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Weak measurability and characterizations of risk

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Author Info

  • M. Ali Khan

    () (Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA)

  • Yeneng Sun

    () (Department of Mathematics, National University of Singapore, Singapore 119260, SINGAPORE)

Abstract

In the context of a continuum of random variables, arising, for example, as rates of return in financial markets with a continuum of assets, or as individual responses in games with a continuum of players, an important economic issue is to show how idiosyncratic risk can be removed through some device of aggregation or diversification when such risk is explicitly introduced into the model. In this paper, we use recent work of Al-Najjar [1] as a general backdrop to provide a review of the basic issues involved when the continuum is formulated as the Lebesgue interval. We present two examples to argue that the fundamental problem of the non-measurability of sample functions, originally identified by Doob, and further elaborated by Feldman, Gilles and Judd in the economic literature, simply cannot be bypassed by reinterpretations of standard results. We also provide an equivalence result in the spirit of Al-Najjar's efforts; but argue that this elementary result does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables, and as such, is incapable of yielding anything of substantive economic interest beyond this law.

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Bibliographic Info

Article provided by Springer in its journal Economic Theory.

Volume (Year): 13 (1999)
Issue (Month): 3 ()
Pages: 541-560

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Handle: RePEc:spr:joecth:v:13:y:1999:i:3:p:541-560

Note: Received: April 23, 1998; revised version: April 28, 1998
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Related research

Keywords: Weak measurability Pettis integrability · Bochner integrability · Decomposition · Law of large numbers · Large games. ·;

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Cited by:
  1. M Ali Khan & Yeneng Sun, 2002. "Exact Arbitrage and Portfolio Analysis in Large Asset Markets," Economics Working Paper Archive 484, The Johns Hopkins University,Department of Economics.
  2. Khan, M. Ali & Sun, Yeneng, 2003. "Exact arbitrage, well-diversified portfolios and asset pricing in large markets," Journal of Economic Theory, Elsevier, vol. 110(2), pages 337-373, June.
  3. Peter Hammond & Yeneng Sun, 2001. "Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The Symmetric Case," Working Papers 01015, Stanford University, Department of Economics.
  4. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
  5. Khan, A. & Sun, Y., 2000. "Asymptotic Arbitrage and the APT with or Without Measure-Theoretic Structures," Papiers d'Economie Mathématique et Applications 2000.81, Université Panthéon-Sorbonne (Paris 1).

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