Friedman / Birnbaum / Lukacs Stochastic Differential Equations and Applications

Volume 1
1. Auflage 2014
ISBN: 978-1-4832-1787-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 248 Seiten, Web PDF

Stochastic Differential Equations and Applications, Volume 1 covers the development of the basic theory of stochastic differential equation systems. This volume is divided into nine chapters. Chapters 1 to 5 deal with the basic theory of stochastic differential equations, including discussions of the Markov processes, Brownian motion, and the stochastic integral. Chapter 6 examines the connections between solutions of partial differential equations and stochastic differential equations, while Chapter 7 describes the Girsanov's formula that is useful in the stochastic control theory. Chapters 8 and 9 evaluate the behavior of sample paths of the solution of a stochastic differential system, as time increases to infinity. This book is intended primarily for 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 2;14
8;Chapter 1. Stochastic Processes;16
8.1;1. The Kolmogorov construction of a stochastic process;16
8.2;2. Separable and continuous processes;21
8.3;3. Martingales and stopping times;24
8.4;Problems;30
9;Chapter 2. Markov Processes;33
9.1;1. Construction of Markov processes;33
9.2;2. The Feller and the strong Markov properties;38
9.3;3. Time-homogeneous Markov processes;45
9.4;Problems;46
10;Chapter 3. Brownian Motion;51
10.1;1. Existence of continuous Brownian motion;51
10.2;2. Nondifferentiability of Brownian motion;54
10.3;3. Limit theorems;55
10.4;4. Brownian motion after a stopping time;59
10.5;5. Martingales and Brownian motion;61
10.6;6. Brownian motion in n dimensions;65
10.7;Problems;68
11;Chapter 4. The Stochastic Integral;70
11.1;1. Approximation of functions by step functions;70
11.2;2. Definition of the stochastic integral;74
11.3;3. The indefinite integral;82
11.4;4. Stochastic integrals with stopping time;87
11.5;5. Itô's formula;93
11.6;6. Applications of Itô's formula;100
11.7;7. Stochastic integrals and differentials in n dimensions;104
11.8;Problems;108
12;Chapter 5. Stochastic Differential Equations;113
12.1;1. Existence and uniqueness;113
12.2;2. Stronger uniqueness and existence theorems;117
12.3;3. The solution of a stochastic differential system as a Markov process;123
12.4;4. Diffusion processes;129
12.5;5. Equations depending on a parameter;132
12.6;6. The Kolmogorov equation;138
12.7;Problems;140
13;Chapter 6. Elliptic and Parabolic Partial Differential Equations and Their Relations to Stochastic Differential Equations;143
13.1;1. Square root of a nonnegative definite matrix;143
13.2;2. The maximum principle for elliptic equations;147
13.3;3. The maximum principle for parabolic equations;149
13.4;4. The Cauchy problem and fundamental solutions for parabolic equations;154
13.5;5. Stochastic representation of solutions of partial differential equations;159
13.6;Problems;165
14;Chapter 7. The Cameron–Martin–Girsanov Theorem;167
14.1;1. A class of absolutely continuous probabilities;167
14.2;2. Transformation of Brownian motion;171
14.3;3. Girsanov's formula;179
14.4;Problems;184
15;Chapter 8. Asymptotic Estimates for Solutions;187
15.1;1. Unboundedness of solutions;187
15.2;2. Auxiliary estimates;189
15.3;3. Asymptotic estimates;195
15.4;4. Applications of the asymptotic estimates;200
15.5;5. The one-dimensional case;203
15.6;6. Counterexample;206
15.7;Problems;208
16;Chapter 9. Recurrent and Transient Solutions;211
16.1;1. Transient solutions;211
16.2;2. Recurrent solutions;215
16.3;3. Rate of wandering out to infinity;218
16.4;4. Obstacles;222
16.5;5. Transient solutions for degenerate diffusion;228
16.6;6. Recurrent solutions for degenerate diffusion;232
16.7;7. The one-dimensional case;234
16.8;Problems;237
17;References;241
18;Index;244