Friedman / Birnbaum / Lukacs Stochastic Differential Equations and Applications

Volume 2
1. Auflage 2014
ISBN: 978-1-4832-1788-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 316 Seiten, Web PDF

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.

Autoren/Hrsg.

Friedman, Avner

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Differential Equations and Applications;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;General Notation;12
7;Contents of Volume 1;14
8;Chapter 10. Auxiliary Results in Partial Differential Equations;16
8.1;1. Schaudere estimates for elliptic and parabolic equations;16
8.2;2. Sobolev's inequality;20
8.3;3. Lp estimates for elliptic equations;23
8.4;4. Lp estimates for parabolic equations;25
8.5;Problems;27
9;Chapter 11. Nonattalnability;29
9.1;1. Basic definitions; a lemma;29
9.2;2. A fundamental lemma;33
9.3;3. The case d(x) = 3;37
9.4;4. The case d(x) = 2;40
9.5;5. M consists of one point and d = 1;46
9.6;6. The case d(x) = 0;50
9.7;7. Mixed case;52
9.8;Problems;54
10;Chapter 12. Stability and Spiraling of Solutions;57
10.1;1. Criterion for stability;57
10.2;2. Stable obstacles;65
10.3;3. Stability of point obstacles;70
10.4;4. The method of descent;73
10.5;5. Spiraling of solutions about a point obstacle;77
10.6;6. Spiraling of solutions about any obstacle;87
10.7;7. Spiraling for linear systems;90
10.8;Problems;93
11;Chapter 13. The DIrichlet Problem for Degenerate Elliptic Equations;95
11.1;1. A general existence theorem;95
11.2;2. Convergence of paths to boundary points;102
11.3;3. Application to the Dirichlet problem;105
11.4;Problems;109
12;Chapter 14. Small Random Perturbations of Dynamical Systems;113
12.1;1. The functional IT(.);113
12.2;2. The first Ventcel–Freidlin estimate;119
12.3;3. The second Ventcel-Freidlin estimate;121
12.4;4. Application to the first initial-boundary value problem;133
12.5;5. Behavior of the fundamental solution as e . 0;135
12.6;6. Behavior of Green's function as e . 0;141
12.7;7. The problem of exit;146
12.8;8. The problem of exit (continued);154
12.9;9. Application to the Dirichlet problem;158
12.10;10. The principal eigenvalue;160
12.11;11. Asymptotic behavior of the principal eigenvalue;163
12.12;Problems;170
13;Chapter 15. Fundamental Solutions for Degenerate Parabolic Equations;175
13.1;1. Construction of a candidate for a fundamental solution;175
13.2;2. Interior estimates;183
13.3;3. Boundary estimates;186
13.4;4. Estimates near infinity;193
13.5;5. Relation between K and a diffusion process;196
13.6;6. The behavior of .(t) near S;201
13.7;7. Existence of a generalized solution in the case of a two-sided obstacle;207
13.8;8. Existence of a fundamental solution in the case of a strictly one-sided obstacle;210
13.9;9. Lower bounds on the fundamental solution;213
13.10;10. The Cauchy problem;215
13.11;Problems;219
14;Chapter 16. Stopping Time Problems and Stochastic Games;220
14.1;Part I. The Stationary Case;220
14.1.1;1. Statement of the problem;220
14.1.2;2. Characterization of saddle points;223
14.1.3;3. Elliptic variational inequalities in bounded domains;227
14.1.4;4. Existence of saddle points in bounded domains;231
14.1.5;5. Elliptic estimates for increasing domains;234
14.1.6;6. Elliptic variational inequalities;244
14.1.7;7. Existence of saddle points in unbounded domains;249
14.1.8;8. The stopping time problem;250
14.2;Part II. The Nonstationary Case;251
14.2.1;9. Characterization of saddle points;251
14.2.2;10. Parabolic variational inequalities;253
14.2.3;11. Parabolic variational inequalities (continued);265
14.2.4;12. Existence of a saddle point;273
14.2.5;13. The stopping time problem;275
14.2.6;Problems;277
15;Chapter 17. Stochastic Differential Games;281
15.1;1. Auxiliary results;281
15.2;2. N-person stochastic differential games with perfect observation;285
15.3;3. Stochastic differential games with stopping time;289
15.4;4. Stochastic differential games with partial observation;294
15.5;Problems;305
15.6;Bibliographical Remarks;307
15.7;References;310
16;Index;314