Stochastic Integrals (eBook)

H. P. McKean (Autor)

Z. W. Birnbaum, E. Lukacs (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
154 Seiten
Elsevier Science (Verlag)
978-1-4832-5923-9 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 20.6.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 1-4832-5923-4 / 1483259234
• ISBN-13: 978-1-4832-5923-9 / 9781483259239