This is a series of four lectures designed to introduce, in a
unified framework, a broad spectrum
of mathematical theory that has grown in connection with the
study of problems of optimization,
equilibrium, control, and stability of linear and nonlinear
systems that naturally arise in a
stochastic environment. With the advent of computers, there has
been a tremendous expansion
of interest in new problem formulations that demand new modes
of analysis but are far from
being covered by classical concepts and classical results. For
those problems, finite-dimensional
spaces alongside of function spaces, and theoretical concerns
go hand in hand with the practical
ones of mathematical modeling and the design of numerical
The presentation will touch on a variety of
stochastic programming problems,
equilibrium problems in a stochastic environment, stochastic
variational inequalities, stochastic
homogenization, financial valuation, flow problems in
heterogeneous media, etc. However, the
primordial goal will be to provide an introduction to the
- A brief review of variational analysis. Functions and their
epigraphs, convexity and semicontinuity.
Set convergence and epigraphical limits. Variational geometry,
and subdifferential calculus.
Random sets. Definition and properties of random sets,
selections. The distribution function
(∼ Choquet capacity) of a random set and convergence in
distribution. The expectation
of a random set and the law of large numbers for random sets.
Average Approximations) of random sets. Application to
stochastic variational inequalities
and related variational problems.
Random lsc functions and expectation functionals. Definition
of random lsc (lower semicontinuous)
functions and calculus. Stochastic processes with lsc paths.
expectation functionals. Almost sure convergence and
convergence in distribution (epigraphical
sense). The Ergodic Theorem for random lsc functions and its
sampled variational problems, approximation, statistical
estimation and homogenization.
Introduction to the calculus of expectation
Decomposable spaces. Fatou’s
lemma for random set and random lsc functions. Interchange of
minimization and (conditional)
expectation. Subdifferentiation of expectation functionals.
and application to financial valuation.