© 2015

Stochastic Control Theory

Dynamic Programming Principle


  • Deals with a quick review of stochastic analysis and stochastic differential equations with random coefficients

  • Deals with viscosity solutions of nonlinear parabolic equation

  • Shows the connection between controlled Zakai equations and control of partially observable diffusions


Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 72)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Makiko Nisio
    Pages 1-30
  3. Makiko Nisio
    Pages 31-78
  4. Makiko Nisio
    Pages 79-115
  5. Makiko Nisio
    Pages 117-151
  6. Makiko Nisio
    Pages 153-207
  7. Makiko Nisio
    Pages 209-244
  8. Back Matter
    Pages 245-250

About this book


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.


dynamic programming principle nonlinear semigroup stochastic differential game stochastic optimal control viscosity solution

Authors and affiliations

  1. 1.Kobe UniversityKobeJapan

Bibliographic information

  • Book Title Stochastic Control Theory
  • Book Subtitle Dynamic Programming Principle
  • Authors Makiko Nisio
  • Series Title Probability Theory and Stochastic Modelling
  • Series Abbreviated Title Probability and Stochastic (formerly: PIA & SMAP)
  • DOI
  • Copyright Information Springer Japan 2015
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-4-431-55122-5
  • Softcover ISBN 978-4-431-56408-9
  • eBook ISBN 978-4-431-55123-2
  • Series ISSN 2199-3130
  • Series E-ISSN 2199-3149
  • Edition Number 2
  • Number of Pages XV, 250
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published in the series ISI Lecture Notes, No 9, by MacMillan India Limited publishers, Delhi
  • Topics Probability Theory and Stochastic Processes
    Functional Analysis
    Partial Differential Equations
  • Buy this book on publisher's site
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