The Numeraire in Stochastic Finance Theory

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。与预期的对数效用最大化问题相对应的最优财富过程可能是最受欢迎和最重要的投资策略，因为它具有丰富的最优化性质。对数最优策略最显著的特征之一是其显著的数值性：每一个非负财富过程，当被对数最优过程折现时，都成为上鞅。出于这最后一个原因，对数最优财富过程通常被称为“数字”。本项目研究金融市场建模中与数字有关的问题。重点讨论了以下几个方面：(1)数值器的存在与市场生存能力之间的深层联系，这将加强和简化随机金融学的理论基础；(2)数值器性质在非传统模型中的几个分支；(3)数值器最优性质列表的扩充；(4)对数投资者具有异质信念的不完备金融市场中一般均衡的闭式表达式；(5)关于市场输入的连续性和可微性，例如统计概率和信息。这个项目提供了一种新的方法来分析理论市场模型的适当性，以成功地反映现实世界的现象，以及研究金融市场的更深层次的性质。这种方法的力量在于其包罗万象的性质和简单性，因为它完全取决于市场中单个非凡投资机会的存在和结构，这就是数字。该项目将产生的更广泛的影响是双重的：(1)它将促进概率和金融学的科学领域之间的相互影响，思想从一个领域到另一个领域的双向流动的丰富互动；(2)它将从一个新的视角改变教学金融数学的教学方法。事实上，在该项目实施后，预计将出现一本将这一新方法纳入随机金融的专著。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The optimal wealth process corresponding to the expected logarithmic utility maximization problem is probably the most popular and important investment strategy, due to the abundance of its optimality properties. One of the most prominent characteristics of the log-optimal strategy is its remarkable numeraire property: every nonnegative wealth process, when discounted by the log-optimal one, becomes a supermartingale. For this last reason, the log-optimal wealth process is commonly referred to as "the numeraire". The present project investigates issues related to the numeraire in financial market modeling. Focus is given in the following areas: (1) deep connections between the existence of the numeraire and market viability, which will result in strengthening and simplifying the theoretical foundations of Stochastic Finance; (2) several ramifications of the numeraire property in non-traditional models; (3) enlargement of the list of optimality properties of the numeraire; (4) closed-form expressions of general equilibria in incomplete financial markets with logarithmic investors possessing heterogeneous beliefs; (5) continuity and differentiability of the numeraire with respect to market inputs, such as statistical probabilities and information.This project provides a new approach in analyzing the appropriateness of theoretical market models to successfully reflect real-world phenomena, as well as to investigate deeper properties of financial markets. The power of this approach is its all-encompassing nature and its simplicity, since it solely depends on the existence and structure of a singe remarkable investment opportunity in the market, which is the numeraire. The broader impact that the project will have is twofold: (1) it will facilitate cross-fertilization between the scientific areas of Probability and Finance, with rich interplay of ideas flowing both ways from one field to the other; (2) it will cast the pedagogical methodology of teaching Financial Mathematics under a new perspective. In fact, a monograph incorporating this new approach in Stochastic Finance is expected to emerge after the project has been carried out.