Ikeda / Watanabe Stochastic Differential Equations and Diffusion Processes

2. Auflage 2014
ISBN: 978-1-4832-9615-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 24, 572 Seiten, Web PDF

Reihe: North-Holland Mathematical Library

Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sections discussing complex (conformal) martingales and Kahler diffusions have been added.

Autoren/Hrsg.

Ikeda, N.

Watanabe, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Differential Equations and Diffusion Processes;4
3;Copyright Page;5
4;Table of Contents;14
5;Dedication;6
6;Preface to the Second Edition;8
7;Preface;10
8;General Notation;16
9;CHAPTER I. Preliminaries;18
9.1;1. Basic notions and notations;18
9.2;2. Probability measures on a metric space;19
9.3;3. Expectations, conditional expectations and regular conditional probabilities;28
9.4;4. Continuous stochastic processes;33
9.5;5. Stochastic processes adapted to an increasing family of sub s -fields;37
9.6;6. Martingales;42
9.7;7. Brownian motions;57
9.8;8. Poisson random measure;59
9.9;9. Point processes and Poisson point processes;60
10;CHAPTER II. Stochastic Integrals and Ito's Formula;62
10.1;1. Itô's definition of stochastic integrals;62
10.2;2, Stochastic integrals with respect to martingales;70
10.3;3. Stochastic integrals with respect to point processes;76
10.4;4. Semi-martingales;80
10.5;5. Itô's formula;83
10.6;6. Martingale characterization of Brownian motions and Poisson point processes;90
10.7;7. Representation theorem for semi-martingales;101
11;CHAPTER III. Stochastic Calculus;114
11.1;1. The space of stochastic differentials;114
11.2;2. Stochastic differential equations with respect to quasimartingales;120
11.3;3. Moment inequalities for martingales;127
11.4;4. Some applications of stochastic calculus to Brownian motions;130
11.5;5. Exponential martingales;166
11.6;6. Conformal martingales;172
12;CHAPTER IV. Stochastic Differential Equations;176
12.1;1. Definition of solutions;176
12.2;2. Existence theorem;184
12.3;3. Uniqueness theorem;195
12.4;4. Solution by transformation of drift and by time change;207
12.5;5. Diffusion processes;219
12.6;6. Diffusion processes generated by differential operators and stochastic differential equations;229
12.7;7. Stochastic differential equations with boundary conditions;234
12.8;8. Examples;249
12.9;9. Stochastic differential equations with respect to Poisson point processes;261
13;CHAPTER V. Diffusion Processes on Manifolds;264
13.1;1. Stochastic differential equations on manifolds;264
13.2;2. Flow of diffeomorphisms;271
13.3;3. Heat equation on a manifold;286
13.4;4. Non-degenerate diffusions on a manifold and their horizontal lifts;292
13.5;5. Stochastic parallel displacement and heat equation for tensor fields;314
13.6;6. The case with boundary conditions;325
13.7;7. Kahler diffusions;358
13.8;8. Malliavin's stochastic calculus of variation for Wiener functionals;366
13.9;9. Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws);392
13.10;10. The case of stochastic differential equations: Applications to heat kernels;408
14;CHAPTER VI. Theorems on Comparison and Approximation and their Applications;454
14.1;1. A comparison theorem for one-dimensional ltd processes;454
14.2;2. An application to an optimal control problem;458
14.3;3. Some results on one-dimensional diffusion processes;463
14.4;4. Comparison theorem for one-dimensional projection of diffusion processes;469
14.5;5. Applications to diffusions on Riemannian manifolds;477
14.6;6. Stochastic line integrals along the paths of diffusion processes;484
14.7;7. Approximation theorems for stochastic integrals and stochastic differential equations;497
14.8;8. The support of diffusion processes;534
14.9;9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve;549
15;Bibliography;558
16;Index;568