Ekeland's Variational Principle for An ${\bar{L}}^0-$Valued Function on A Complete Random Metric Space

Abstract

Motivated by the recent work on conditional risk measures, this paper studies the Ekeland's variational principle for a proper, lower semicontinuous and lower bounded ${\bar{L}}^0-$valued function, where ${\bar{L}}^0$ is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekeland's variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekeland's variational principle for such a local function on a complete random normed module. Finally, as applications, we establish the Bishop-Phelps theorem in a complete random normed module under the framework of random conjugate spaces.

26 pages

References in corpus (1)

• Recent progress in random metric theory and its applications to conditional risk measures