Convergence of stochastic infima: Equi-semicontinuity.

*(English)*Zbl 0593.90064
Stochasticc optimization, Proc. Int. Conf., Kiev/USSR 1984, Lect. Notes Control Inf. Sci. 81, 561-575 (1986).

[For the entire collection see Zbl 0585.00014.]

Many stochastic optimization problems focus the attention on the stochastic infimum \(\omega\) \(\to \inf_{x}f(x,\omega)\) of a normal integrand f(x,\(\omega)\). Often the only possible approach to derive the probability distribution of the stochastic infimum is via approximations which inevitably lead to the convergence in distribution of normal integrands. This is based on the notion of epi-convergence (epigraphs convergence) and epi-topology for the structural relations between infima and epigraphs.

On the other hand the same notion of epi-convergence revealed to be the appropriate notion to study convergence of probability measures.

Convergence of infima and convergence of probability measures, as key ingredients of convergence of stochastic infima, find then naturally in the epi-convergence an appropriate and fruitful tool to determine the minimal setting for convergence of stochastic infima.

From a more general point of view, the convergence theory for normal integrands can be regarded itself as an extension of the classical convergence of stochastic processes which, together with the alternative approach to weak convergence of probability measures, gives an extended setting to deal with convergence of functionals of stochastic processes.

Many stochastic optimization problems focus the attention on the stochastic infimum \(\omega\) \(\to \inf_{x}f(x,\omega)\) of a normal integrand f(x,\(\omega)\). Often the only possible approach to derive the probability distribution of the stochastic infimum is via approximations which inevitably lead to the convergence in distribution of normal integrands. This is based on the notion of epi-convergence (epigraphs convergence) and epi-topology for the structural relations between infima and epigraphs.

On the other hand the same notion of epi-convergence revealed to be the appropriate notion to study convergence of probability measures.

Convergence of infima and convergence of probability measures, as key ingredients of convergence of stochastic infima, find then naturally in the epi-convergence an appropriate and fruitful tool to determine the minimal setting for convergence of stochastic infima.

From a more general point of view, the convergence theory for normal integrands can be regarded itself as an extension of the classical convergence of stochastic processes which, together with the alternative approach to weak convergence of probability measures, gives an extended setting to deal with convergence of functionals of stochastic processes.

##### MSC:

90C15 | Stochastic programming |