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

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

By
David M. Kreps, Walter Schachermayer
Mathematical Finance. June
2020

We examine Kreps’ conjecture that optimal expected utility in the classic Black-Scholes-Merton 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 0, variance 1, 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 E3] > 0.