Abstract. This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated.Three main contributions seem to be reached. Firstly, necessary and su¢ cient optimality conditions are given. Secondly, the optimal contract is often a bang-bang solution, i:e:, the sensitivity between the retained risk and the total claims saturates the imposed constraints. For some special cases the optimal contract might not be bang-bang, but there is always a bang-bang contract as close as desired to the optimal one. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems are easy to solve in practice, since there are very e¢ cient algorithms.

MSC:Keywords: Risk minimization Saddle point condition Actuarial and financial applications a b s t r a c tThe minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.

a b s t r a c tThe paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Condi tional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valu ation of volatility linked derivatives are discussed.

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The minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging. There are two major reasons. Firstly, heavy tails and the lack of symmetry in the returns of many assets provokes that the classical opti mization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several general risk or dispersion measures. The representation the orems of risk functions are applied to transform the general risk minimization problem in a minimax problem, and later in a linear pro gramming problem between infinite dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex like algorithm is developed. The algorithm solves the dual problem and provides both optimal portfolios and their sensitivities.The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are spe cially analyzed. A final real data numerical example illustrates the practical performance of the proposed methodology.

a b s t r a c tThe paper deals with optimal portfolio choice problems when risk levels are given by coherent risk mea sures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as (strong) compatibility between prices and risks. Surprisingly, the lack of bounded optimal risk and/or return levels arises for important pricing models (Black and Scholes) and risk measures (VaR, CVaR, absolute deviation, etc.). Bounded problems present a Market Price of Risk and generate a pair of benchmarks. From these bench marks we introduce APT and CAPM like analyses, in the sense that the level of correlation between every available security and some economic factors explains the security expected return. The risk level non correlated with these factors has no influence on any return, despite the fact that we are dealing with risk functions beyond the standard deviation.

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Abstract. This paper has considered a risk measure ρ and a (maybe incomplete and/or imperfect) arbitrage-free market with pricing rule Π. They are said to be compatible if there are no reachable strategies y such that Π(y) is bounded and ρ(y) is close to −∞. We show that the lack of compatibility leads to meaningless situations in financial or actuarial applications.The presence of compatibility is characterized by properties connecting the Stochastic Discount Factor of Π and the sub-gradient of ρ. Consequently, several examples pointing out that the lack of compatibility may occur in very important pricing models are yielded. For instance the CVaR is not compatible with the Black and Scholes model or the CAPM.We prove that for a given incompatible couple (Π, ρ) we can construct a minimal risk measure M (Π,ρ) compatible with ρ and such that ρ ≤ M (Π,ρ) . This result is particularized for the CVaR and the CAPM and the Black and Scholes model. Therefore we construct the Compatible Conditional Value at Risk (CCVaR). It seems that the CCVaR preserves the good properties of the CVaR and overcomes its shortcomings. Compatibilidad entre reglas de valoración y medidas de riesgo: el CCVaRResumen. Consideraremos una medida de riesgo ρ y un mercado libre de arbitraje (puede ser que incompleto o imperfecto) con regla de valoración Π.Éstos serán compatibles si no hay estrategias alcanzables y tales que Π(y) permanece acotado y ρ(y) se acerca a −∞. Veremos que la falta de compatibilidad conduce a situaciones sin sentido económico en las aplicaciones actuariales o financieras.La compatibilidad será caracterizada mediante propiedades que ligan al Factor de Descuento Estocástico de Π y al sub-gradiente de ρ. Consecuentemente, se podrán dar importantes ejemplos en los que hay falta de compatibilidad. Por ejemplo, el CVaR no es compatible con el modelo de Black-Scholes o el CAPM.Probaremos que para cualquier par incompatible (Π, ρ) se puede construir una medida de riesgo minimal M (Π,ρ) compatible con ρ, y tal que ρ ≤ M (Π,ρ) . Este resultado se particularizará para el CVaR y el CAPM y el modelo de Black-Scholes. Por tanto, construiremos el CVaR Compatible (CCVaR). El CCVaR parece preservar las buenas propiedades del CVaR y superar sus deficiencias.