Set-valued risk measures on L p d with 0 ≤ p ≤ ∞ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Scalar risk measures with multiple eligible assets also turn out to be a special case within the set-valued framework.

Abstract. Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004) 1. Introduction. The concept of set-valued coherent measures of risk has been introduced recently by Jouini, Meddeb, and Touzi [16]. The basic question is how to evaluate the (financial) risk of a multivariate random outcome in terms of more than one reference instrument, for example if the regulator accepts deposits in more than one currency. This is of particular importance if transaction costs have to be paid for each transaction between assets including the reference instruments.This question became unexpectedly topical as the European Central Bank decreed [7] that temporarily, until the end of 2009, "the list of assets eligible as collateral in Eurosystem credit operations will be expanded" by "marketable debt instruments denominated in other currencies than the euro, namely the US dollar, the British pound and the Japanese yen, and issued in the euro area. These instruments will be subject to a uniform haircut add-on of 8%." See also [8].The following exposition is based on the model used in [16]; in particular, we assume the presence of proportional transaction costs modeled via a closed convex cone which goes back to [17]. A special feature of this model is that the risk of a d-dimensional random variable is evaluated in terms of m reference instruments with 1 ≤ m ≤ d. Usually m d will be true, but the case m = d is not excluded (and also, of course, neither m = 1, d > 1, nor m = d = 1).We shall make a few generalizations compared to [16]. First, we do not assume a "substitutability condition"; i.e., we do not assume that everything could be transferred into one distinguished currency. Second, we consider a general subspace M of R d as the collection

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre-Fenchel conjugates for setvalued functions is introduced and a Moreau-Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.

New versions and extensions of Benson's outer approximation algorithm for solving linear vector optimization problems are presented. Primal and dual variants are provided in which only one scalar linear program has to be solved in each iteration rather than two or three as in previous versions. Extensions are given to problems with arbitrary pointed solid polyhedral ordering cones. Numerical examples are provided, one of them involving a new set-valued risk measure for multivariate positions.

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for setand vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems.