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Derivation and Martingales

  • Charles A. Hayes
  • Christian Y. Pauc

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 49)

Table of contents

  1. Front Matter
    Pages I-VII
  2. Introduction

    1. Charles A. Hayes, Christian Y. Pauc
      Pages 1-2
  3. Pointwise Derivation

    1. Front Matter
      Pages 3-3
    2. Charles A. Hayes, Christian Y. Pauc
      Pages 5-13
    3. Charles A. Hayes, Christian Y. Pauc
      Pages 41-77
    4. Charles A. Hayes, Christian Y. Pauc
      Pages 78-109
    5. Charles A. Hayes, Christian Y. Pauc
      Pages 110-119
  4. Martingales and Cell Functions

    1. Front Matter
      Pages 123-123
    2. Charles A. Hayes, Christian Y. Pauc
      Pages 125-148
    3. Charles A. Hayes, Christian Y. Pauc
      Pages 148-167
    4. Charles A. Hayes, Christian Y. Pauc
      Pages 167-172
    5. Charles A. Hayes, Christian Y. Pauc
      Pages 172-182
  5. Back Matter
    Pages 187-205

About this book

Introduction

In Part I of this report the pointwise derivation of scalar set functions is investigated, first along the lines of R. DE POSSEL (abstract derivation basis) and A. P. MORSE (blankets); later certain concrete situations (e. g. , the interval basis) are studied. The principal tool is a Vitali property, whose precise form depends on the derivation property studied. The "halo" (defined at the beginning of Part I, Ch. IV) properties can serve to establish a Vitali property, or sometimes produce directly a derivation property. The main results established are the theorem of JESSEN-MARCINKIEWICZ-ZYGMUND (Part I, Ch. V) and the theorem of A. P. MORSE on the universal derivability of star blankets (Ch. VI) . . In Part II, points are at first discarded; the setting is somatic. It opens by treating an increasing stochastic basis with directed index sets (Th. I. 3) on which premartingales, semimartingales and martingales are defined. Convergence theorems, due largely to K. KRICKEBERG, are obtained using various types of convergence: stochastic, in the mean, in Lp-spaces, in ORLICZ spaces, and according to the order relation. We may mention in particular Th. II. 4. 7 on the stochastic convergence of a submartingale of bounded variation. To each theorem for martingales and semi-martingales there corresponds a theorem in the atomic case in the theory of cell (abstract interval) functions. The derivates concerned are global. Finally, in Ch.

Keywords

Derivation Martingal Martingale Semimartingale function theorem

Authors and affiliations

  • Charles A. Hayes
    • 1
  • Christian Y. Pauc
    • 2
  1. 1.University of CaliforniaDavisUSA
  2. 2.University of NantesNantesFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-86180-2
  • Copyright Information Springer-Verlag Berlin Heidelberg 1970
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-86182-6
  • Online ISBN 978-3-642-86180-2
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site