Derivation and Martingales

ISBN-13: 9783642861826 / Angielski / Miękka / 2012 / 206 str.

Charles A. Hayes; C. y. Pauc; Charles A. Hayes

Opis

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.

Spis treści

I Pointwise Derivation.- I: Derivation Bases.- 1. Setting and general notation.- 2. dePossel’s derivation basis.- 3. Examples of bases.- 4. Pretopological notions.- 5. Comparison lemmas.- II: Derivation Theorems for ?-additive Set Functions under Assumptions of the Vitali Type.- 1. The individual Vitali assumption.- 2. The individual full derivation theorem for Radon or ?-fmite ?-integrals.- 3. The individual full derivation theorem for Radon measures.- 4. Class derivation theorems.- 5. Relation to Younovitch’s derivation theorem.- 6. The strong Vitali property.- 7. Half-regular and regular branches of a derivation basis.- III: The Converse Problem I: Covering Properties Deduced from Derivation Properties of ?-additive Set Functions.- 1. dePossel’s equivalence theorem.- 2. A necessary and sufficient condition for a weak derivation basis to derive a ?-finite ?-measure (Radon measure) ?.- 3. Younovitch’s equivalence theorem.- 4. A converse theorem for bases deriving the ?(q)-functions, q ? 1.- IV: Halo Assumptions in Derivation Theory. Converse Problem II.- 1. A. P. Morse’s halo properties.- 2. Abstract version of the strong Vitali theorem modelled after Banach.- 3. Abstract version of the strong Vitali theorem modelled after Carathéodory.- 4. Weak halo evanescence condition.- 5. Further criteria for the validity of the Density Theorem involving the weak halo.- 6. An individual derivability condition of Busemann-Feller type.- 7. The weak halo property in general bases.- 8. Product invariance of a weak halo property.- V: The Interval Basis. The Theorem of Jessen-Marcin-Kiewicz-Zygmund.- 1. The interval basis as a weak derivation basis.- 2. Theorem of Jessen-Marcinkiewicz-Zygmund.- 3. Properties of the halo function as consequences of derivation properties.- 4. Saks’ counterexample.- 5. The parallelepipedon basis.- 6. Saks’ “rarity” theorem.- VI: A. P. Morse’s Blankets.- 1. Nets.- 2. Hives.- 3. Fundamental covering theorems.- 4. Star blankets.- II Martingales and Cell Functions.- I: Theory without an Intervening Measure.- 1. Additive functions.- 2. ?-additive functions.- 3. Premartingales, semi-martingales, and martingales.- 4. Ordered space of martingales of basis(??).- 5. Integrals of premartingales.- 6. Martingales and additive functions.- 7. ?-additive martingales.- 8. Induced martingales.- 9. Premartingales and cell functions.- 10. Integrals of cell functions.- 11. Convergence theorems for martingales of bounded variation when ? is a measure algebra.- II: Theory in a Measure Space without Vitali Conditions.- 1. Preliminaries.- 2. Absolutely continuous and singular premartingales.- 3. Stochastic processes.- 4. Stochastic convergence.- 5. Mean convergence of order 1.- 6. Convergence in Orlicz spaces.- 7. Cell functions.- III: Theory in a Measure Space with Vitali Conditions.- 1. Preliminaries and definitions.- 2. Vitali conditions.- 3. Order convergence of martingales.- 4. Necessity of the Vitali conditions.- 5. Order convergence of submartingales.- 6. Order convergence of cell functions.- IV: Applications.- 1. Pointwise setting.- 2. Specifically pointwise concepts and results. Convergence almost everywhere.- 3. Martingales in the classical sense.- 4. Product spaces.- 5. The Radon- Nikodym integrand defined as a derivate.- 6. Representation of the spaces Lx as spaces of cell functions.- 7. Pointwise derivation of cell functions.- 8. Examples of concrete cell bases.- 9. Stochastic bases on a group.- Complements.- 1°. Derivation of vector-valued integrals.- 2°. Functional derivatives.- 3°. Topologies generated by measures.- 4°. Vitali’s theorem for invariant measures.- 5°. Global derivatives in locally compact topological groups..- 6°. Submartingales with decreasing stochastic bases.- 7°. Vector-valued martingales and derivation.- 9°. Derivation of measures.