Skip to main content
SpringerLink
Log in

Derivation and Martingales

  • Book
  • © 1970

Overview

Authors:
  1. Charles A. Hayes

    1. University of California, Davis, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  2. Christian Y. Pauc

    1. University of Nantes, Nantes, France

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 49)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (11 chapters)

  1. Front Matter

    Pages I-VII
    Download chapter PDF

  2. Introduction

    1. Introduction

      • Charles A. Hayes, Christian Y. Pauc
      Pages 1-2

  3. Pointwise Derivation

    1. Front Matter

      Pages 3-3
      Download chapter PDF

    2. Derivation Bases

      • Charles A. Hayes, Christian Y. Pauc
      Pages 5-13

    3. Derivation Theorems for σ-additive Set Functions under Assumptions of the Vitali Type

      • Charles A. Hayes, Christian Y. Pauc
      Pages 13-30

    4. The Converse Problem I: Covering Properties Deduced from Derivation Properties of σ-additive Set Functions

      • Charles A. Hayes, Christian Y. Pauc
      Pages 30-40

    5. Halo Assumptions in Derivation Theory. Converse Problem II

      • Charles A. Hayes, Christian Y. Pauc
      Pages 41-77

    6. The Interval Basis. The Theorem of Jessen-Marcinkiewicz-Zygmund

      • Charles A. Hayes, Christian Y. Pauc
      Pages 78-109

    7. A. P. Morse’s Blankets

      • Charles A. Hayes, Christian Y. Pauc
      Pages 110-119

  4. Martingales and Cell Functions

    1. Front Matter

      Pages 123-123
      Download chapter PDF

    2. Theory without an Intervening Measure

      • Charles A. Hayes, Christian Y. Pauc
      Pages 125-148

    3. Theory in a Measure Space without Vitali Conditions

      • Charles A. Hayes, Christian Y. Pauc
      Pages 148-167

    4. Theory in a Measure Space with Vitali Conditions

      • Charles A. Hayes, Christian Y. Pauc
      Pages 167-172

    5. Applications

      • Charles A. Hayes, Christian Y. Pauc
      Pages 172-182

  5. Back Matter

    Pages 187-205
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • University of California, Davis, USA

    Charles A. Hayes

  • University of Nantes, Nantes, France

    Christian Y. Pauc

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation