Titlebook: Mathematical Risk Analysis; Dependence, Risk Bou Ludger Rüschendorf Book 2013 The Editor(s) (if applicable) and The Author(s), under exclus

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Fréchet Classes, Risk Bounds, and Duality Theoryctional .(.), i.e. to determine the range of .(.) under all possible dependence structures. When the distribution of .. is given by .., 1 ≤ . ≤ . then the set of all possible dependence structures is given by the Fréchet class ..

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Convex Order, Excess of Loss, and Comonotonicityof the excess of loss. Two different approaches to this result are given. The first approach due to Meilijson and Nadas (.) is based on a simple duality argument and uses just a monotonicity property of the inverse distribution function.

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Risk Measures for Real Risks postulate is that measures of risk should include the aspect of securization of risk, i.e. the possibility to transfer risks by hedging actions or by buying or selling derivatives to the market. The price of the hedging action or of the derivatives is one part of the risk of a position.

Essential

Risk Measures for Portfolio Vectorsisk of . caused by the variation of the components and by their dependence. From this point of view an important property of risk measures is consistency w.r.t. various classes of convex orders and of dependence orders.

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Optimal Allocations and Pareto Equilibrium05/2008, 006a). This game theoretic notion is naturally associated with properties of some risk measures which describe the optimal risk sharing problem in a market with . traders as described in the introduction.

Arrhythmia

Characterization and Examples of Optimal Risk Allocations for Convex Risk Functionalschange in the context of expected utility (see Borch (1962), Bühlmann (1970), and Gerber (1979)) and as a consequence results in the derivation of optimal (reinsurance) contracts between . companies. A general differential characterization of optimal risk allocations for convex premium principles was given in Deprez and Gerber (1985).

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