Segal W, Segal I E
Office of Policy Development and Research, Department of Housing and Urban Development, 451 7th Street, SW, Room 8212, Washington, DC 20410, USA.
Proc Natl Acad Sci U S A. 1998 Mar 31;95(7):4072-5. doi: 10.1073/pnas.95.7.4072.
A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.
量子效应为金融市场价格演变中的极端不规则性提供了一种自然的解释。相关变量缺乏同时可观测性以及尝试观测对这些变量值的干扰就是此类效应。交易者和经济学家已经注意到了这些特征，并且它们似乎是市场动态的内在属性。本文从维纳过程的相应推广及其在布莱克 - 斯科尔斯 - 默顿理论中的作用方面对这一解释进行探讨。证明了维纳过程作为维纳空间上平方可积函数的希尔伯特空间中一个稠密域上的半双线性形式的可微性，并将其推广到量子情形。这为伊藤随机积分理论的相应推广提供了基础。推导了布莱克 - 斯科尔斯期权定价公式到量子情形的扩展。
