JEL classification: G22 G11 MSC: 91B30 91B28 90C48 Keywords:Optimal reinsurance Risk measure and deviation measure Optimality conditions a b s t r a c t This paper studies the optimal reinsurance problem when risk is measured by a general risk measure. Necessary and sufficient optimality conditions are given for a wide family of risk measures, including deviation measures, expectation bounded risk measures and coherent measures of risk. The optimality conditions are used to verify whether the classical reinsurance contracts (quota-share, stop-loss) are optimal essentially, regardless of the risk measure used. The paper ends by particularizing the findings, so as to study in detail two deviation measures and the conditional value at risk.

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.

The current literature does not reach a consensus on which risk measures should be used in practice. Our objective is to give at least a partial solution to this problem. We study properties that a risk measure must satisfy to avoid inadequate portfolio selections. The properties that we propose for risk measures can help avoid the problems observed with popular measures, like Value at Risk (VaR α ) or Conditional VaR α (CVaR α ). This leads to the definition of two new families: complete and adapted risk measures. Our focus is on risk measures generated by distortion functions. Two new properties are put forward for these: completeness, ensuring that the distortion risk measure uses all the information of the loss distribution, and adaptability, forcing the measure to use this information adequately.

The main object of this paper is to prove that for a linear or convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set. ᮊ

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.