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SUMMARY Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables to the real line. Economically, a risk measure should capture the preferences of the decision-maker. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin's rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial mathematics literature disregard the utility concept (i.e., correspond to linear utilities), restricting their applicability. Some alternatives proposed in the literature are discussed.

SUMMARY The purpose of our paper is to link some results on the Choquet integrals with the theory of coherent risk measures. Using this link we establish some properties of dilatation monotone and comonotonic coherent measures of risk. In particular it is shown that on an atomless probability space dilatation monotone and comonotonic additive coherent risk measures have to be law invariant.

SUMMARY Distorted measures have been used in pricing of insurance contracts for a long time. This paper reviews properties of related acceptability functionals in risk management, called distortion functionals. These functionals may be characterized by being mixtures of average values-at-risk. We give a dual representation of these functionals and show how they may be used in portfolio optimization. An iterative numerical procedure for the solution of these portfolio problems is given which is based on duality.

SUMMARY We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

SUMMARY The class of all lawinvariant, convex risk measures for portfolio vectors is characterized. The building blocks of this class are shown to be formed by the maximal correlation risk measures. We further introduce some classes of multivariate distortion risk measures and relate them to multivariate quantile functionals and to an extension of the average value at risk measure.

SUMMARY We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with negative risk aversion and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.

SUMMARY This paper considers a class of one dimensional calculus of variations problems with monotonicity and comonotonicity constraints arising in economic and financial models where law invariant concave criteria (or law invariant convex measures of risk) are used. Existence solutions, optimality conditions, sufficient conditions for the regularity of solutions are established. Applications to risk sharing with convex comonotone law invariant risk measures or with robust utilities are given.

SUMMARY The optimal risk allocation problem or equivalently the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures ϱ 1 , …, ϱ n . This problem has a long history in mathematical economics and insurance. In the first part of the paper we review some mathematical tools and discuss their applications to various problems on risk measures related to the allocation problem like to monotonicity properties of optimal allocations, to optimal investment problems or to an appropriate definition of the conditional value at risk. We then consider the risk allocation problem for convex risk measures ϱ i . In general the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. We formulate ameaningful modification of the optimal risk allocation problem also formarkets without assuming the equilibrium condition and characterize optimal solutions. The basic idea is to restrict the class of admissible allocations in a proper way.

SUMMARY For a monetary utility functional U and a coherent risk measure ρ, both with compact scenario sets in L q , we optimize the ratio α( V ): = U ( V )/ρ( V ) over an (arbitrage-free) linear sub-space V ⊆ L p , 1 ≤ p ≤ ∞, of attainable returns in an incomplete market model such that ρ > 0 on V \ {0}. If a solution Vˆ ∈ V with α(Vˆ) = α¯ V : = sup V∈V α( V )∈[0,∞) exists, then the first order optimality condition allows to construct an absolutely continuous martingale measure for V as a convex combination Q ¯+α¯ V Q /1+α¯ V of two probability measures Q ¯, Q from the respective scenario sets defining U and ρ. Conversely, if α¯ V ∈ [0,∞), then α¯ V equals the smallest a ∈[0,∞) such that Q ¯+ a Q /1+ a is an absolutely continuous martingale measure for V for some probability measures Q ¯, Q from the scenario sets defining U , ρ, and α¯ V = ∞ holds iff such a convex combination does not exist.

SUMMARY This paper presents an analysis of the tracking problems of multiple indices with multidimensional performance criterion consisting of mean wealth and the tracking errors. We evaluate the performance of portfolios via the vector inequalities defined by convex cones, which enable us to describe various preference relations for investors. In Brownian market models with deterministic coefficients, we completely determine the set of efficient portfolios as well as the efficient frontier in our context. As a product of our analysis, we exhibit a version of Tobin's mutual fund theorem.