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Foundations of infinitesimal stochastic analysis. (English) Zbl 0624.60052
Studies in Logic and the Foundations of Mathematics, Vol. 119. Amsterdam- New York-Oxford: North-Holland. XII, 478 p. $ 47.50; Dfl. 150.00 (1986).
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 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 properties of processes. In particular continuous, decent, Lebesgue and Borel.
Chapter 6 is a significant introduction to the time evolution of stochastic processes. The authors include progressive and previsible measurability, nonanticipating processes, measurable and internal stopping, martingales, predictable processes. Finally, they extend some of these ideas to [0,\(\infty)\). In Chapter 7, the authors investigate pathwise integration of a process that is a sum of a martingale and a process of bounded variation by means of infinitesimal analysis.
The authors conclude with an extensive list of references and the above mentioned appendices. This book should certainly be included in the library of any mathematician familiar with infinitesimal analysis and stochastic processes. It would be useful for a graduate level course in the application of infinitesimal analysis to stochastic processes assuming that the students have a good intuitive comprehension of how stochastic processes relate to collections.
Reviewer: R.A.Herrmann

60G05 Foundations of stochastic processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
03H10 Other applications of nonstandard models (economics, physics, etc.)