This e-book deals a scientific advent to the optimum stochastic keep an eye on idea through the dynamic programming precept, that is a robust instrument to research keep watch over problems.

First we think about thoroughly observable regulate issues of finite horizons. utilizing a time discretization we build a nonlinear semigroup concerning the dynamic programming precept (DPP), whose generator presents the Hamilton–Jacobi–Bellman (HJB) equation, and we symbolize the worth functionality through the nonlinear semigroup, along with the viscosity resolution conception. after we keep an eye on not just the dynamics of a procedure but additionally the terminal time of its evolution, control-stopping difficulties come up. This challenge is handled within the comparable frameworks, through the nonlinear semigroup. Its effects are acceptable to the yankee choice cost problem.

Zero-sum two-player time-homogeneous stochastic differential video games and viscosity strategies of the Isaacs equations bobbing up from such video games are studied through a nonlinear semigroup with regards to DPP (the min-max precept, to be precise). utilizing semi-discretization arguments, we build the nonlinear semigroups whose turbines supply decrease and higher Isaacs equations.

Concerning in part observable keep watch over difficulties, we consult with stochastic parabolic equations pushed by means of coloured Wiener noises, particularly, the Zakai equation. The life and specialty of recommendations and regularities in addition to Itô's formulation are acknowledged. A keep watch over challenge for the Zakai equations has a nonlinear semigroup whose generator offers the HJB equation on a Banach area. the price functionality seems to be a different viscosity resolution for the HJB equation less than light conditions.

This version offers a extra generalized remedy of the subject than does the sooner e-book *Lectures on Stochastic regulate Theory* (ISI Lecture Notes 9), the place time-homogeneous situations are handled. the following, for finite time-horizon keep watch over difficulties, DPP used to be formulated as a one-parameter nonlinear semigroup, whose generator presents the HJB equation, by utilizing a time-discretization process. The semigroup corresponds to the price functionality and is characterised because the envelope of Markovian transition semigroups of responses for consistent keep watch over methods. in addition to finite time-horizon controls, the e-book discusses control-stopping difficulties within the comparable frameworks.