zbMATH — the first resource for mathematics

Hyperfinite models of adapted probability logic. (English) Zbl 0601.03005
This paper deals with the model theory of adapted probability logic, a logic appropriate to the study of continuous time stochastic processes. The paper shows that a certain class of models, the hyperfinite models, has certain useful homogeneity properties. These properties allow the hyperfinite models to serve similar functions to those of saturated models in first order logic.
Reviewer: H.E.Kyburg

03B48 Probability and inductive logic
60A05 Axioms; other general questions in probability
03H99 Nonstandard models
Full Text: DOI
[1] Anderson, R.M., A nonstandard representation of Brownian motion and ito integration, Israel J. math., 25, 15-46, (1976) · Zbl 0353.60052
[2] Billingsley, P., Convergence on probability measures, (1968), Wiley New York · Zbl 0172.21201
[3] Chang, C.C.; Keisler, H.J., Model theory, (1973), North-Holland Amsterdam · Zbl 0276.02032
[4] Cutland, N., Nonstandard measure theory and its applications, Bull. London math. soc., 15, 529-589, (1983) · Zbl 0529.28009
[5] S. Fajardo, Completeness theorems for the general theory of processes, Proc. Sixth Latin American Logic Conference, to appear. · Zbl 0595.03014
[6] Fajardo, S., Probability logic with conditional expectation, Ann. pure appl. logic, 28, 137-161, (1985) · Zbl 0564.03019
[7] Henson, C.W.; Kaufmann, M.; Keisler, H.J., The strength of nonstandard methods in arithmetic, J. symbolic logic, 49, 1039-1058, (1984) · Zbl 0587.03048
[8] Hoover, D., Probability logic, Ann. math. logic, 14, 282-313, (1978) · Zbl 0394.03033
[9] Hoover, D.; Keisler, H.J., Adapted probability distributions, Trans. amer. math. soc., 286, 159-201, (1984) · Zbl 0548.60019
[10] Keisler, H.J., Hyperfinite probability theory and probability logic, () · Zbl 0167.01403
[11] Keisler, H.J., Probability quantifiers, (), 509-556, Chapter 14
[12] Keisler, H.J., Model theory for infinitary logic, (), x+208
[13] Keisler, H.J., An infinitesimal approach to stochastic analysis, Amer. math. soc. memoirs, 397, x + 184, (1984) · Zbl 0529.60062
[14] Keisler, H.J., A completeness proof for adapted probability logic, Ann. pure appl. logic, 31, 61-70, (1986) · Zbl 0601.03004
[15] Kreisel, G., Axiomatizations of nonstandard analysis that are conservative extensions of formal systems for classical standard analysis, (), 93-106
[16] Loeb, P.A., Conversion from standard to non-standard measure spaces and applications in probability theory, Trans. amer. math. soc., 211, 113-122, (1975) · Zbl 0312.28004
[17] Loeb, P.A., A functional approach to nonstandard measure theory, (), 251-261 · Zbl 0533.28008
[18] Rodenhausen, H., The completeness theorem for adapted probability logic, Doctoral dissertation, (1982), Heidelberg
[19] Rodenhausen, H., A characterization of nonstandard liftings of measurable functions and stochastic processes, Israel J. math., 43, 1-22, (1982) · Zbl 0504.60003
[20] Stroyan, K.D.; Bayod, J.M., Foundations of infinitesimal stochastic analysis, (1986), North-Holland Amsterdam · Zbl 0624.60052
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.