Probability Theory and Stochastic Modelling

Stochastic Control Theory

Dynamic Programming Principle

Authors: Nisio, Makiko

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  • 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
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Hardcover $109.99
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Softcover $99.99
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  • ISBN 978-4-431-56408-9
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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.

Table of contents (6 chapters)

Table of contents (6 chapters)

Buy this book

eBook $79.99
price for USA in USD (gross)
  • ISBN 978-4-431-55123-2
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.99
price for USA in USD
  • ISBN 978-4-431-55122-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $99.99
price for USA in USD
  • ISBN 978-4-431-56408-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Stochastic Control Theory
Book Subtitle
Dynamic Programming Principle
Authors
Series Title
Probability Theory and Stochastic Modelling
Series Volume
72
Copyright
2015
Publisher
Springer Japan
Copyright Holder
Springer Japan
eBook ISBN
978-4-431-55123-2
DOI
10.1007/978-4-431-55123-2
Hardcover ISBN
978-4-431-55122-5
Softcover ISBN
978-4-431-56408-9
Series ISSN
2199-3130
Edition Number
2
Number of Pages
XV, 250
Additional Information
Originally published in the series ISI Lecture Notes, No 9, by MacMillan India Limited publishers, Delhi
Topics