Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Market
29 Pages Posted: 11 Jul 2019 Last revised: 6 Feb 2020
Date Written: July 1, 2019
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.
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