Publication: Minimizing measures of risk by saddle point conditions
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2010-09
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Elsevier
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Abstract
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.
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Keywords
Risk minimization, Saddle point condition, Actuarial and finantial aplications
Bibliographic citation
Journal of Computational and applied mathematics, 2010, v. 234, nº 10, pp. 2924-2931