Nonlinear Functionals in Stochastic Programming; a Note on Stability and Empirical Estimates

Economic processes are very often influenced simultaneously by a decision parameter (that can be chosen according to conditions) and a random factor. Since mostly it is necessary to determine the decision parameter without knowledge of a random element realization, a deterministic optimization problem has to be defined. This deterministic problem can usually depend on an “underlying” probability measure corresponding to the random element. The investigation of such types problems often belong to the stochastic programming field. The great attention has been focus on the problems in which objective functions depend “linearly” on the probability measure. This note is focus on the cases when the above mentioned assumption is not fulfilled; see e.g. Markowitz functionals or some risk measures. We try to cover static (one stage problems) as well as dynamic approaches (multistage stochastic programming case).