Stochastic Optimization Theory

This final chapter deals with a class of stochastic optimization problems. For this purpose we introduce a measure theoretic structure on top of the topological structure, and the resulting interplay brings a new set of questions interesting on their own as well. The measure theory is nonclassical in that the measures are only finitely additive on the field of cylinder sets, the canonical example being the Gauss measure. The notion of a weak random variable suffices for the stochastic extension of the control problems of the previous chapter, a crucial notion being that of “white noise,” leading to a treatment that is novel with this book, of filtering and control problems embracing in particular linear stochastic partial differential equations. Important tools in the development are the Krein factorization theorem and the Riccati equation. For nonlinear operations we develop a “nonlinear” white noise theory in which the notion of a physical random variable plays a crucial role, as in the calculation of the Radon-Nikodym derivative of finitely additive Gaussian measures. Within the scope of the present work we can but touch upon the general theory of nonlinear stochastic differential equations.