McShane / Birnbaum / Lukacs Stochastic Calculus and Stochastic Models

1. Auflage 2014
ISBN: 978-1-4832-1877-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 252 Seiten, Web PDF

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Stochastic Calculus and Stochastic Models focuses on the properties, functions, and applications of stochastic integrals. The publication first ponders on stochastic integrals, existence of stochastic integrals, and continuity, chain rule, and substitution. Discussions focus on differentiation of a composite function, continuity of sample functions, existence and vanishing of stochastic integrals, canonical form, elementary properties of integrals, and the Itô-belated integral. The book then examines stochastic differential equations, including existence of solutions of stochastic differential equations, linear differential equations and their adjoints, approximation lemma, and the Cauchy-Maruyama approximation. The manuscript takes a look at equations in canonical form, as well as justification of the canonical extension in stochastic modeling; rate of convergence of approximations to solutions; comparison of ordinary and stochastic differential equations; and invariance under change of coordinates. The publication is a dependable reference for mathematicians and researchers interested in stochastic integrals.

Autoren/Hrsg.

McShane, E. J.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Calculus and Stochastic Models;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;8
6;Acknowledgments;10
7;Chapter I. Introduction;14
7.1;0 Motivation, and a Forward Look;14
7.2;1 Random Variables;17
7.3;2 Conditional Expectations;24
7.4;3 Stochastic Processes;29
8;Chapter II. Stochastic Integrals;39
8.1;1 Stochastic Models and Properties They Should Possess;39
8.2;2 Definition of the Integral;42
8.3;3 The Canonical Form;55
8.4;4 Elementary Properties of the Integral;58
8.5;5 The Itô-Belated Integral;64
9;Chapter III. Existence of Stochastic Integrals;72
9.1;1 Fundamental Lemma;72
9.2;2 Existence of the Stochastic Integral: First Theorem;74
9.3;3 Second Existence Theorem;79
9.4;4 Third and Fourth Existence Theorems;84
9.5;5 The Vanishing of Certain Integrals;87
9.6;6 Special Cases;92
9.7;7 Examples: Brownian Motions; Point Processes;96
9.8;8 Extension to the Itô-Belated Integral;102
10;Chapter IV. Continuity, Chain Rule, and Substitution;115
10.1;1 Continuity of Sample Functions;115
10.2;2 Differentiation of a Composite Function;127
10.3;3 Applications of Itô's Differentiation Formula;138
10.4;4 Substitution;147
10.5;5 Extension to Itô-Belated Integrals;156
11;Chapter V. Stochastic Differential Equations;165
11.1;1 Existence of Solutions of Stochastic Differential Equations;165
11.2;2 Linear Differential Equations and Their Adjoints;173
11.3;3 An Approximation Lemma;178
11.4;4 The Cauchy-Maruyama Approximation;189
12;Chapter VI. Equations in Canonical Form;193
12.1;1 Invariance under Change of Coordinates;193
12.2;2 Runge-Kutta Approximations;199
12.3;3 Comparision of Ordinary and Stochastic Differential Equations;204
12.4;4 Rate of Convergence of Approximations to Solutions;216
12.5;5 Continuous Dependence of the Solution on the Disturbance;227
12.6;6 Justification of the Canonical Extension in Stochastic Modeling;241
13;References;248
14;Subject Index;250