Derivation and Martingales

1. Front Matter

   Pages I-VII

   PDF

2. Introduction

   1. Introduction

      • Charles A. Hayes, Christian Y. Pauc

      Pages 1-2

3. Pointwise Derivation

   1. Front Matter

      Pages 3-3

      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

      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

   PDF

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.

• Derivation
• Martingal
• Martingale
• Semimartingale
• function
• theorem

• University of California, Davis, USA

  Charles A. Hayes

• University of Nantes, Nantes, France

  Christian Y. Pauc

Policies and ethics