Stochastic Differential Equations and Applications

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