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Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets

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  • David M. Kreps
  • Walter Schachermayer

Abstract

We examine Kreps' (2019) conjecture that optimal expected utility in the classic Black--Scholes--Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that "approach" the BSM economy in a natural sense: The $n$th discrete-time economy is generated by a scaled $n$-step random walk, based on an unscaled random variable $\zeta$ with mean zero, variance one, and bounded support. We confirm Kreps' conjecture if the consumer's utility function $U$ has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function $U$ with asymptotic elasticity equal to 1, for $\zeta$ such that $E[\zeta^3] > 0.$

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  • David M. Kreps & Walter Schachermayer, 2019. "Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets," Papers 1907.11424, arXiv.org, revised Feb 2020.
    • Handle: RePEc:arx:papers:1907.11424
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    References listed on IDEAS

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    Cited by:

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    2. Friedrich Hubalek & Walter Schachermayer, 2020. "Convergence of Optimal Expected Utility for a Sequence of Binomial Models," Papers 2009.09751, arXiv.org.

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