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 procedures. The presentation will touch on a variety of applications: 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 mathematical foundations. Plan
- A brief review of variational analysis. Functions and their epigraphs, convexity and semicontinuity. Set convergence and epigraphical limits. Variational geometry, subgradients 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. SAA (Sample. 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. Properties of expectation functionals. Almost sure convergence and convergence in distribution (epigraphical sense). The Ergodic Theorem for random lsc functions and its applications: sampled variational problems, approximation, statistical estimation and homogenization.
- Introduction to the calculus of expectation functionals. Decomposable spaces. Fatou’s lemma for random set and random lsc functions. Interchange of minimization and (conditional) expectation. Subdifferentiation of expectation functionals. Martingale integrands and application to financial valuation.