Continuous Exponential Martingales and BMO
In three chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this book explains in detail the beautiful properties of continuous exponential martingales that play an essential role in various questions concerning the absolute con
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1579
Norihiko Kazamaki
Continuous Exponential Martingales and BMO
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Norihiko Kazamaki Department of Mathematics Faculty of Science Toyama University Gofuku, Toyama 930, Japan
Mathematics Subject Classification (1991): 60G
ISBN 3-540-58042-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58042-5 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready by author/editor SPIN: 10130035 46/3140-543210 - Printed on acid-free paper
Preface This book consists of three chapters and we shall deal entirely with continuous local martingales. Let M be a continuous local martingale and let
E(M)
= exp ( M
-
where (M) denotes the increasing process associated with M. As is well-known, it is a local martingale which plays an essential role in various questions concerning the absolute continuity of probability laws of stochastic processes. Our purpose here is to make a full report on the exciting results about BMO in the theory of exponential local martingales. BMO denotes the class of all uniformly integrable martingales M = (Mt , F t ) such that
where the supremum is taken over all stopping times T. A martingale in BMO is a probabilistic version of a function of bounded mean oscillation introduced in [31] by F. John and L. Nirenberg. In Chapter 1 we shall explain in detail the beautiful properties of an exponential local martingale. In Chapter 2 we shall collect the main tools to study various properties about continuous BMO-martingales. The fundamentally important result is that the following are equivalent:
(a)
ME BMO.
(b) E(M) is a uniformly integrable martingale which satisfies the reverse Holder inequality :
for some p > 1, where T is an arbitrary stopping time. (c)
(A p )
E(M) satisfies the condition: sUPT
liE
1100
1. These three conditions were originally introduced in the classical analysis. For example, the (A p ) condition is a probabilistic version of the one introduced in [62] by B. Muckenhoupt. In Chapter 3 we shall prove that it is a necessary and sufficient
Preface
Vl
condition for the validity of some weighted norm inequalities for martingales. Furthermore, we shall study two important subclasses of BMO, namely, the class L oo of all bounded martingales and the class H oo of all martingales M
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