Abstract:We introduce a family of notions of equivalence for stochastic processes on spaces with an underlying filtration. These refine the notion of having the same distribution by taking account of the relation of the processes to their underlying filtrations. The weakest of these notions is the same as the notion of synonymity introduced by Aldous. Analysis of the strongest equivalence property leads to spaces with a strong universality property for adapted stochastic processes, which we call saturation. Spaces having this property contain ’strong’ solutions to a large class of stochastic integral equations.
D. Aldous, Weak convergence and the general theory of processes, preprint.
- Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
- M. T. Barlow, Construction of a martingale with given absolute value, Ann. Probab. 9 (1981), no. 2, 314–320. MR 606994
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- 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. MR 678335, DOI 10.1090/S0002-9947-1983-0678335-1 H. J. Keisler, Hyperfinite probability theory and probability logic, Lecture Notes, Univ. of Wisconsin (unpublished).
- H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297
- H. J. Keisler, Probability quantifiers, Model-theoretic logics, Perspect. Math. Logic, Springer, New York, 1985, pp. 509–556. MR 819545
- Frank B. Knight, A predictive view of continuous time processes, Ann. Probability 3 (1975), no. 4, 573–596. MR 383513, DOI 10.1214/aop/1176996302
- A. U. Kussmaul, Stochastic integration and generalized martingales, Research Notes in Mathematics, No. 11, Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1977. MR 0488281
- 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. MR 556680
- Dorothy Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108–111. MR 6595, DOI 10.1073/pnas.28.3.108
- Dorothy Maharam, Decompositions of measure algebras and spaces, Trans. Amer. Math. Soc. 69 (1950), 142–160. MR 36817, DOI 10.1090/S0002-9947-1950-0036817-8
- Michel Métivier and Jean Pellaumail, Stochastic integration, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. MR 578177
- 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. MR 648092, DOI 10.1090/S0002-9947-1982-0648092-2 H. Rodenhausen, The completeness theorem for adapted probability logic, Ph.D. Thesis, Heidelberg University.
- 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. MR 849100
- Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 286 (1984), 159-201
- MSC: Primary 60G05; Secondary 60E05
- DOI: https://doi.org/10.1090/S0002-9947-1984-0756035-8
- MathSciNet review: 756035