Minimizing Measures of Risk by Saddle Point Conditions Articles uri icon

publication date

  • September 2010

start page

  • 2924

end page

  • 2931

issue

  • 10

volume

  • 234

international standard serial number (ISSN)

  • 0377-0427

electronic international standard serial number (EISSN)

  • 1879-1778

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.