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Stochastic Control Theory

Makiko Nisio - Personal Name;

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.

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Series Title: XV, 250
Call Number: -
Publisher: : ., 2015
Collation: -
Language: English
ISBN/ISSN: 978-4-431-55123-2
Classification: NONE
Content Type: text
Media Type: computer
Carrier Type: -
Edition: -
Subject(s): stochastic optimal control
Specific Detail Info: -
Statement of Responsibility: 2

Other Information

Cataloger: Suwardi
Source: https://link.springer.com/book/10.1007/978-4-431-55123-2
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Stochastic Control Theory
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