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

29 Pages Posted: 11 Jul 2019 Last revised: 6 Feb 2020

See all articles by David M. Kreps

David M. Kreps

Stanford Graduate School of Business

W. Schachermayer

University of Vienna

Date Written: July 1, 2019

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 nth discrete-time economy is generated by a scaled n-step random walk based on an unscaled random variable ζ 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 ζ such that Ε [ζ3]>0.

Suggested Citation

Kreps, David M. and Schachermayer, W., Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Market (July 1, 2019). Stanford University Graduate School of Business Research Paper No. 3802, 2019. Available at SSRN: https://ssrn.com/abstract=3417898 or http://dx.doi.org/10.2139/ssrn.3417898

David M. Kreps (Contact Author)

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

W. Schachermayer

University of Vienna

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