Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets in Initially Enlarged Filtrations

Abstract

In this paper, we extend Kreps' 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 in initially enlarged filtrations converge to the BSM economy in an initially enlarged filtration in a "strong" sense. The n-th discrete-time economy is generated by a scaled n-step random walk, based on an unscaled random variable with mean 0, variance 1, and bounded support. Moreover, the informed insider knows each functional generating the enlarged filtrations path-by-path. We confirm Kreps' conjecture in initially enlarged filtrations when the consumer's utility function U has asymptotic elasticity strictly less than one.