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arXiv · 1907.11424

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

David M. Kreps·Walter Schachermayer
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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. 2019-07-26. Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets. https://arxiv.org/abs/1907.11424

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