Rich and saturated adapted spaces
The paper is a continuation of the author's series of works on nonstandard methods in various fields of mathematics. The main goal of the paper is to investigate the relationship between such properties of adapted spaces as richness and saturation. The notion of an adapted distribution and of a saturated adapted probability space were introduced by \textit{D. N. Hoover} and the author [Trans. Am. Math. Soc. 286, 159-201 (1984; Zbl 0548.60019)]. For a wide variety of existence problems every existence theorem which holds on some adapted space holds on a saturated adapted space. \textit{S. Fajardo} and the author [Adv. Math. 120, No. 2, 191-257 (1996; Zbl 0860.60028)] developed a new method for proving existence theorems in probability theory using the notion of a neocompact set. The author introduces the notion of a law mapping which is the special case of the notion of a law structure. Some particular law mappings which correspond to the distribution of a random variable and adapted distribution of a stochastic process are investigated. The main results of the paper show that saturation is equivalent to richness for probability spaces, adapted spaces with finite time sets, and adapted spaces with infinite time sets. It is proved that every rich adapted space with rational times induces a saturated right continuous adapted space with real times.
scientific article; zbMATH DE number 1044180
scientific article