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
July 2019Working Paper No. 3802

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 has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function with asymptotic elasticity equal to 1, for ζ such that Ε [ζ3]>0.