R}^{M\times N}_{+}\hskip 2.0pt:\hskip
2.0pt\mathbb{E}[\mu]=1\\}$, it is easy to see that $\mathcal{M}_{1}$ can be
identified with $\Delta^{M\times N}$. The same comments about topology and
Fatou continuity from the previous section also apply in this section.

For $k\in[N]$ define $E^{k}\in\mathbb{R}^{M\times N}$ by
$E^{k}_{mn}=\delta_{kn}$, where $\delta_{kn}$ is the Kronecker delta. Let
$\mathcal{E}=\left\\{E^{k}\hskip 2.0pt:\hskip 2.0ptk\in[N]\right\\}$, clearly
$\mathcal{E}$ is a basis for $L^{\infty}(\mathcal{G})$. Define the map
$\mathbb{L}:\mathbb{R}^{M\times
N}\to\mathbb{R}^{N}:X\mapsto[\mathbb{E}[X|\mathcal{G}]]_{\mathcal{E}}$. Given
some $X\in\mathbb{R}^{N}$ we will define $\tilde{X}\in\mathbb{R}^{M\times N}$
by $\tilde{X}=\sum_{k=1}^{N}X_{k}E^{k}$. A set
$\mathcal{R}\subseteq\mathbb{R}^{M\times N}$ is $\mathcal{G}$-law invariant if
$\mathbb{L}(\mathcal{R})$ is permutation invariant.

The following proposition gives necessary and sufficient conditions for the
supporting set of an $\mathcal{G}$-law invariant coherent risk measure.

######  Proposition 18.

Let $\rho:\mathbb{R}^{M\times N}\to\mathbb{R}$ be a coherent risk measure.
Then $\rho$ is $\mathcal{G}$-law invariant if and only if $\mathcal{S}_{\rho}$
is $\mathcal{G}$-law invariant.

###### Proof.

Define the mapping

$\displaystyle\tilde{\rho}:\mathbb{R}^{N}\to\mathbb{R}:X\mapsto\rho(\tilde{X}).$

It is easy to verify that this mapping is a coherent risk measure. Let
$\mathcal{S}_{\tilde{\rho}}\subseteq\Delta^{N}$ be the supporting set for
$\tilde{\rho}$. It follows that $\rho$ is $\mathcal{G}$-law invariant if and
only if $\tilde{\rho}$ is law invariant. For $X\in\mathbb{R}^{N}$ we have that

$\displaystyle\tilde{\rho}(X)=\rho(\tilde{X})=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\mu\tilde{X}\right]$
$\displaystyle=\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\mathbb{E}[\mu|\mathcal{G}]\tilde{X}\right]$
$\displaystyle=\frac{1}{N}\sup_{\mu\in\mathcal{S}_{\rho}}\mathbb{L}(\mu)\cdot
X=\frac{1}{N}\sup_{\mu\in\mathbb{L}(\mathcal{S}_{\rho})}\mu\cdot X.$

Thus $\mathbb{L}(\mathcal{S}_{\rho})\subseteq\mathcal{S}_{\tilde{\rho}}$ by
Proposition 2. Since $\mathbb{L}(\mathcal{S}_{\rho})$ is compact and convex
(image of a compact and convex set under a continuous linear map), the same
theorem states that
$\mathbb{L}(\mathcal{S}_{\rho})=\mathcal{S}_{\tilde{\rho}}$. We then have

$\displaystyle\rho\text{ is }\mathcal{G}\text{-law invariant }$
$\displaystyle\iff\tilde{\rho}\text{ is law invariant}$
$\displaystyle\iff\mathcal{S}_{\tilde{\rho}}\text{ is permutation invariant}$
$\displaystyle\iff\mathbb{L}(\mathcal{S}_{\rho})\text{ is permutation
invariant}$ $\displaystyle\iff\mathcal{S}_{\rho}\text{ is
}\mathcal{G}\text{-law invariant}.$

Thus proving that $\rho$ is $\mathcal{G}$-law invariant if and only if
$\mathcal{S}_{\rho}$ is $\mathcal{G}$-law invariant. ∎

###  A.3 Representation results

This section provides a representation result for $\mathcal{G}$-law-invariant
coherent risk measures. Define the set $\Delta_{\mathcal{G}}^{M\times
N}=\Delta^{M\times N}\cap L^{\infty}(\mathcal{G})$, which can be identified
with $\mathcal{M}_{1}(\mathcal{G})$.

######  Theorem 5.

The mapping $\rho:\mathbb{R}^{M\times N}\to\mathbb{R}$ is a coherent risk
measure if and only if there exists a unique compact and convex set
$\mathcal{R}\subseteq\Delta_{\mathcal{G}}^{M\times N}$ and a CA-assignment
$(\tau_{\mu})_{\mu\in\mathcal{R}}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{R}}\left(\mathbb{E}\left[\mu
X\right]+\tau_{\mu}(X)\right),\hskip 3.0ptX\in L^{\infty}.$ (12)

Moreover, in (12), $\rho$ is $\mathcal{G}$-law invariant if and only if
$\mathcal{R}$ is $\mathcal{G}$-law invariant.

###### Proof.

Let $\rho$ be a coherent risk measure. Define the set
$\mathcal{R}=\left\\{\mathbb{E}[\mu|\mathcal{G}]:\mu\in\mathcal{S}_{\rho}\right\\}$,
it is easy to see that $\mathcal{R}\subseteq\Delta_{\mathcal{G}}^{M\times N}$.
The set $\mathcal{R}$ is compact and convex since it is the image of a compact
convex set under a (continuous) linear map. For $\mu\in\mathcal{R}$ define the
set
$\mathcal{S}(\mu)=\left\\{\nu\in\mathcal{S}_{\rho}:\mathbb{E}[\nu|\mathcal{G}]=\mu\right\\}$.
It is clear that
$\mathcal{S}_{\rho}=\bigcup_{\mu\in\mathcal{R}}\mathcal{S}(\mu)$. We have

$\displaystyle\rho(X)=\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}[\mu
X]=\sup_{\mu\in\mathcal{R}}\sup_{\nu\in\mathcal{S}(\mu)}\mathbb{E}[\nu X]$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\sup_{\nu\in\mathcal{S}(\mu)}(\mathbb{E}[\mu
X]+\mathbb{E}[(\nu-\mu)X])$
$\displaystyle=\sup_{\mu\in\mathcal{R}}\left(\mathbb{E}[\mu
X]+\tau_{\mu}(X)\right)$

for all $X\in L^{\infty}$, where
$\tau_{\mu}(X)=\sup_{\nu\in\mathcal{S}(\mu)}\mathbb{E}[(\nu-\mu)X]$. The fact
that $\tau_{\mu}\in\mathrm{CA}(\mu)$ follows a similar argument as the proof
of Proposition 10. Since $\mathcal{R}$ supports
$\rho|_{L^{\infty}(\mathcal{G})}$ and is closed and convex, it is unique from
Proposition 2. For the converse, note that (9) is the supremum of coherent
risk measures (by Proposition 9; see (8)) and thus a coherent risk measure.

To prove the last claim, let
$\tilde{\rho}:\mathbb{R}^{N}\to\mathbb{R}:X\mapsto\rho(\tilde{X})$ and
$\mathcal{S}_{\tilde{\rho}}\subseteq\Delta^{N}$. We have

$\displaystyle\tilde{\rho}(X)=\rho(\tilde{X})=\sup_{\mu\in\mathcal{R}}\mathbb{E}[\mu\tilde{X}]=\frac{1}{N}\sup_{\mu\in\mathbb{L}(\mathcal{R})}\mu\cdot
X$

for all $X\in\mathbb{R}^{N}$. Therefore
$\mathbb{L}(\mathcal{R})=\mathcal{S}_{\tilde{\rho}}$, and

$\displaystyle\rho\text{ is }\mathcal{G}\text{-law
invariant}\iff\tilde{\rho}\text{ is law invariant}$
$\displaystyle\iff\mathcal{S}_{\tilde{\rho}}\text{ is permutation invariant}$
$\displaystyle\iff\mathcal{R}\text{ is }\mathcal{G}\text{-law invariant}.$

Thus proving that $\rho$ is $\mathcal{G}$-law invariant if and only if
$\mathcal{R}$ is $\mathcal{G}$-law invariant. ∎

Following the same argument in Remark 5, we obtain a representation in the
form of Theorem 1; that is, …

##  Appendix B Some additional results accompanying Section 3

######  Proposition 19.

The mapping $\mathbb{E}[\cdot|\mathcal{G}]:L^{\infty}\to
L^{\infty}(\mathcal{G})$ is
$w\left(L^{\infty},\mathcal{M}\right)/w\left(L^{\infty}(\mathcal{G}\right),\mathcal{M}(\mathcal{G}))$-continuous.

###### Proof.

Let $X_{\lambda}\to X$ in $w\left(L^{\infty},\mathcal{M}\right)$ and
$\mu\in\mathcal{M}(\mathcal{G})$. We have

$\displaystyle\mathbb{E}^{\mu}[\mathbb{E}[X_{\lambda}|\mathcal{G}]]=\mathbb{E}[\mathbb{E}[X_{\lambda}|\mathcal{G}]\eta_{\mu}]=\mathbb{E}[X_{\lambda}\eta_{\mu}]\to\mathbb{E}[X\eta_{\mu}]=\mathbb{E}^{\mu}[\mathbb{E}[X|\mathcal{G}]].$

Showing that $\mathbb{E}[X_{\lambda}|\mathcal{G}]\to\mathbb{E}[X|\mathcal{G}]$
in $w\left(L^{\infty}(\mathcal{G}\right),\mathcal{M}(\mathcal{G}))$. ∎

######  Proposition 20.

If $X_{\lambda}\to X$ in $w\left(L^{1},L^{\infty}\right)$ then
$\mathbb{E}[X_{\lambda}|\mathcal{G}]\to\mathbb{E}[X|\mathcal{G}]$ in
$w\left(L^{1},L^{\infty}\right)$.

###### Proof.

Let $X_{\lambda}\to X$ in $w\left(L^{1},L^{\infty}\right)$ and $Y\in
L^{\infty}$, we have

$\displaystyle\mathbb{E}[\mathbb{E}[X_{\lambda}|\mathcal{G}]Y]=\mathbb{E}[X_{\lambda}\mathbb{E}[Y|\mathcal{G}]]\to\mathbb{E}[X\mathbb{E}[Y|\mathcal{G}]]=\mathbb{E}[\mathbb{E}[X|\mathcal{G}]Y].$

Implying that
$\mathbb{E}[X_{\lambda}|\mathcal{G}]\to\mathbb{E}[X|\mathcal{G}]$ in
$w\left(L^{1},L^{\infty}\right)$. ∎

######  Corollary 3.

The set $L^{1}(\mathcal{G})$ is $w\left(L^{1},L^{\infty}\right)$-closed in
$L^{1}$.

###### Proof.

Let $X_{\lambda}\in L^{1}(\mathcal{G})$ and $X_{\lambda}\to X$ in
$w\left(L^{1},L^{\infty}\right)$. This also implies that
$X_{\lambda}\to\mathbb{E}[X|\mathcal{G}]$ in $w\left(L^{1},L^{\infty}\right)$
by Proposition 20. Since $w\left(L^{1},L^{\infty}\right)$ is Hausdorff,
$X=\mathbb{E}[X|\mathcal{G}]\implies X\in L^{1}(\mathcal{G})$. Therefore
$L^{1}(\mathcal{G})$ is $w\left(L^{1},L^{\infty}\right)$-closed in $L^{1}$. ∎

##  Appendix C Additional results and proofs accompanying Section 4

######  Proposition 21.

The mapping
$R:\mathcal{M}\to\mathcal{M}(\mathcal{G}):\mu\mapsto\mu|_{\mathcal{G}}$ is
$w(\mathcal{M},L^{\infty})/w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$-continuous.

###### Proof.

Let $\mu_{\lambda},\mu\in\mathcal{M}$ with $\lim_{\lambda}\mu_{\lambda}=\mu$
in $w(\mathcal{M},L^{\infty})$. Fix some $X\in L^{\infty}(\mathcal{G})$. We
have

$\displaystyle\mathbb{E}^{R(\mu_{\lambda})}[X]=\mathbb{E}^{\mu_{\lambda}}[X]\to\mathbb{E}^{\mu}[X]=\mathbb{E}^{R(\mu)}[X].$

Therefore $R(\mu_{\lambda})\to R(\mu)$ in
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$. ∎

###### Proof of Proposition 10.

$(\mathrm{i})\Rightarrow(\mathrm{ii})$: Assume that $\tau$ is a Fatou
$\mu$-coherent adjustment, by Proposition 9,
$\rho:L^{\infty}\to\mathbb{R}:X\mapsto\mathbb{E}[\eta_{\mu}X]+\tau(X)$ is a
Fatou coherent risk measure. We have

$\displaystyle\mathbb{E}\left[\eta_{\mu}X\right]+\tau(X)=\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\eta_{\nu}X\right],\hskip
3.0ptX\in L^{\infty}.$

By Proposition 7, $\mathcal{S}^{\mathcal{G}}_{\rho}$ must be contained in the
supporting set of
$\rho|_{L^{\infty}(\mathcal{G})}=\mathbb{E}^{\mu}|_{L^{\infty}(\mathcal{G})}$,
and therefore $\mathcal{S}^{\mathcal{G}}_{\rho}=\\{\mu\\}$ (meaning
$\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$ for all
$\nu\in\mathcal{S}_{\rho}$). Note that

$\displaystyle\mathbb{E}[\eta_{\mu}X]+\tau(X)=\mathbb{E}[\eta_{\mu}X]+\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}[(\eta_{\nu}-\eta_{\mu})X],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty},$

implying
$\tau(X)=\sup_{\nu\in\mathcal{S}_{\rho}}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X\right]$,
$X\in L^{\infty}$. Let $\mathcal{S}_{\tau}=\mathcal{S}_{\rho}$, we have

$\displaystyle\mathcal{S}_{\tau}=\mathcal{S}_{\rho}$
$\displaystyle=\left\\{\nu\in\mathcal{M}_{1}\mid\mathbb{E}\left[\eta_{\nu}X\right]\leq\rho(X)\text{
for all }X\in L^{\infty}\right\\}$
$\displaystyle=\left\\{\nu\in\mathcal{M}_{1}\mid\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X\right]\leq\tau(X)\text{
for all }X\in L^{\infty}\right\\}.$

The uniqueness follows directly from Proposition 2.
$(\mathrm{ii})\Rightarrow(\mathrm{iii})$: Obvious.
$(\mathrm{iii})\Rightarrow(\mathrm{i})$: Let
$\mathcal{R}\subseteq\mathcal{M}_{1}$ have the stated properties. Proving
$\tau(X)=\sup_{\nu\in\mathcal{R}}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X\right]$
is a $\mu$-coherent adjustment is straightforward. Let
$(X_{n})_{n=1}^{\infty}$ be a bounded sequence that converges
$\mathbb{P}$-a.s. to $X$. By the DCT,
$\lim_{n\to\infty}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X_{n}\right]=\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X\right]$.
Therefore,

$\displaystyle\tau(X)=\sup_{\nu\in\mathcal{R}}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X\right]=\sup_{\nu\in\mathcal{R}}\liminf\limits_{n\to\infty}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X_{n}\right]$
$\displaystyle\leq\liminf\limits_{n\to\infty}\sup_{\nu\in\mathcal{R}}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X_{n}\right]$
$\displaystyle=\liminf\limits_{n\to\infty}\tau(X_{n}),$

proving the Fatou continuity of $\tau$. ∎

###### Proof of Corollary 1.

$(\mathrm{i})\Rightarrow(\mathrm{ii})$: This follows from the representation
given in Proposition 7 and by the fact that the supremum of continuous
functions is lower semicontinuous. $(\mathrm{ii})\Rightarrow(\mathrm{iv})$:
The sum of lower semicontinuous functions is lower semicontinuous.
$(\mathrm{iv})\Rightarrow(\mathrm{iii})$: This follows since
$\mathbb{E}^{\mu}+\tau$ is a coherent risk measure that is lower
$w(L^{\infty},\mathcal{M})$-lower semicontinuous.
$(\mathrm{iii})\Rightarrow(\mathrm{i})$: Since $\mathbb{E}^{\mu}+\tau$ is
Fatou, we can use a similar argument as Proposition 7 to show that $\tau$ has
the representation Proposition 10 (ii). Thus by Proposition 7, the mapping
$\tau$ is Fatou. ∎

The next proposition, parallel to Proposition 10, characterizes Lebesgue-
continuous $\mu$-coherent adjustments.

######  Proposition 22.

Let $\tau:L^{\infty}\to\mathbb{R}$ and $\mu\in\mathcal{M}_{1}(\mathcal{G})$.
The following are equivalent.

  1. (i)

The mapping $\tau$ is a Lebesgue-continuous $\mu$-coherent adjustment;

  2. (ii)

there exists a $w(\mathcal{M},L^{\infty})$-compact and convex set
$\mathcal{S}_{\tau}\subseteq\mathcal{M}_{1}$ such that
$\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$ for all
$\nu\in\mathcal{S}_{\tau}$ and
$\tau(X)=\sup_{\nu\in\mathcal{S}_{\tau}}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\right\\}$,
$X\in L^{\infty}$;

  3. (iii)

there exists $\mathcal{R}\subseteq\mathcal{M}_{1}$ that is
$w(\mathcal{M},L^{\infty})$-precompact such that
$\mathbb{E}[\eta_{\nu}|\mathcal{G}]=\eta_{\mu}$ for all $\nu\in\mathcal{R}$
and
$\tau(X)=\sup_{\nu\in\mathcal{R}}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\right\\}$,
$X\in L^{\infty}$.

In case (ii) holds, the set $\mathcal{S}_{\tau}$ is uniquely given by
$\left\\{\nu\in\mathcal{M}_{1}:\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\leq\tau(X)\text{
for all }X\in L^{\infty}\right\\}$.

###### Proof.

Everything follows the same line of reasoning as Proposition 10, except for
proving Lebesgue continuity in $(\mathrm{iii})\implies(\mathrm{i})$. Let
$(X_{n})_{n=1}^{\infty}$ be a sequence in $L^{\infty}$ of uniformly bounded
random variables that converge $\mathbb{P}$-a.s. to $X\in L^{\infty}$. Since
$X\mapsto\sup_{\nu\in\mathcal{R}}\mathbb{E}[\eta_{\nu}X]$ is a Lebesgue-
continuous coherent risk measure, we have

$\displaystyle\lim_{n\to\infty}\tau(X_{n})=\lim_{n\to\infty}\sup_{\nu\in\mathcal{R}}\mathbb{E}\left[\left(\eta_{\nu}-\eta_{\mu}\right)X_{n}\right]$
$\displaystyle=\lim_{n\to\infty}\sup_{\nu\in\mathcal{R}}\mathbb{E}\left[\eta_{\nu}X_{n}\right]-\lim_{n\to\infty}\mathbb{E}[\eta_{\mu}X_{n}]$
$\displaystyle=\sup_{\nu\in\mathcal{R}}\mathbb{E}[\eta_{\nu}X]-\mathbb{E}[\eta_{\mu}X]=\tau(X).$

Proving the Lebesgue continuity of $\tau$. ∎

###### Proof of Proposition 12.

Let $\tilde{\rho}=\rho|_{L^{\infty}(\mathcal{G})}$, by Proposition 8, we have
that $\mathcal{S}^{\mathcal{G}}_{\rho}=\mathcal{S}_{\tilde{\rho}}$ is compact.
Let $(\tau_{\mu})_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}$ be the same as in
the construction of representation (9) in the proof of Theorem 2. We claim
that for all $\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$, $\mathcal{S}(\mu)$ is
$w(\mathcal{M},L^{\infty})$-compact and convex. Convexity of
$\mathcal{S}(\mu)$ for all $\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$ follows
from convexity of $\mathcal{S}_{\rho}$ and its definition. For compactness
note that $\mathcal{S}(\mu)$ is $w(\mathcal{M},L^{\infty})$-precompact
($\mathcal{S}(\mu)\subseteq\mathcal{S}_{\rho}$) for all
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$. Therefore, compactness of
$\mathcal{S}(\mu)$ would follow if we prove that $\mathcal{S}(\mu)$ is
$w(\mathcal{M},L^{\infty})$-closed for all
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$. This is a consequence of the fact
that
$\mathcal{S}(\mu)=\mathcal{S}_{\rho}\cap\\{\nu\in\mathcal{M}:\nu|_{\mathcal{G}}=\mu\\}$
for all $\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$, which are both
$w(\mathcal{M},L^{\infty})$-closed. Recall that for all
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$,
$\tau_{\mu}(X)=\sup_{\nu\in\mathcal{S}(\mu)}\left\\{\mathbb{E}^{\nu}[X]-\mathbb{E}^{\mu}[X]\right\\}$,
$X\in L^{\infty}$, we have by Proposition 22 that $\tau_{\mu}$ is Lebesgue
continuous for all $\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$.

We will now show that $(\tau_{\mu})_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}$
is regular. First, to show concavity (a), let
$\mu_{1},\mu_{2}\in\mathcal{S}_{\rho}^{\mathcal{G}}$,
$\nu_{1}\in\mathcal{S}(\mu_{1}),\nu_{2}\in\mathcal{S}(\mu_{2})$, $X\in
L^{\infty}$, and $\lambda\in(0,1)$. Since $\mathcal{S}_{\rho}$ is convex we
have
$\lambda\nu_{1}+(1-\lambda)\nu_{2}\in\mathcal{S}(\lambda\mu_{1}+(1-\lambda)\mu_{2})$,
and thus

$\displaystyle\mathbb{E}\left[\left(\eta_{\nu_{1}}-\eta_{\mu_{1}}\right)\lambda
X\right]+\mathbb{E}\left[\left(\eta_{\nu_{2}}-\eta_{\mu_{2}}\right)(1-\lambda)X\right]\leq\tau_{\lambda\mu_{1}+(1-\lambda)\mu_{2}}(X).$
(13)

Taking the supremum of the left side of (13) over
$(\nu_{1},\nu_{2})\in\mathcal{S}(\mu_{1})\times\mathcal{S}(\mu_{2})$ obtains
the result. To show semiconitinuity (b), let
$(\mu_{\lambda})_{\lambda\in\Lambda}$ be a net in
$\mathcal{S}_{\rho}^{\mathcal{G}}$ and
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$ such that $\mu_{\lambda}\to\mu$ in
$w(\mathcal{M}(\mathcal{G}),L^{\infty}(\mathcal{G}))$ and let $X\in
L^{\infty}$. For each $\lambda$ choose
$\nu_{\lambda}^{*}\in\mathcal{S}(\mu_{\lambda})$ such that
$\tau_{\mu_{\lambda}}(X)=\mathbb{E}[(\eta_{\nu_{\lambda}^{*}}-\eta_{\mu_{\lambda}})X]$,
which is possible since $\mathcal{S}(\mu_{\lambda})$ is
$w(\mathcal{M},L^{\infty})$-compact. Since the net
$(\nu_{\lambda}^{*})_{\lambda\in\Lambda}$ takes values in the
$w(\mathcal{M},L^{\infty})$-compact set $\mathcal{S}_{\rho}$, there exists a
convergent subnet $(\nu_{\gamma}^{*})_{\gamma\in\Gamma}$. Denote the limit of
this subnet by $\nu\in\mathcal{S}_{\rho}$. For $Y\in L^{\infty}(\mathcal{G})$,
by using the tower property and the congverence, we get

$\displaystyle\mathbb{E}\left[\mathbb{E}[\eta_{\nu}|\mathcal{G}]Y\right]=\mathbb{E}\left[\eta_{\nu}Y\right]=\lim_{\gamma}\mathbb{E}\left[\eta_{\nu_{\gamma}^{*}}Y\right]=\lim_{\gamma}\mathbb{E}\left[\eta_{\mu_{\gamma}}Y\right]=\mathbb{E}[\eta_{\mu}Y].$

Thus $\nu\in\mathcal{S}(\mu)$ and

$\displaystyle\tau_{\mu}(X)\geq\mathbb{E}\left[(\eta_{\nu}-\eta_{\mu})X\right]=\lim_{\gamma}\mathbb{E}\left[\left(\eta_{\nu^{*}_{\gamma}}-\eta_{\mu_{\gamma}}\right)X\right]=\lim_{\gamma}\tau_{\mu_{\gamma}}(X)$
$\displaystyle=\liminf_{\gamma}\tau_{\mu_{\gamma}}(X)\geq\liminf_{\lambda}\tau_{\mu_{\lambda}}(X).$

This shows that $(\tau_{\mu})_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}$ is
regular. Now we prove the uniqueness of
$(\tau_{\mu})_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}$ by assuming that there
exists another regular Lebesgue-continuous CA-assignment
$(\tilde{\tau}_{\mu})_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}$ such that

$\displaystyle\rho(X)=\sup_{\mu\in\mathcal{S}^{\mathcal{G}}_{\rho}}\left\\{\mathbb{E}^{\mu}[X]+\tilde{\tau}_{\mu}(X)\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

We will show that $\tau_{\mu}=\tilde{\tau}_{\mu}$ for all
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$. For each
$\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}$, let
$\mathcal{R}(\mu)\subseteq\mathcal{M}_{1}$ be the
$w(\mathcal{M},L^{\infty})$-compact convex set that gives the representation
in Proposition 22 for $\tilde{\tau}_{\mu}$. Let
$\tilde{\mathcal{S}}=\bigcup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\mathcal{R}(\mu)\subseteq\mathcal{M}_{1}$.
Note that

$\displaystyle\sup_{\nu\in\tilde{\mathcal{S}}}\mathbb{E}[\eta_{\nu}X]=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\left\\{\sup_{\nu\in\mathcal{R}(\mu)}\mathbb{E}[\eta_{\nu}X]\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\left\\{\mathbb{E}[\eta_{\mu}X]+\sup_{\nu\in\mathcal{R}(\mu)}\mathbb{E}[\left(\eta_{\nu}-\eta_{\mu}\right)X]\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{S}_{\rho}^{\mathcal{G}}}\left\\{\mathbb{E}[\eta_{\mu}X]+\tilde{\tau}_{\mu}(X)\right\\}=\rho(X),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X\in L^{\infty}.$

Therefore $\operatorname*{conv}(\tilde{\mathcal{S}})$ is
$w(\mathcal{M},L^{\infty})$-dense in $\mathcal{S}_{\rho}$ by Propostion 2. We