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Polyhedral risk measures in stochastic programming. (English) Zbl 1114.90077

The authors define the class of polyhedral risk measures as optimal values of certain linear stochastic programs with recourse where the arguments appear on the right-hand sides of the dynamic constraints. They provide conditions implying that polyhedral risk measures are coherent and consistent with second order stochastic dominance. For the one-period case it has been shown that well-known risk measures are contained in the class of polyhedral risk measures introduced: Conditional-Value- at Risk/quantile dispersion, and expected loss. For the multiperiod case, five polyhedral (coherent) risk measures are suggested.
In the last section, the authors shown that several properties of expectation - based stochastic programs remain valid for stochastic programs with polyhedral risk measures as objective (or, alternatively, with an objective consisting of a linear combination of an expectation and a polyhedral risk measure). In particular, they present stability results for two-stage stochastic programs with polyhedral risk measures and show that dual decomposition structures are maintained.

MSC:

90C15 Stochastic programming
91B30 Risk theory, insurance (MSC2010)
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