Abstract
A natural and general way to model information in a microeconomic system is to specify it as a sub-σ-field (of the σ-field of events, or measurable subsets of the set of states of the world). Several apparently different definitions of convergence have been introduced. For continuous time trading models, relationships between continuous information (assumed to be increasing and right continuous) and various continuity properties of asset prices have been studied by Harrison and Huang. For a static general equilibrium pure exchange economy, the information topology advocated by Allen yields continuity of microeconomic behavior (i.e., continuous dependence of agents’ demand functions on their state dependent utilities, initial endowments, and information) and of the value of information. Additional convergence notions for σ-fields appear in the probability theory literature. This paper compares these convergence notions and analyzes when it can be said that one is weaker than another.
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References
Allen, Beth, 1983a, Neighboring information and distributions of agents’ characteristics under uncertainty, Journal of Mathematical Economics, forthcoming.
Allen, Beth, 1983b, General equilibrium with information sales, mimeo, University of Bonn.
Billingsley, Patrick, 1968, Convergence of Probability Measures (New York: Wiley).
Boylan, Edward S., 1971, Equiconvergence of martingales, Annals of Mathematical Statistics, 42, 552–559.
Cotter, Kevin, 1983, Consumer choice of information with a “weak” topology, mimeo, University of Minnesota, Minneapolis.
Dunford, Nelson and Jacob T. Schwartz, 1958, Linear Operators Part I: General Theory (New York: Interscience).
Fetter, Helga, 1977, On the continuity of conditional expectations, Journal of Mathematical Analysis and Applications, 61, 227–231.
Halmos, Paul R., 1950, Measure Theory (New York: Van Nostrand).
Harrison, J. Michael, 1982, “Stochastic Calculus and Its Applications,” Lecture Notes, Graduate School of Business, Stanford University.
Harrison, J. Michael and Stanley R. Pliska, 1981, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11, 215–260.
Hirsch, Morris W., 1976, Differential Topology (New York: Springer Verlag).
Huang, Chi-fu, 1983, Information structure and equilibrium asset prices, mimeo, Graduate School of Business, Stanford University.
Kudö, Hirokichi, 1974, A note on the strong convergence of σ-algebras, Annals of Probability, 2, 76–83.
Neveu, Jacques, 1965, Mathematical Foundations of the Calculus of Probability (San Francisco: Holden-Day).
Neveu, J., 1972, Note on the tightness of the metric on the set of complete sub-σ-algebras of a probability space, Annals of Mathematical Statistics, 43, 1369–1371.
Rogge, L., 1974, Uniform inequalities for conditional expectations, Annals of Probability, 2, 486–489.
Rudin, Walter, 1973, Functional Analysis (New York: McGraw-Hill).
Schwartz, Laurent, 1973, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures (Oxford University Press).
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© 1984 Springer-Verlag Berlin Heidelberg
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Allen, B. (1984). Convergence of σ-Fields and Applications to Mathematical Economics. In: Hammer, G., Pallaschke, D. (eds) Selected Topics in Operations Research and Mathematical Economics. Lecture Notes in Economics and Mathematical Systems, vol 226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45567-4_11
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DOI: https://doi.org/10.1007/978-3-642-45567-4_11
Publisher Name: Springer, Berlin, Heidelberg
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