Premium
Convergence of optimal expected utility for a sequence of discrete‐time markets
Author(s) -
Kreps David M.,
Schachermayer Walter
Publication year - 2020
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12277
Subject(s) - counterexample , conjecture , mathematics , bounded function , limit of a sequence , discrete time and continuous time , mathematical economics , random walk , sequence (biology) , convergence (economics) , economics , limit (mathematics) , function (biology) , econometrics , combinatorics , mathematical analysis , statistics , evolutionary biology , biology , genetics , economic growth
We examine 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 that “approach” the BSM economy in a natural sense: The n th 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 E [ ζ 3 ] > 0 .