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

TL;DR

This paper proves that optimal expected utility in a sequence of discrete-time markets converges to the continuous Black-Scholes-Merton model within initially enlarged filtrations, confirming Kreps' conjecture under certain utility conditions.

Contribution

It extends Kreps' conjecture to initially enlarged filtrations, demonstrating convergence of discrete-time optimal utilities to the continuous model with insider information.

Findings

Convergence of discrete-time utilities to continuous BSM utility in enlarged filtrations.

Validation of Kreps' conjecture under utility functions with elasticity less than one.

The model incorporates insider knowledge and bounded support random variables.

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.