Adapted probability distributions. (English) Zbl 0548.60019

Hoover, Douglas N.; Keisler, H. Jerome

Trans. Am. Math. Soc. 286, 159-201 (1984).

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

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
[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
[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