# zbMATH — the first resource for mathematics

Adapted probability distributions. (English) Zbl 0548.60019
Let x be a stochastic process on a probability space endowed with a filtration $$FB_ t$$, $$t\geq 0$$. Consider the family of new stochastic processes in finitely many time variables obtained from x by iterating the following two operations: (1) composition by continuous functions, and (2) conditional expectation with respect to $$FB_ t$$. Two stochastic processes x and y are said to have the same adapted distribution if each pair of new processes obtained from x and y by iterating (1) and (2) have the same finite dimensional distributions. By restricting to n iterations of (2) one obtains a sequence of stronger and stronger equivalence relations $$x\equiv_ ny$$, $$n=0,1,2,..$$. between stochastic processes. For each $$n\equiv_{n+1}$$ strictly refines $$\equiv_ n$$. The weakest relation $$x\equiv_ 0y$$ amounts to identity of the finite dimensional distributions of x and y. The next relation $$x\equiv_ 1y$$ amounts to synonimity of x and y (in the sense of D. Aldous). Finally, the strongest relation, $$x\equiv_ ny$$ for all n, means that x and y have the same adapted distribution.
Analysis of the strongest equivalence relation leads to probability spaces with a strong universality property for adapted stochastic processes, called saturation. On probability spaces having this property, there exist ’strong’ solutions to a large class of stochastic integral equations.
Reviewer: M.Iosifescu

##### MSC:
 60E05 Probability distributions: general theory 60G05 Foundations of stochastic processes 60A10 Probabilistic measure theory 60G48 Generalizations of martingales 03B48 Probability and inductive logic
##### Keywords:
finite dimensional distributions
Full Text:
##### References:
 [1] D. Aldous, Weak convergence and the general theory of processes, preprint. · Zbl 0372.60032 [2] Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15 – 46. · Zbl 0353.60052 · doi:10.1007/BF02756559 · doi.org [3] M. T. Barlow, Construction of a martingale with given absolute value, Ann. Probab. 9 (1981), no. 2, 314 – 320. · Zbl 0484.60037 [4] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0271.60009 [5] Douglas N. Hoover and Edwin Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), no. 1, 1 – 36, 37 – 58. · Zbl 0533.60063 [6] H. J. Keisler, Hyperfinite probability theory and probability logic, Lecture Notes, Univ. of Wisconsin (unpublished). · Zbl 0601.03005 [7] H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. · Zbl 0529.60062 · doi:10.1090/memo/0297 · doi.org [8] H. J. Keisler, Probability quantifiers, Model-theoretic logics, Perspect. Math. Logic, Springer, New York, 1985, pp. 509 – 556. [9] Frank B. Knight, A predictive view of continuous time processes, Ann. Probability 3 (1975), no. 4, 573 – 596. · Zbl 0317.60018 [10] A. U. Kussmaul, Stochastic integration and generalized martingales, Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1977. Research Notes in Mathematics, No. 11. · Zbl 0355.60045 [11] Peter A. Loeb, An introduction to nonstandard analysis and hyperfinite probability theory, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 105 – 142. · Zbl 0441.03027 [12] Dorothy Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 108 – 111. · Zbl 0063.03723 [13] Dorothy Maharam, Decompositions of measure algebras and spaces, Trans. Amer. Math. Soc. 69 (1950), 142 – 160. · Zbl 0041.18002 [14] Michel Métivier and Jean Pellaumail, Stochastic integration, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Probability and Mathematical Statistics. · Zbl 0463.60004 [15] Edwin Perkins, On the construction and distribution of a local martingale with a given absolute value, Trans. Amer. Math. Soc. 271 (1982), no. 1, 261 – 281. · Zbl 0506.60043 [16] H. Rodenhausen, The completeness theorem for adapted probability logic, Ph.D. Thesis, Heidelberg University. · Zbl 0714.60071 [17] K. D. Stroyan and José Manuel Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, vol. 119, North-Holland Publishing Co., Amsterdam, 1986. · Zbl 0624.60052 [18] Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York; North-Holland Publishing Co., Amsterdam-New York, 1978. · Zbl 0494.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.