Two-Parameter Martingales and Their Quadratic Variation
This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Me
- PDF / 7,567,378 Bytes
- 182 Pages / 439.37 x 666.142 pts Page_size
- 51 Downloads / 181 Views
1308 Peter lmkeller
Two-Parameter Martingales and Their Quadratic Variation
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Peter lmkeller Mathematisches lnstitut der Ludwig-Maximilians-Universitat Munchen Theresienstr. 39, 8000 Munchen, Federal Republic of Germany
Mathematics Subject Classification (1980): 60G07, 60G44, 60E 15, 60G42, 60G48, 60G55, 60G60, 60H05 ISBN 978-3-540-19233-6 Springer-Verlag Berlin Heidelberg New York ISBN 978-0-387-19233-8 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. ©Springer-Verlag Berlin Heidelberg 1988 Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Contents
Introduction 0. Notations and conventions
23
I. Basics; processes depending on a parameter
28
1. Regular functions and their discontinuities
29
2. Measurability and basic processes
34
3. One-directional stopping; a section theorem
42
4. One-directional projection theorems
47
5. One-directional decomposition theorems
52
6. Regularity of one-directional projections
59
II. Two-parameter processes
66
7. Two-directional stopping; a predictable section theorem 8. Martingale inequalities and applications
67 72
9. A stochastic integral and the regularity of martingales
80
10. Two-directional projection theorems
86
11. Two-directional decomposition theorems
92
III. Jumps of martingales and their compensation
99
12. Simple sets and increasing processes
101
13. The decomposition of simple sets
109
14. The jumps of regular martingales
116
15. The compensation of martingale jumps
124
IV. Quadratic variation and structure of martingales
131
16. Some general results on quadratic variation
132
17. Quadratic variations of compensated components
139
18. Orthogonality of compensated components
147
19. The structure of square integrable martingales
151
JV
20. The quadratic variation of square and L log+Lintegrable martingales
157
21. Inequalities for the quadratic variation1
a counterexample
162
References
169
Index of definitions
174
Index of special symbols
176
Introduction
There are many fields of stochastics where multi-parameter processes can be encountered. For example, to register the positions of the spins of a ferromagnetic substance at a fixed instant of time, one has to attach an appropriate state space to every point of a three-dimensional lattice. Mathematically this leads to a family of random variables indexed by a subset of
m3 ,
a
special case of a so-cal
Data Loading...
