Adapted probability distributions
Authors:
Douglas N. Hoover and H. Jerome Keisler
Journal:
Trans. Amer. Math. Soc. 286 (1984), 159201
MSC:
Primary 60G05; Secondary 60E05
MathSciNet review:
756035
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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.
 [1981]
D. Aldous, Weak convergence and the general theory of processes, preprint.
 [1976]
Robert
M. Anderson, A nonstandard representation for Brownian motion and
Itô integration, Israel J. Math. 25 (1976),
no. 12, 15–46. MR 0464380
(57 #4311)
 [1981]
M.
T. Barlow, Construction of a martingale with given absolute
value, Ann. Probab. 9 (1981), no. 2,
314–320. MR
606994 (82j:60079)
 [1968]
Patrick
Billingsley, Convergence of probability measures, John Wiley
& Sons, Inc., New YorkLondonSydney, 1968. MR 0233396
(38 #1718)
 [1982]
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 (85d:60111), http://dx.doi.org/10.2307/1999003
 [1979]
H. J. Keisler, Hyperfinite probability theory and probability logic, Lecture Notes, Univ. of Wisconsin (unpublished).
 [1982]
H.
Jerome Keisler, An infinitesimal approach to stochastic
analysis, Mem. Amer. Math. Soc. 48 (1984),
no. 297, x+184. MR 732752
(86c:60086), http://dx.doi.org/10.1090/memo/0297
 [1983]
H.
J. Keisler, Probability quantifiers, Modeltheoretic logics,
Perspect. Math. Logic, Springer, New York, 1985, pp. 509–556. MR
819545
 [1975]
Frank
B. Knight, A predictive view of continuous time processes,
Ann. Probability 3 (1975), no. 4, 573–596. MR 0383513
(52 #4394)
 [1977]
A.
U. Kussmaul, Stochastic integration and generalized
martingales, Pitman Publishing, LondonSan Francisco,
Calif.Melbourne, 1977. Research Notes in Mathematics, No. 11. MR 0488281
(58 #7841)
 [1979]
Peter
A. Loeb, An introduction to nonstandard analysis and hyperfinite
probability theory, Probabilistic analysis and related topics, Vol. 2,
Academic Press, New YorkLondon, 1979, pp. 105–142. MR 556680
(80m:60005)
 [1942]
Dorothy
Maharam, On homogeneous measure algebras, Proc. Nat. Acad.
Sci. U. S. A. 28 (1942), 108–111. MR 0006595
(4,12a)
 [1950]
Dorothy
Maharam, Decompositions of measure algebras and
spaces, Trans. Amer. Math. Soc. 69 (1950), 142–160. MR 0036817
(12,167b), http://dx.doi.org/10.1090/S00029947195000368178
 [1980]
Michel
Métivier and Jean
Pellaumail, Stochastic integration, Academic Press [Harcourt
Brace Jovanovich, Publishers], New YorkLondonToronto, Ont., 1980.
Probability and Mathematical Statistics. MR 578177
(82b:60060)
 [1982]
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
(83h:60044), http://dx.doi.org/10.1090/S00029947198206480922
 [1982]
H. Rodenhausen, The completeness theorem for adapted probability logic, Ph.D. Thesis, Heidelberg University.
 [1983]
K.
D. Stroyan and José
Manuel Bayod, Foundations of infinitesimal stochastic
analysis, Studies in Logic and the Foundations of Mathematics,
vol. 119, NorthHolland Publishing Co., Amsterdam, 1986. MR 849100
(87m:60001)
 [1978]
Claude
Dellacherie and PaulAndré
Meyer, Probabilities and potential, NorthHolland Mathematics
Studies, vol. 29, NorthHolland Publishing Co., AmsterdamNew York;
NorthHolland Publishing Co., AmsterdamNew York, 1978. MR 521810
(80b:60004)
 [1981]
 D. Aldous, Weak convergence and the general theory of processes, preprint.
 [1976]
 R. M. Anderson, A nonstandard representation of Brownian motion and Itô integration, Israel J. Math. 25 (1976), 1546. MR 0464380 (57:4311)
 [1981]
 M. T. Barlow, Construction of a martingale with a given absolute value, Ann. Probab. 9 (1981), 314320. MR 606994 (82j:60079)
 [1968]
 P. Billingsley, Convergence of probability measure, Wiley, New York, 1968. MR 0233396 (38:1718)
 [1982]
 D. N. Hoover and E. Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), 158. MR 678335 (85d:60111)
 [1979]
 H. J. Keisler, Hyperfinite probability theory and probability logic, Lecture Notes, Univ. of Wisconsin (unpublished).
 [1982]
 , An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. (to appear). MR 732752 (86c:60086)
 [1983]
 , Probability quantifiers, Abstract Model Theory and Logics of Mathematical Concepts (J. Barwise and S. Feferman, eds.), SpringerVerlag, Berlin and New York (to appear). MR 819545
 [1975]
 F. B. Knight, A predictive view of continuous time processes, Ann. Probab. 3 (1975), 573596. MR 0383513 (52:4394)
 [1977]
 A. U. Kussmaul, Stochastic integration and generalized martingales, Pitman, New York, 1977. MR 0488281 (58:7841)
 [1979]
 P. A. Loeb, An introduction to nonstandard analysis and hyperfinite probability theory, Probabilistic Analysis and Related Topics , (Bharucha and Reid, eds.), Academic Press, New York, 1979, pp. 105142. MR 556680 (80m:60005)
 [1942]
 D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108111. MR 0006595 (4:12a)
 [1950]
 , Decompositions of measure algebras and spaces, Trans. Amer. Math. Soc. 69 (1950), 142160. MR 0036817 (12:167b)
 [1980]
 M. Metivier and J. Pellaumail, Stochastic integration, Academic Press, New York, 1980. MR 578177 (82b:60060)
 [1982]
 E. Perkins, On the construction and distribution of a local martingale with a given absolute value, Trans. Amer. Math. Soc. 271 (1982), 261281. MR 648092 (83h:60044)
 [1982]
 H. Rodenhausen, The completeness theorem for adapted probability logic, Ph.D. Thesis, Heidelberg University.
 [1983]
 K. D. Stroyan and J. M. Bayod, Foundations of infinitesimal stochastic analysis, NorthHolland, Amsterdam (to appear). MR 849100 (87m:60001)
 [1978]
 P. A. Meyer and C. Dellacherie, Probabilities and potential, NorthHolland Mathematical Studies No. 29, NorthHolland, Amsterdam, 1978. MR 521810 (80b:60004)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407560358
PII:
S 00029947(1984)07560358
Article copyright:
© Copyright 1984
American Mathematical Society
