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 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 .

中文翻译：

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一系列离散时间市场的最优预期效用的收敛


我们研究了 Kreps 的猜想，即经典 Black-Scholes-Merton (BSM) 经济中的最优预期效用是自然意义上“接近”BSM 经济的一系列离散时间经济体的最优预期效用的极限：离散时间经济是通过缩放的n步随机游走生成的，基于均值为 0、方差 1 和有界支持的未缩放随机变量 ζ。如果消费者的效用函数U的渐近弹性严格小于 1，我们证实了 Kreps 的猜想，并且我们为效用函数U的渐近弹性等于 1 的猜想提供了一个反例，对于 ze 。