Stochastic Differential Equations and Applications - 1st Edition - ISBN: 9780122682025, 9781483217888

Stochastic Differential Equations and Applications

1st Edition

Volume 2

Authors: Avner Friedman
Editors: Z. W. Birnbaum E. Lukacs
eBook ISBN: 9781483217888
Imprint: Academic Press
Published Date: 28th January 1976
Page Count: 316
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Description

Stochastic Differential Equations and Applications, Volume 2 is an eight-chapter text that focuses on the practical aspects of stochastic differential equations.

This volume begins with a presentation of the auxiliary results in partial differential equations that are needed in the sequel. The succeeding chapters describe the behavior of the sample paths of solutions of stochastic differential equations. These topics are followed by a consideration of an issue whether the paths can hit a given set with positive probability, as well as the stability of paths about a given manifold and with spiraling of paths about this manifold. Other chapters deal with the applications to partial equations, specifically with the Dirichlet problem for degenerate elliptic equations. These chapters also explore the questions of singular perturbations and the existence of fundamental solutions for degenerate parabolic equations. The final chapters discuss stopping time problems, stochastic games, and stochastic differential games.

This book is intended primarily to undergraduate and graduate mathematics students.

Table of Contents


Preface

General Notation

Contents of Volume 1

10 Auxiliary Results in Partial Differential Equations

1. Schaudere Estimates for Elliptic and Parabolic Equations

2. Sobolev's Inequality

3. Lp Estimates for Elliptic Equations

4. Lp Estimates for Parabolic Equations

Problems

11 Nonattainability

1. Basic Definitions; A Lemma

2. A Fundamental Lemma

3. The Case d(x)⋝3

4. The Case d(x)⋝2

5. M Consists of One Point and d = 1

6. The Case d(x) = 0

7. Mixed Case

Problems

12 Stability and Spiraling of Solutions

1. Criterion for Stability

2. Stable Obstacles

3. Stability of Point Obstacles

4. The Method of Descent

5. Spiraling of Solutions About a Point Obstacle

6. Spiraling of Solutions About any Obstacle

7. Spiraling for Linear Systems

Problems

13 The Dirichlet Problem for Degenerate Elliptic Equations

1. A General Existence Theorem

2. Convergence of Paths to Boundary Points

3. Application to the Dirichlet Problem

Problems

14 Small Random Perturbations of Dynamical Systems

1. The Functional IT(φ)

2. The First Ventcel-Freidlin Estimate

3. The Second Ventcel-Freidlin Estimate

4. Application to the First Initial-Boundary Value Problem

5. Behavior of the Fundamental Solution as ε→0

6. Behavior of Green's Function as ε→0

7. The Problem of Exit

8. The Problem of Exit (Continued)

9. Application to the Dirichlet Problem

10. The Principal Eigenvalue

11. Asymptotic Behavior of the Principal Eigenvalue

Problems

15 Fundamental Solutions for Degenerate Parabolic Equations

1. Construction of a Candidate for a Fundamental Solution

2. Interior Estimates

3. Boundary Estimates

4. Estimates Near Infinity

5. Relation Between K and a Diffusion Process

6. The Behavior of ξ(t) near S

7. Existence of a Generalized Solution in the Case of a Two-Sided Obstacle

8. Existence of a Fundamental Solution in the Case of a Strictly One-Sided Obstacle

9. Lower Bounds on the Fundamental Solution

10. The Cauchy Problem

Problems

16 Stopping Time Problems and Stochastic Games

Part I. The Stationary Case

1. Statement of the Problem

2. Characterization of Saddle Points

3. Elliptic Variational Inequalities in Bounded Domains

4. Existence of Saddle Points in Bounded Domains

5. Elliptic Estimates for Increasing Domains

6. Elliptic Variational Inequalities

7. Existence of Saddle Points in Unbounded Domains

8. The Stopping Time Problem

Part II. The Nonstationary Case

9. Characterization of Saddle Points

10. Parabolic Variational Inequalities

11. Parabolic Variational Inequalities (Continued)

12. Existence of a Saddle Point

13. The Stopping Time Problem

Problems

17 Stochastic Differential Games

1. Auxiliary Results

2. N-Person Stochastic Differential Games with Perfect Observation

3. Stochastic Differential Games with Stopping Time

4. Stochastic Differential Games with Partial Observation

Problems

Bibliographical Remarks

References

Index

Details

No. of pages:
316
Language:
English
Copyright:
© Academic Press 1976
Published:
Imprint:
Academic Press
eBook ISBN:
9781483217888

About the Author

Avner Friedman

About the Editor

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University