McKean / Birnbaum / Lukacs Stochastic Integrals

1. Auflage 2014
ISBN: 978-1-4832-5923-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 154 Seiten, Web PDF

Stochastic Integrals discusses one area of diffusion processes: the differential and integral calculus based upon the Brownian motion. The book reviews Gaussian families, construction of the Brownian motion, the simplest properties of the Brownian motion, Martingale inequality, and the law of the iterated logarithm. It also discusses the definition of the stochastic integral by Wiener and by Ito, the simplest properties of the stochastic integral according to Ito, and the solution of the simplest stochastic differential equation. The book explains diffusion, Lamperti's method, forward equation, Feller's test for the explosions, Cameron-Martin's formula, the Brownian local time, and the solution of dx=e(x) db + f(x) dt for coefficients with bounded slope. It also tackles Weyl's lemma, diffusions on a manifold, Hasminski's test for explosions, covering Brownian motions, Brownian motions on a Lie group, and Brownian motion of symmetric matrices. The book gives as example of a diffusion on a manifold with boundary the Brownian motion with oblique reflection on the closed unit disk of R squared. The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics.

Autoren/Hrsg.

McKean, H. P.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Integrals;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;8
6;List of Notations;12
7;Chapter 1. Brownian Motion;18
7.1;Introduction;18
7.2;1.1 Gaussian Families;20
7.3;1.2 Construction of the Brownian Motion;22
7.4;1.3 Simplest Properties of the Brownian Motion;26
7.5;1.4 A Martingale Inequality;28
7.6;1.5 The Law of the Iterated Logarithm;29
7.7;1.6 Lévy's Modulus;31
7.8;1.7 Several-Dimensional Brownian Motion;34
8;Chapter 2. Stochastic Integrals and Differentials;37
8.1;2.1 Wiener's Definition of the Stochastic Integral;37
8.2;2.2 Itô's Definition of the Stochastic Integral;38
8.3;2.3 Simplest Properties of the Stochastic Integral;41
8.4;2.4 Computation of a Stochastic Integral;45
8.5;2.5 A Time Substitution;46
8.6;2.6 Stochastic Differentials and Itô's Lemma;49
8.7;2.7 Solution of the Simplest Stochastic Differential Equation;52
8.8;2.8 Stochastic Differentials under a Time Substitution;58
8.9;2.9 Stochastic Integrals and Differentials for Several-Dimensional Brownian Motion;60
9;Chapter 3. Stochastic Integral Equations (d = 1);67
9.1;3.1 Diffusions;67
9.2;3.2 Solution of dx = e(x) db + f(x) dt for Coefficients with Bounded Slope;69
9.3;3.3 Solution of dx = e(x) db + f(x) dt for General Coefficients Belonging to C1(R1);71
9.4;3.4 Lamperti's Method;77
9.5;3.5 Forward Equation;78
9.6;3.6 Feller's Test for Explosions;82
9.7;3.7 Cameron-Martin's Formula;84
9.8;3.8 Brownian Local Time;85
9.9;3.9 Reflecting Barriers;88
9.10;3.10 Some Singular Equations;94
10;Chapter 4. Stochastic Integral Equations (d = 2);99
10.1;4.1 Manifolds and Elliptic Operators;99
10.2;4.2 Weyl's Lemma;102
10.3;4.3 Diffusions on a Manifold;107
10.4;4.4 Explosions and Harmonic Functions;115
10.5;4.5 Hasminskii's Test for Explosions;119
10.6;4.6 Covering Brownian Motions;125
10.7;4.7 Brownian Motions on a Lie Group;132
10.8;4.8 Injection;134
10.9;4.9 Brownian Motion of Symmetric Matrices;140
10.10;4.10 Brownian Motion with Oblique Reflection;143
11;References;150
12;Subject Index;156