A new view on risk measures associated with acceptance sets

Scalarization and separation of sets are important topics for many fields of research in mathematics and mathematical economics like multiobjective optimization, risk theory, optimization under uncertainty and financial mathematics. For a given real vector space $\mathcal{X}$ over $\mathbb{R}$, nonlinear translation invariant functionals $\varphi_{\mathcal{A},K}:\mathcal{X}\rightarrow\mathbb{R}\cup\{-\infty\}\cup\{+\infty\}$ are considered with

$$\varphi_{\mathcal{A},K}(X):=\inf\{t\in{\mathbb{R}}\mid X\in tK+\mathcal{A}\}$$

(1.1)

where $\varnothing\neq\mathcal{A}\subseteq\mathcal{X}$ and $K\in\mathcal{X}\backslash\{0\}$ such that $\mathcal{A}-\mathbb{R}_{+}K\subseteq\mathcal{A}$. Functionals given by (1.1) are used by Gerstewitz in [23] for deriving separation theorems for not necessarily convex sets as separating functional and as scalarization functional of vector optimization problems. Formulations of $\varphi_{\mathcal{A},K}$ convenient in risk theory are subject of this paper. Relationships between coherent risk measures and the functional (1.1) are studied by Artzner et al. [1] and Jaschke, Küchler [30] (see also [27], [26], [38], [22] and references therein). For a given acceptance set $\mathcal{A}$ in the space of financial positions $\mathcal{X}$, we consider the functional

$\displaystyle\mathcal{X}\ni X\mapsto\rho_{\mathcal{A},\mathcal{M},\pi}(X):=\inf\{\pi(Z)\mid Z\in\mathcal{M},X+Z\in\mathcal{A}\}$

(1.2)

following Farkas et al. [15] and Baes et al. [3], where $\pi\colon\mathcal{M}\rightarrow\mathbb{R}$ is a pricing functional defined on a set of allowed movements $\mathcal{M}\subseteq\mathcal{X}$ by which an investor can change the current financial position $X$. The functional (1.2) is a risk measure describing the costs for satisfying regulatory preconditions. It is a generalization of (1.1) through the simultaneous allowance of a set of directions instead of one fixed direction $K\in\mathcal{X}$ and the valuation of the movements by a general functional $\pi$. However, in the so called Reduction Lemma in [15], Farkas et al. showed that $\rho_{\mathcal{A},\mathcal{M},\pi}$ can be reduced to a functional from type (1.1). It is shown that this class of functionals from (1.1) coincides with the class of translation invariant functions and, thus, such functionals are employed as coherent risk measures in risk theory (see [1]), since translation invariance is a basic property of risk measures.

Considering risk measures with respect to acceptance sets is from special interest in financial mathematics. Since the financial crisis revealed many deficits in risk taking and management of financial institutions, regulation acquired intensively greater importance in recent years, leading to restrictions for business activities like minimum capital requirements for credit risk of banks, see e.g. Basel III preconditions in [6] and [7]. That leads to partly massive restrictions of possible investment stategies and to reduced gains by a risk-return trade-off, see e.g. [4]. Thus, finding optimal portfolios under given regulatory acceptance conditions is essential for the survive of an institution. In [34], we followed Baes et al. in [3] and considered a portfolio optimization problem with the aim of fulfill regulatory preconditions (we also say “reaching acceptability” because they are described by an acceptance set $\mathcal{A}$) under minimal costs for closed acceptance sets.

The paper is organized as follows: After a short overview about the needed mathematical background, we describe the basic financial model, which we consider in Sections 3 and 4. This includes the vector space of financial positions $\mathcal{X}$, the set of eligible payoffs $\mathcal{M}\subseteq\mathcal{X}$ and the linear pricing functional $\pi\colon\mathcal{M}\rightarrow\mathbb{R}$. Here, we give a definition for acceptance sets $\mathcal{A}\subseteq\mathcal{X}$ by Baes et al. in [3], but we do not additionally assume closedness. Moreover, we state some very important examples from a practical point of view. Afterwards, we study properties of the risk measure $\rho_{\mathcal{A},\mathcal{M},\pi}$ in Section 5.

In this article, certain standard notions and terminology are used. Let $\mathcal{X}$ be a vector space over $\mathbb{R}$. The extended set of real numbers is denoted by $\overline{\mathbb{R}}:=\mathbb{R}\cup\{-\infty,+\infty\}$, the set of non-negative real numbers by $\mathbb{R}_{+}$ , the set of positive real numbers by $\mathbb{R}_{>}$ and the set of non-positive real numbers by $\mathbb{R}_{-}$. Consider subsets $\mathcal{A},B\subseteq\mathcal{X}$. Then,

$\displaystyle\mathcal{A}+\mathcal{B}:=\{X+Y\mid X\in\mathcal{A},Y\in\mathcal{B}\}$

is the Minkowksi sum of $\mathcal{A}$ and $\mathcal{B}$. For simplicity, we replace $\{X\}+\mathcal{A}$ by $X+\mathcal{A}$ for the sum of a set $\mathcal{A}$ and a single element $X\in\mathcal{X}$. We set

$\displaystyle\lambda\mathcal{A}=\{\lambda X\mid X\in\mathcal{A}\}\text{ with }\lambda\in\mathbb{R},$

which leads for the Minkowski sum of $\mathcal{A}$ and $-\mathcal{B}$ to

$\displaystyle\mathcal{A}-\mathcal{B}=\{X-Y\mid X\in\mathcal{A},Y\in\mathcal{B}\}.$

The cardinality of $\mathcal{A}$ is denoted by $\left|\mathcal{A}\right|$. $\mathcal{A}$ is star-shaped (around $0$) if

$\displaystyle\forall X\in\mathcal{A},\forall\lambda\in[0,1]:\quad\lambda X\in\mathcal{A}.$

and $\mathcal{A}$ is convex if

$\displaystyle\forall X,Y\in\mathcal{A},\forall\lambda\in[0,1]:\quad\lambda X+(1-\lambda)Y\in\mathcal{A}.$

It is obvious that $\mathcal{A}$ is star-shaped if it is convex and $0\in\mathcal{A}$. A nonempty set $\mathcal{A}$ is called a cone if

$\displaystyle\forall\ X\in\mathcal{A},\forall\lambda\geq 0:\ \lambda X\in\mathcal{A}.$

Let $\mathcal{C}\subseteq\mathcal{X}$ be a cone. Then, $\mathcal{C}$ is star-shaped. Furthermore, $\mathcal{C}$ is said to be proper or nontrivial if $\{0\}\subsetneq\mathcal{C}\neq\mathcal{X}$. We call $\mathcal{C}$ pointed if $\mathcal{C}\cap(-\mathcal{C})=\{0\}$ and reproducing if $\mathcal{C}-\mathcal{C}=\mathcal{X}$. For each cone $\mathcal{C}$ holds

$\displaystyle\mathcal{C}\text{ convex }\quad\Longleftrightarrow\quad\mathcal{C}+\mathcal{C}\subseteq\mathcal{C}.$

A cone from special interest is the recession cone of a subset $\mathcal{A}\subset\mathcal{X}$, i.e.,

$\displaystyle\mathrm{rec}\hskip 1.70709pt\mathcal{A}:=\{X\in\mathcal{X}\mid Y+\lambda X\in\mathcal{A}\ \text{ for all }Y\in\mathcal{A},\lambda\in\mathbb{R}_{+}\}.$

The following properties will be useful and are collected from [41]:

Now, we suppose that $(\mathcal{X},\tau)$ is a topological vector space over $\mathbb{R}$. Note that we just write $\mathcal{X}$ for $(\mathcal{X},\tau)$ if $\tau$ is obvious or not from interest. Elements of $\tau$ are called open sets and elements of $\mathcal{X}\backslash\tau$ are called closed. If $\{0\}$ is closed, $\mathcal{X}$ is called Hausdorff. We say that $\mathcal{X}$ is locally convex if it exists $\{\mathcal{U}_{i}\}_{i\in I}\subseteq\tau$ of $0$ such that the sets $\mathcal{U}_{i}$ are convex. For $\mathcal{A}\subseteq\mathcal{X}$, $\mathrm{cl}\hskip 1.70709pt(\mathcal{A})$ denotes the closure, $\mathrm{bd}\hskip 1.70709pt(\mathcal{A})$ the boundary and $\mathrm{int}\hskip 1.70709pt(\mathcal{A})$ the interior of $\mathcal{A}$. The following lemma gives a connection between topological and directional properties of $\mathcal{A}$:

We call $(X,\left\|\cdot\right\|)$ a normed vector space where $\mathcal{X}$ is a vector space (here assumed to be) over $\mathbb{R}$ and $\left\|\cdot\right\|$ denotes a norm on $\mathcal{X}$, i.e., a map $\left\|\cdot\right\|\colon\mathcal{X}\rightarrow\mathbb{R}_{+}$ such that for every $X,Y\in\mathcal{X}$ and $\alpha\in\mathbb{R}$ the following holds: (i) $\left\|X\right\|=0\Rightarrow X=0$, (ii) $\left\|\alpha X\right\|=\left|\alpha\right|\left\|X\right\|$ and (iii) the triangle inequality $\left\|X+Y\right\|\leq\left\|X\right\|+\left\|Y\right\|$. We write just $\mathcal{X}$ for $(\mathcal{X},\left\|\cdot\right\|)$ if the norm is obvious or not from interest. A sequence $(X_{k})_{k\in\mathbb{N}}\subseteq\mathcal{X}$ is said to converge in $\mathcal{X}$ to the limit $X\in\mathcal{X}$ (and we write $X_{k}\rightarrow X$ for $k\rightarrow+\infty$) if $\lim\limits_{k\rightarrow+\infty}\left\|X_{k}-X\right\|=0$. We say $(\mathcal{X},\left\|\cdot\right\|)$ is complete if each Cauchy sequence has a limit in $\mathcal{X}$. A complete normed vector space is called Banach space. The set

$\displaystyle\mathcal{B}_{r}(Z):=\left\{W\in\mathcal{X}\mid\left\|W-Z\right\|\leq r\right\}$

denotes the closed ball with center $Z\in\mathcal{X}$ and radius $r>0$.

Consider a map $f\colon\mathcal{X}\rightarrow\overline{\mathbb{R}}$ where $\mathcal{X}$ is a vector space over $\mathbb{R}$. Then,

$\displaystyle\mathrm{dom}f:=\{X\in\mathcal{X}\mid f(X)<+\infty\}$

is the domain of $f$ and

$\displaystyle\mathrm{epi}f:=\{(X,t)\in\mathcal{X}\times\mathbb{R}\mid f(X)\leq t\}$

the epigraph of $f$. We call $f\colon\mathcal{X}\rightarrow\overline{\mathbb{R}}$ proper if $\mathrm{dom}f\neq\varnothing$ and $f(X)>-\infty$ for all $X\in\mathcal{X}$. If $\mathrm{epi}f$ is convex, then $f$ is said to be convex. $f$ is said to be linear if

$\displaystyle\forall X,Y\in\mathcal{X},\forall\lambda,\mu\in\mathbb{R}:\quad f(\lambda X+\mu Y)=\lambda f(X)+\mu f(Y).$

The kernel or null space of a linear map $f$ is the subspace

$\displaystyle\ker f:=\{X\in\mathcal{X}\mid f(X)=0\}.$

We call

$\displaystyle\mathrm{lev}\hskip 1.70709pt_{f,=}(\alpha):=\{X\in\mathcal{X}\mid f(X)=\alpha\}$

level line,

$\displaystyle\mathrm{lev}\hskip 1.70709pt_{f,\leq}(\alpha):=\{X\in\mathcal{X}\mid f(X)\leq\alpha\}$

sublevel set and

$\displaystyle\quad\mathrm{lev}\hskip 1.70709pt_{f,<}(\alpha):=\{X\in\mathcal{X}\mid f(X)<\alpha\}$

strict sublevel set of $f$ to the level $\alpha\in\mathbb{R}$. For short, we denote each of the sets $\mathrm{lev}\hskip 1.70709pt_{f,=}(\alpha)$, $\mathrm{lev}\hskip 1.70709pt_{f,\leq}(\alpha)$ and $\mathrm{lev}\hskip 1.70709pt_{f,<}(\alpha)$ level set.

Now, we suppose that $\mathcal{X}$ is a topological vector space over $\mathbb{R}$. $f$ is called lower semicontinuous if $\mathrm{epi}f$ is closed. The set

$\displaystyle\mathcal{X}^{*}:=\{\varphi\colon\mathcal{X}\rightarrow\mathbb{R}\mid\varphi\text{ linear and continuous}\}$

denotes the topological dual space. If $\mathcal{X}$ is a locally convex Hausdorff space with $\mathcal{X}\neq\{0\}$, there is some non-trivial linear continuous functional, i.e., $\mathcal{X}^{*}\neq\{0\}$. Equivalently,

$\displaystyle\forall X\in\mathcal{X}\backslash\{0\}\ \exists\varphi\in\mathcal{X}^{*}:\qquad\varphi(X)>0.$

Consequently, for given $X,Y\in\mathcal{X}$ with $X\neq Y$, there exists $\varphi\in\mathcal{X}^{*}$ with $\varphi(X)\neq\varphi(Y)$.

In vector optimization and other applications it is necessary to compare elements $X,Y\in\mathcal{X}$. Hence, let $\mathcal{R}\subseteq\mathcal{X}\times\mathcal{X}$ be a binary relation on $\mathcal{X}$ and we set $X\mathcal{R}Y$ equivalently for $(X,Y)\in\mathcal{R}$, compare [24] for standard terminology and examples. For our purposes we remember that $\mathcal{R}$ is a partial order if it is reflexive, antisymmetric and transitive. $\mathcal{X}$ is said to be partially ordered by $\mathcal{R}$ if $\mathcal{R}$ is a partial order on $\mathcal{X}$. In the following, we write $\leq$ for $\mathcal{R}$.

Cones in $\mathcal{X}$ are suitable for describing binary relations on $\mathcal{X}$ and, especially, partial orders. The following theorem specifies that:

By Theorem 2.7, $\leq_{\mathcal{C}}$ from (2.1) is a partial order if and only if $\mathcal{C}\subseteq\mathcal{X}$ is a convex, pointed cone. Thus, we call a convex, pointed cone $\mathcal{C}$ an ordering cone. The corresponding partial order $\leq_{\mathcal{C}}$ is given by

$\displaystyle\forall X,Y\in\mathcal{X}:\ X\leq_{\mathcal{C}}Y:\Longleftrightarrow Y-X\in\mathcal{C}.$

(2.2)

For example, if $\mathcal{X}$ is partially ordered by $\leq$, the natural ordering cone in $\mathcal{X}$ is the positive cone

$\displaystyle\mathcal{X}_{+}=\{X\in\mathcal{X}\mid 0\leq X\}$

(2.3)

and the corresponding partial order is simplified denoted $\leq$ instead $\leq_{\mathcal{X}_{+}}$. Every element $X\in\mathcal{X}$ is called positive. One example for $\leq$ is the component-wise ordering on $\mathcal{X}=\mathbb{R}^{n}$.

Our basic market model refers to Baes et al. [3] and Farkas, Koch-Medina, Munari [15]. We consider an one-period-model of a financial market, where an investor choices his or her portfolio at the time $t=0$, which results in a capital position with some (in general) random payoff at the future time $t=1$. In this section, we present this model in more detail.

Throughout this paper, we consider a real vector space $\mathcal{X}$ called the space of capital positions, usually a space of random variables. Furthermore, let $\Omega$ be the set of possible states in $t=1$, $\mathcal{F}$ be a $\sigma$-Algebra on $\Omega$ and $\mathbb{P}\colon\mathcal{F}\rightarrow[0,1]$ a probability measure.

While Baes et al. [3] assume a locally convex Hausdorff topological vector space over $\mathbb{R}$ fulfilling the first axiom of countability and Farkas et al. [15] assume a topological vector space over $\mathbb{R}$, we suppose a real vector space $\mathcal{X}$, which we only extend to be equipped with a topology and further properties where necessary. Sometimes we speak of financial positions instead of capital positions $X\in\mathcal{X}$. Nevertheless, $X$ is the capital of an investor in the future time $t=1$ and is given by the residuum of assets and liabilities, i.e., positive outcomes are gains and negative outcomes are losses. $\mathcal{X}$ is typically chosen as a space like $\mathcal{L}^{p}$ with $1\leq p<+\infty$, see Example 2.6. In some situations, we consider a finite set $\Omega=(\omega_{1},\dots,\omega_{n})$, i.e., vectors $X\in\mathbb{R}^{n}$ with $X_{k}=X(\omega_{k})$ ($k=1,\dots,n$). For each event $\mathcal{B}\in\mathcal{F}$, we set

$\displaystyle X\in\mathcal{B}\ \mathbb{P}-\text{a.s.}\quad:=\quad\mathbb{P}(X\in\mathcal{B})=\mathbb{P}(\{\omega\in\Omega\mid X(\omega)\in\mathcal{B}\})=1,$

where ”a.s.” means ”almost surely”. If $X,Y\in\mathcal{X}$ are random variables, we write $X=Y$ and $X\neq Y$ if and only if $\mathbb{P}(X=Y)=1$ and $\mathbb{P}(X=Y)=0$, respectively. Note that $\mathcal{X}$ could be any normed vector space. Furthermore, we suppose $\mathcal{X}$ to be partially ordered by the pointed convex cone $\mathcal{X}_{+}$. The cone is represented by the order relation $\leq$ as given in (2.3). If we consider a space of random variables, e.g. $\mathcal{X}=\mathcal{L}^{p}(\Omega,\mathcal{F},\mathbb{P})$, we understand $0\leq X$ in the sense of $\mathbb{P}$-a.s.

In the following the superscript ^T always denotes transposed vectors. As mentioned in the beginning, an investor can invest into a finite set $\mathcal{S}$ of eligible assets

$\displaystyle S^{i}=(S_{0}^{i},S_{1}^{i})^{T},\quad i\in\{0,1,\dots,N\}$

(3.1)

with $N\in\mathbb{N}$ in time $t=0$. Here, $S_{0}^{i}\in\mathbb{R}$ denotes the price in $t=0$ for one unit of the liquid asset and $S_{1}^{i}\in\mathcal{X}$ denotes the (in general random) payoff in $t=1$ for each unit. We set

$\displaystyle S^{0}:=(1,(1+r)\mathds{1}_{\Omega})^{T}=(1,1+r)^{T}.$

(3.2)

$S^{0}$ describes a secure investment opportunity with interest rate $r\in\mathbb{R}_{+}$. Secure means that the payoff is a constant. For every constant random variable $X=c\cdot\mathds{1}_{\Omega}$, i.e., $\mathbb{P}(X=c)=1$, with $c\in\mathbb{R}$ arbitrary, we just write $X=c$. For a collection of all prices or payoffs, respectively, we set

$\displaystyle S_{j}:=(S_{j}^{0},S_{j}^{1},\dots,S_{j}^{N})^{T},\qquad j\in\{0,1\}.$

(3.3)

There has been many research concerning the case of one eligible asset (also, how to choose it), see e.g. Farkas et al. for a defaultable bond $\mathcal{X}=\mathcal{B}(\Omega)$ in [13] and [14]. Other spaces have been studied, too, see e.g. Kaina and Rüschendorf in [31] for $\mathcal{X}=\mathcal{L}^{p}$ or Cheridito and Li in [10] for Orlicz spaces. The actions a decision maker can take into account are described by suitable portfolios of the assets $S^{i}$. The subspace

$$\mathcal{M}:=\mathrm{span}\hskip 1.70709pt(1,S_{1}^{1},\dots,S_{1}^{N})$$

(3.4)

of $\mathcal{X}$ spanned by the secure payoff and $S_{1}^{i}$ with $i=1,\dots,N$ is called space of the eligible payoffs. We assume linear independent eligible payoffs. Consequently,

$\displaystyle 1<\dim(\mathcal{M})<+\infty.$

(3.5)

Every $Z\in\mathcal{M}$ describes the random payoff related to a portfolio of those eligible assets. To ensure $S_{1}^{0}=1\in\mathcal{M}\subseteq\mathcal{X}$ for cases like $\mathcal{X}=\mathcal{L}^{p}(\Omega,\mathcal{F},\mu)$, we have to assume a finite measure space, i.e., $\mu(\Omega)<+\infty$, which is guaranteed here by a probability measure $\mu=\mathbb{P}$, i.e., $\mathbb{P}(\Omega)=1$ for arbitrary $\Omega$. If $\mathcal{X}$ is a topological vector space, then $\mathcal{M}$ is equipped with the relative topology induced by $\mathcal{X}$ which is normable through $\dim\mathcal{M}<+\infty$. In the following, the fixed norm in $\mathcal{M}$ for topological vector spaces $\mathcal{X}$ is given by $\left\|\cdot\right\|$.

As we consider a financial market, we make the following typical assumptions:

Portfolios $x=(x_{0},x_{1},\dots,x_{N})^{T}\in\mathbb{R}^{N+1}$ with the same payoff have the same initial price by Law of One Price, which, especially, leads to a unique price for every eligible payoff $Z\in\mathcal{M}$. Following Baes et al. [3], we define a pricing functional $\pi\colon\mathcal{M}\rightarrow\mathbb{R}$ (with $\mathcal{M}$ given by (3.4)) as

$\displaystyle\pi(Z):=S_{0}^{T}x\ \text{ for all }x\in\mathbb{R}^{N+1}:\ Z=S_{1}^{T}x.$

(3.8)

Of course, $\pi$ is a linear operator. If $\mathcal{X}$ is a topological vector space, then $\pi$ is also continuous, since $\dim\mathcal{M}<+\infty$. An important property is the monotonicity of $\pi$. In contrast to [34], where the absence of good deals was required, we show that $\pi$ is always monotonically increasing, i.e.,

$\displaystyle\forall Z_{1},Z_{2}\in\mathcal{M}:\quad Z_{2}-Z_{1}\in\mathcal{X}_{+}\quad\Longrightarrow\quad\pi(Z_{1})\leq\pi(Z_{2}).$

(3.9)

All eligible assets with the same price $m\in\mathbb{R}$ are summarized by

$\displaystyle\pi_{m}:=\{Z\in\mathcal{M}\mid\pi(Z)=m\}.$

(3.10)

If there are $Z_{1},Z_{2}\in\mathcal{M}$ with $Z_{2}-Z_{1}\in\mathcal{X}_{+}$ and $Z_{1}\neq Z_{2}$, then $\mathcal{M}\cap\mathcal{X}_{+}\neq\{0\}$. Especially, we can consider $Z_{1}=0$. As in [3], we assume the existence of $U\in\mathcal{M}\cap\mathcal{X}_{+}$ with strict positive price, i.e.,

$\displaystyle\mathcal{X}_{+}\cap\bigcup_{m>0}\pi_{m}\neq\varnothing.$

Since $\pi$ is a linear functional, we can simplified assume $\pi(U)=1$.

Note that the Law of One Price from Assumption 1 is automatically fulfilled by Assumption 2 because the existence of $\pi$ is required in Assumption 2.

As observed in [15] for a topological vector space $\mathcal{X}$ and proved in [34], we can rewrite (3.10) by Assumption 2 for $\mathcal{X}$ being a vector space, since no topological properties are required in the proof:

Let $\mathcal{X}$ be a vector space above $\mathbb{R}$, $\mathcal{M}\subseteq\mathcal{X}$ the space of eligible payoffs $S^{i}$ from (3.4) fulfilling (3.5) and $\pi\colon\mathcal{M}\rightarrow\mathbb{R}$ a pricing functional given by (3.8). The so called acceptance set specifies those positions which are allowed to be occupied. The decision maker has to decide which actions can be undertaken to modify the current financial position such that the new one is acceptable if the current one is not. On the other hand, if the current position is already acceptable, the decision maker could set money free while the position is remaining acceptable and the money could be used otherwise. A suitable set for describing all capital positions $X\in\mathcal{X}$ being acceptable capitalized with respect to regulatory constraints can be defined in the following way (see Baes et al. [3] and Artzner et al. [1]):

Now, after we completed our financial setting, we consider the financial model (FM) in the space of capital positions $\mathcal{X}$ with the probability space ($\Omega,\mathcal{F},\mathbb{P}$) (see Section 3) throughout the paper:

$\displaystyle(\mathrm{FM}):\quad$ $\displaystyle S^{i}\in\mathbb{R}\times\mathcal{X}\ (i=0,1,\dots,N)\text{ is the }i\text{-th eligible asset from }\eqref{def:eligible_assets}$ $\displaystyle\text{with }S^{0}\text{ being the secure investment opportunity from \eqref{def:eligible_assets_secure}},$ $\displaystyle\mathcal{M}\text{ is the subspace of }\mathcal{X}\text{ given by }\eqref{calM}\text{ fulfilling }\eqref{eq:dim_M},$ $\displaystyle\pi\colon\mathcal{M}\rightarrow\mathbb{R}\text{ is the pricing functional given by }\eqref{def:pi_Z},$ $\displaystyle\mathcal{A}\subseteq\mathcal{X}\text{ is an acceptance set according to Definition \ref{def:acceptanceset}}$ and Assumption 1 is fulfilled.

The definition is motivated by Baes et al. [3], but we do not assume $\mathcal{A}$ to be closed, in general.

Acceptance sets are mostly given by risk measures in practice. An axiomatic approach to define (coherent) risk measures is introduced by Artzner et al. in [1] and generalized to convex risk measures by Föllmer and Schied in [18] and Frittelli and Rosazza Gianin in [21]:

Convex risk measures are important because only these take diversification into account. More exactly, diversification means that the decision maker invests a portion $\lambda\in[0,1]$ into a possible strategy or investment opportunity with output $X\in\mathcal{X}$ and the remaining part into another one with output $Y\in\mathcal{X}$. The convexity of the risk measure (see (4.2)) implies that diversification should not increase the risk, which can be expressed by the (for monetary risk measures as given by Definition 4.4) equivalent condition of quasi-convexity (see [16]), i.e.,

$\displaystyle\forall X,Y\in\mathcal{X},\forall\lambda\in[0,1]:\quad\rho(\lambda X+(1-\lambda)Y)\leq\max\{\rho(X),\rho(Y)\}.$

As mentioned in Example 2.6, typical spaces of capital positions are $\mathcal{L}^{p}$-space, especially with $1\leq p\leq+\infty$, see e.g. [21] and [18]. Biagini and Frittelli showed in [8] that there are no finite convex risk measures for $\mathcal{X}=\mathcal{L}^{p}$ with $0\leq p<1$ which are not constant.