Stochastic Control Theory: Dynamic Programming Principle

ISBN-13: 9784431551225 / Angielski / Twarda / 2014 / 250 str.

Makiko Nisio

Opis

This book provides an introduction to stochastic controls, via the method of dynamic programming, formulated by nonlinear semigroup. The dynamic programming principle, originated by R. Bellman in 1950s, is known as the two stage optimization procedure and gives a powerful tool to analyze stochastic control problems. Through the dependence of value function on its terminal cost function, we construct a nonlinear two parameter semigroup which formulates the dynamic programming principle and whose generator provides Hamilton--Jacobi--Bellman equation. Here we mainly concerned with finite time horizon stochastic controls. But we also apply the semigroup approach to control-stopping problems and stochastic differential games together with examples in financial market models. This book is organized as follows. Chapters 1--4 deal with completely observable finite dimensional controlled diffusions. Chapters 5 and 6 are concerned with Hilbert space valued stochastic processes, related to partially observable control problems.

Blurb

This book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems.

First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton–Jacobi–Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem.

Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations.

Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions.

This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as a one-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. Besides finite time-horizon controls, the book discusses control-stopping problems in the same frameworks.