Optimization by Vector Space Methods
Optimization by Vector Space Methods
Dual Stochastic Dominance and Related Mean-Risk Models
SIAM Journal on Optimization
Optimality conditions in portfolio analysis with general deviation measures
Mathematical Programming: Series A and B
Optimization of Convex Risk Functions
Mathematics of Operations Research
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The 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.