Controlled Diffusion Processes / Edition 1

Hardcover (Print)
Buy New
Buy New from
Used and New from Other Sellers
Used and New from Other Sellers
from $74.00
Usually ships in 1-2 business days
(Save 60%)
Other sellers (Hardcover)
  • All (11) from $74.00   
  • New (6) from $139.51   
  • Used (5) from $74.00   


This book deals with the optimal control of solutions of fully observable Ito-type stochastic differential equations. The validity of the Bellman differential equation for payoff functions is proved and rules for optimal control strategies are developed.
Read More Show Less

Editorial Reviews

From the Publisher

From the reviews:

“The book treats a large class of fully nonlinear parabolic PDEs via probabilistic methods. … The monograph may be strongly recommended as an excellent reading to PhD students, postdocs et al working in the area of controlled shastic processes and/or nonlinear partial differential equations of the second order. … recommended to a wider audience of all students specializing in shastic analysis or shastic finance starting from MSc level.” (Alexander Yu Veretennikov, Zentralblatt MATH, Vol. 1171, 2009)

Read More Show Less

Product Details

Table of Contents

Notation xi

1 Introduction to the Theory of Controlled Diffusion Processes 1

1 The Statement of Problems-Bellman's Principle-Bellman's Equation 2

2 Examples of the Bellman Equations-The Normed Bellman Equation 7

3 Application of Optimal Control Theory-Techniques for Obtaining Some Estimates 16

4 One-Dimensional Controlled Processes 22

5 Optimal Stopping of a One-Dimensional Controlled Process 35

Notes 42

2 Auxiliary Propositions 45

1 Notation and Definitions 45

2 Estimates of the Distribution of a Stochastic Integral in a Bounded Region 51

3 Estimates of the Distribution of a Stochastic Integral in the Whole Space 61

4 Limit Behavior of Some Functions 67

5 Solutions of Stochastic Integral Equations and Estimates of the Moments 77

6 Existence of a Solution of a Stochastic Equation with Measurable Coefficients 86

7 Some Properties of a Random Process Depending on a Parameter 91

8 The Dependence of Solutions of a Stochastic Equation on a Parameter 102

9 The Markov Property of Solutions of Stochastic Equations 110

10 Ito's Formula with Generalized Derivatives 121

Notes 128

3 General Properties of a Payoff Function 129

1 Basic Results 129

2 Some Preliminary Considerations 140

3 The Proof of Theorems 1.5-1.7 147

4 The Proof of Theorems 1.8-1.11 for the Optimal Stopping Problem 152

Notes 161

4 The Bellman Equation 163

1 Estimation of First Derivatives of Payoff Functions 165

2 Estimation from Below of Second Derivatives of a Payoff Function 173

3 Estimation from Above of Second Derivatives of a Payoff Function 181

4 Estimation of a Derivative of a Payoff Function with Respect to t 188

5 Passage to the Limit in the Bellman Equation193

6 The Approximation of Degenerate Controlled Processes by Nondegenerate Ones 200

7 The Bellman Equation 203

Notes 211

5 The Construction of [epsilon]-Optimal Strategies 213

1 [epsilon]-Optimal Markov Strategies and the Bellman Equation 213

2 [epilson]-Optimal Markov Strategies. The Bellman Equation in the Presence of Degeneracy 218

3 The Payoff Function and Solution of the Bellman Equation: The Uniqueness of the Solution of the Bellman Equation 228

Notes 243

6 Controlled Processes with Unbounded Coefficients: The Normed Bellman Equation 245

1 Generalizations of the Results Obtained in Section 3.1 245

2 General Methods for Estimating Derivatives of Payoff Functions 254

3 The Normed Bellman Equation 266

4 The Optimal Stopping of a Controlled Process on an Infinite Interval of Time 275

5 Control on an Infinite Interval of Time 285

Notes 291


1 Some Properties of Stochastic Integrals 293

2 Some Properties of Submartingales 299

Bibliography 303

Index 307

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star


4 Star


3 Star


2 Star


1 Star


Your Rating:

Your Name: Create a Pen Name or

Barnes & Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation


  • - By submitting a review, you grant to Barnes & and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Terms of Use.
  • - Barnes & reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)