Foundations of infinitesimal stochastic analysis
type: Item
instance of: scholarly article
title: Foundations of infinitesimal stochastic analysis
zbMATH Open document ID: 0624.60052
author: José Manuel Bayod
author: Keith Stroyan
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This appears to be the first book relative to infinitesimal stochastic analysis specifically designed as a textbook. The authors have included selected exercises as part of their exposition. I give a brief description of the chapter contents. In Chapter 0, the authors motivate the possibility that nonstandard analysis could be useful for studies in probability by investigating finite probability problems. They then turn their attention to very basic concepts in nonstandard analysis. These concepts are discussed more fully in Appendix I and II. They give only the most minimal number of definitions and propositions that are needed for the remaining applications. This includes saturation, at least, to the point where the nonstandard superstructure model becomes comprehensive. For these purposes they utilize polyenlargements obtained by the usual direct limit process. Chapter 1 gives a general outline of the basic results now known for limited (finite) and unlimited (infinite) hyperfinite measures in the sense of Loeb. The authors have simplified Loeb's original approach, however. The chapter then concludes by examining relations between hyperfinite sums and the integral relative to hyperfinite measures. Chapter 2 presents the relation between Borel and hyperfinite measures. They first apply previous results and methods to classical Lebesgue measure making the necessary wide application of properties associated with the standard part map. The chapter concludes with applications to the more general Borel measure. Chapter 3 discusses Fubini-type results relative to products of hyperfinite measures and hyperfinite product measures. In particular this chapter leads to the important Fubini-type theorem of \textit{H. J. Keisler} [see An infinitesimal approach to stochastic analysis. Mem. Am. Math. Soc. 297, 184 p. (1984; Zbl 0529.60062)]. Chapter 4 deals with the infinitesimal view of distributions, laws and independence. Chapter 5 is devoted to some interesting path properti…
Mathematics Subject Classification ID: 60G05
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 03H10
zbMATH DE Number: 4013657
zbMATH Keywords: infinitesimal stochastic analysis
zbMATH Keywords: textbook
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: nonstandard superstructure model
zbMATH Keywords: hyperfinite sums
zbMATH Keywords: products of hyperfinite measures
zbMATH Keywords: path properties of processes
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4013657