This paper aims at giving a possible axiomatic foundation to the **risk** assess- ment of final payoffs when additional information is available. This is the case, for example, when the riskiness of a payoff occurring at time T is quantified at an intermediate date t ∈ (0, T ). We define conditional **convex** **risk** **measures** as maps, satisfying some natural axioms, which associate to every payoff, rep- resented by a random variable X, its riskiness ρ(X) which is itself a random variable, depending on the available information. Furthermore, under a mild technical assumption, we give a characterization of these maps as worst condi- tional expected loss with respect to a given set of probabilistic models, maybe corrected by some random penalty function. A new regularity property is in- troduced and several equivalent formulations are presented; this property, which is economically plain, states that ρ(X) should not depend on that part of the future which is ruled out by the additional information. As an example for con- ditional **convex** **risk** **measures**, the class of entropic **risk** **measures**, as defined in [7], is generalized to the conditional setting. These **risk** **measures** are first defined as capital requirements with respect to an **utility**-**based** acceptability criterion. Then their penalty functions are identified as the conditional relative entropy between the considered probabilistic models and a reference model. The last part of the paper is devoted to a study of dynamic **convex** **risk** **measures**, i.e. families of conditional **convex** **risk** **measures**, describing the **risk** assessment of a final payoff at successive dates. We introduce two economically motivated prop- erties of time consistency that relate different components of a dynamic **convex** **risk** measure. Finally, we provide some characterizations of these properties in terms of the family of penalty functions of their components.

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https://doi.org/10.1515/strm-2019-0002 Received February 2, 2019; accepted June 21, 2019
Abstract: The family of admissible positions in a transaction costs model is a random closed set, which is **convex** in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions **based** on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of **risk** **measures** of non-**convex** portfolios are presented.

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E Q [−X] .
The contribution of this paper is twofold. First we derive precise connections between **risk** measurement under the theories of variational, homothetic and multiple priors preferences
— (1.4) — and **risk** measurement using **convex** **measures** of **risk** — (1.5). In particular, we identify two subclasses of **convex** **risk** **measures** that we call entropy coherent and entropy **convex** **measures** of **risk**, and that include all coherent **risk** **measures**. We show that, under technical conditions, negative certainty equivalents under variational, homothetic, and multiple priors preferences are translation invariant if and only if they are **convex**, entropy **convex**, and entropy coherent **measures** of **risk**, respectively. It entails that **convex**, entropy **convex** and entropy coherent **measures** of **risk** induce linear or exponential **utility** functions in the theories of variational, homothetic and multiple priors preferences. We show further that, under a normalization condition, this characterization remains valid when the condition of translation invariance is replaced by requiring convexity. The mathematical details in the proofs of these characterization results are delicate.

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1 Introduction
The purpose of this short paper is to expose an extremely interesting relationship between Hyperbolic Absolute **Risk** Aversion (**HARA**) **utility**, disparity minimization, and shortfall.
Specifically, we show that the entire family of **HARA** **utility** functions has a minimium- divergence, shortfall-**based** representation, which means that **HARA** **utility** can be under- stood through the simple notion that the decision maker seeks the allocation that minimizes the probability of realizing an outcome below some pre-determined reference level. This result bridges the behavioral notion of shortfall minimization, first espoused by Roy (1952), with the now-familiar expected **utility** idea in a broad way. Specifically, we extend the en- dogenous **utility** arguments of Stutzer (2000, 2003), showing that his findings are special cases of a much more expansive relationship between shortfall, disparity, and conventional expected **utility**.

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combines a fund f with riskless lending or borrowing **based** on a quadratic **utility** function. The quadratic Jensen measure as well as the quadratic Treynor measure can be interpreted as situations with a restriction y = ε > 0, but small, and – once again – quadratic **utility**. Cubic Sharpe, Jensen and Treynor measure describe decision situations with corresponding settings for y but cubic **utility**. For all 25 portfolio selection problems from July 1997 to July 1999 we determine optimal portfolios **based** on the rules just described and compute resulting certainty equivalents for an investor whose **utility** function is actually cubic (and of the **HARA** type). Certainly, there are greater certainty equivalents achievable by portfolio selection according to the optimized cubic performance measure and with a restriction y ∈ [0, 1] instead of y = 1 or y = ε so that we express all resulting certainty equivalents as percentages of this attainable maximum value. Besides quadratic and cubic Sharpe, Jensen, and Treynor measure we also consider the optimized quadratic measure for which we assume portfolio selection with a re- striction y ∈ [0, 1] **based** on quadratic **utility**. Furthermore, we consider portfolio selection **based** on expected excess returns. Since **risk** neutrality would not lead to an inner solution for an investor’s riskless lending or borrowing we assume **risk** neutral fund selection and the choice y = 1 but a quadratic **utility** for the determination of the amount of the riskless invest- ment.

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1. Introduction
In their seminal paper Artzner et.al. [ADEH99] presented an axiomatic foundation of coherent **risk** **measures** to quantify and compare uncertain future cash-flows within financial institutions. Recently, F¨ollmer et.al. [FS02a] extended the notion of coher- ent **risk** **measures** to **convex** **risk** **measures**. It is evident from the axiomatic structure referred above that **convex** analysis plays a crucial role. Indeed, as we will outline in this paper, **convex** optimization with its embedded duality is the underlying operational and computational technique for its applications. This was implicitly addressed by Artzner et.al. [ADEH99], Delbaen [De00] and F¨ollmer et.al. [FS02a] in their proofs of the fundamental representation theorems. But to our knowledge, the fundamental concepts have never been presented explicitly under, at least from a computational point of view, “natural" perspective of **convex** optimization as we will do in this paper for the restricted but important finite dimensional case. Independently Ruszczy´nski et.al. [RS04] studied the intimate relation between **convex** **risk** functions and duality structure in topological vector spaces of measurable functions. Indeed, the spirit of their work is very similar to ours but they use advanced theory of **convex** analysis of measurable functions. Restrict- ing the analysis to the finite dimesional case, the representation results can be derived by applying elementary concepts of **convex** analysis **based** on the impressive work of R. T. Rockafellar. At the same time, this perspective opens up the computational aspects

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A sensible axiomatic approach to quantify **risk** was first mentioned in [Artzner et al., 99] for a static setting: The authors introduced the notion of coherent **risk** **measures** assessing **risk** of projects considered as real valued random variables. Several other references as [Delbaen, 02] advanced upon this approach for more general probability spaces. The approach to coherent **risk** **measures** is **based** on four quite intuitive axioms and leads to a simple and hence applicable robust representation that we encounter later. We will rigorously introduce the underlying notion of **risk** **measures** in the respective chapters of this thesis. However, for the sake of completeness and an intuitive understanding at this stage, the four axioms for a **risk** measure to be coherent are given by monotonicity, cash invariance, sub-additivity and positive homo- geneity of degree one. The major advantage of coherent **risk** **measures** is their simple and intuitive robust representation in terms of maximized expected loss as elaborated below. Furthermore, coherent **risk** **measures** do not ne- cessitate a specific probabilistic model and hence help to significantly reduce model **risk** in applications. However, coherent **risk** **measures** have two ma- jor shortcomings: First, they overestimate **risk** as they lead to a worst-case approach by virtue of robust representation: An issue that has to be scoped with from point of view of financial institutions having an intrinsic interest in assessing **risk** not too conservatively when calculating minimal capital re- quirements. Secondly, due to the assumption of homogeneity, coherent **risk** **measures** do not take into account liquidity **risk** as one of the major problems in the current financial crisis.

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2010 Mathematics Subject Classication: 91B14, 91B16, 03C20, 03C98 Journal of Economic Literature classication: D71, G11
∗ This work has been partially supported by a German Research Foundation (DFG) grant
while the author visited the Mathematics Department of Princeton University. A talk **based** on this paper was presented at the 10th Society for the Advancement of Economic Theory (SAET) Conference on Current Trends in Economics in Singapore, August 2010. I would like to thank Daniel Eckert, Edward Nelson, Konrad Podczeck and Frank Riedel for discussions and comments.

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Furthermore, the values of the financial assets are supposed to depend on a set of stochastic state variables and a stochastic inflation **risk** is considered.
In the literature about the optimal portfolio rules two main fields of research can be found. On the one hand some authors concentrate on establishing the ex- istence (and uniqueness) of a viscosity solution for the Hamilton-Jacobi-Bellman equation deriving from the stochastic optimal control approach (see for instance Crandall et al., 1992; and Buckdahn and Ma, 2001a, 2001b). On the other, some authors oﬀer an algebraic closed form solution to the optimal portfolio compo- sition. In particular, we refer to the works of Kim and Omberg (1996), Wachter (1998), Chacko and Viceira (1999), Deelstra et al. (2000), Boulier et al. (2001), Zariphopoulou (2001) and Menoncin (2002). The two last works use a solution approach **based** on the Feynman-Kaˇc theorem, 1 in an incomplete market and in a complete market with a background **risk** respectively.

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2 .
In this paper we consider a slightly more general, and completely nat- ural, concept of stochastic precedence and analyze its relations with the notions of stochastic ordering. Motivations for our study arise from differ- ent fields, in particular from the frame of Target-**Based** Approach in deci- sions under **risk**. Although this approach has been mainly developed under the assumption of stochastic independence between Targets and Prospects, our analysis concerns the case of stochastic dependence, that we model by means of a special class of copulas, introduced for the purpose. Examples are provided to better explain the behavior of the target-**based** model un- der changes in the connecting copulas of the random variables, especially regarding their properties of symmetry and dependence.

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Key words: **Convex** analysis, stochastic optimization, **risk** **measures**, mean-variance models, duality.
1 Introduction
Comparison of uncertain outcomes is central for decision theory. If the outcomes have a probabilistic description, a wealth of concepts and techniques from the theory of probability can be employed. We can mention here the expected **utility** theory, stochastic ordering, and various mean–**risk** models. Our main objective is to con- tribute to this direction of research, by exploiting relations between **risk** models and optimization theory.

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The paper is organized as follows. First, the literature concerning functional forms for the **utility** function is briefly reviewed. The CARA, CRRA, **HARA**, EP, PRT, and FTP functional forms are described. Following this, the procedure used to obtain a **utility** function from a **risk** aversion measure is discussed in some detail. There are three main steps in this procedure, with the final step being the transformation of the marginal **utility** function into the **utility** function. It is observed that this final step is the one that most severely restricts the **risk** preferences that can be represented by a **utility** function. Section 4 observes that in theory marginal **utility** is a complete representation of the **risk** preferences of an EU decision maker. In addition, however, this section presents a series of EU **based** decisions, and indicates how marginal **utility** is sufficient to determine and analyze the particular decision being discussed.

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determines the relevant linear parameters of optimal behavior. The **HARA** form itself implies lin- ear **risk** tolerance. We discuss the connections between linear scale transformations, linear optimal solutions and linear **risk** tolerance.
It needs to be stressed at the outset that the idea is not just to re-derive a well-known set of results. Rather, the aim is to show that the **HARA** form is inherent to the economic optimization problem, one which is a fundamental one, appearing in many Macro and Finance contexts. Hence, the analysis does not aim to provide another solution method to a problem that had been solved, but rather to show what restrictions on preferences are embodied in economic reasoning about this fundamental economic problem.

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we provide a representation for these **risk** **measures** as worst conditional loss with respect to a set of probabilistic models and a penalty function. The main difference in comparison with the unconditional setting is provided by the ran- dom nature of these two objects. This is natural, since they describe, in some sense, the degree of trustworthiness towards different models, which depends on available information and thus may change in time. In the representation we propose, additional information is reflected both in the conditional nature of the expectations and in the penalty function. This issue is particularly important when successive **risk** measurements of the same payoff are performed or, in our terminology, when a dynamic **risk** measure has to be constructed. In this case, a penalty process has to be chosen, describing how the degree of trustworthiness of different models evolves through time. In the last section it is shown how this choice is constrained by some basic natural consistency properties. Notwith- standing, in our opinion the class of penalty processes is still too large from an economic viewpoint, so that other consistency properties have to be discussed even in connection with the theory of updating information. Finally, a complete economic interpretation of the penalty term still lacks, even in the classical set- ting. This interpretation could be related to some sort of preference structure in the dual space, that of probabilistic models. We leave this important issue to further investigation.

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This particular argument demonstrates some difficulties in the use of con- vex **risk** **measures** in **risk** management. In the case of distortion-exponential **measures**, using the coherent measure ρ g,a=0 creates a stable portfolio. How- ever the introduction of even a slight dependence on the the scale of losses, by using ρ g,a even with an arbitrarily small a, produces an incentive for infinite fragmentation of portfolios. Note that this still happens when we start with a particular pair of sub-portfolios (e.g. business lines) X 1 , X 2 such that X 1 + X 2 = X and ρ(X) ≤ ρ(X 1 ) + ρ(X 1 ). That is, even if the initial configuration of the portfolio is such that benefits from pooling risks occur, once splitting without any constraints is allowed, fragmentation of the portfolio is inevitable.

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This particular argument demonstrates some difficulties in the use of con- vex **risk** **measures** in **risk** management. In the case of distortion-exponential **measures**, using the coherent measure ρ g,a=0 creates a stable portfolio. How- ever the introduction of even a slight dependence on the the scale of losses, by using ρ g,a even with an arbitrarily small a, produces an incentive for infinite fragmentation of portfolios. Note that this still happens when we start with a particular pair of sub-portfolios (e.g. business lines) X 1 , X 2 such that X 1 + X 2 = X and ρ(X) ≤ ρ(X 1 ) + ρ(X 1 ). That is, even if the initial configuration of the portfolio is such that benefits from pooling risks occur, once splitting without any constraints is allowed, fragmentation of the portfolio is inevitable.

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Key words: equilibria, generalised subgradients, law-invariant **convex** **risk** **measures**, optimal capital and **risk** allocations.
1 Introduction
In [17] we established that every law-invariant **convex** **risk** measure on L ∞ is σ(L ∞ , L ∞ )-lower semi-continuous and thus canonically extended to a law- invariant closed **convex** **risk** measure on L 1 . There are several advantages of the model space L 1 : in contrast to L ∞ , the model space L 1 includes important **risk** models such as normally distributed. Moreover, L 1 is in some sense maximal amongst the law-invariant model spaces bearing a locally **convex** topology and thus allowing for **convex** duality. Other attempts to extending the model space beyond L ∞ suggest spaces which depend on the **risk** measure, in terms of being chosen such that some given **risk** measure stays real valued. But when studying optimal **risk** allocations and equilibria which involves more than one **risk** mea- sure, the model space should preferably be independent of these **risk** **measures**.

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Abstract. The numerical representation of **convex** **risk** **measures** beyond essentially bounded financial positions is an important topic which has been the theme of recent literature. In other direction, it has been discussed the assessment of essentially bounded risks taking explicitly new information into account, i.e., conditional **convex** **risk** **measures**. In this paper we combine these two lines of research. We discuss the numerical representation of conditional **convex** **risk** **measures** which are defined in a space L p (F, R), for p ≥ 1, and take values in L 1 (G, R) (in this sense, real-valued ).

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Concerning the possibility of (~ localizing >> the classical integral-geometric for- mulae for convex bodies by introducing certain locally defined measures, the [r]

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under this metric can be characterized by the following: for any G n , G ∈ Q, G n → G if and only if G n (z) → G(z) at any continuity points of G.
Most of the time, we work with quantile functions that are continuous on (0, 1) in order to avoid irregularities due to the presence of atoms. In practice, it is not restrictive to focus on continuous quantile functions. Indeed, people do assume the continuity of quantile functions in many applications, e.g., when computing the Value at **Risk**. The study of discontinuous quantile functions is more technical and of little interest, so we choose not to pursue in this direction. In the following, we denote by Q c the set of all continuous quantile functions. We also denote by Q ∞ the set of all bounded quantile functions, and Q ∞ c the set of all bounded continuous quantile functions.

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