Inverse Problems in Mathematical Finance: Exploring Bayesian Model Selection Algorithms

本项目探索解决数学金融模型选择问题的信息理论思想。通常，这些问题包括扩散度量的规范，或者更一般地说，路径空间的度量，它描述了市场的未来状态。反转中使用的数据由函数的期望值组成，它对应于“基准”证券的观察价格。从偏微分方程和蒙特卡罗模拟的角度研究了这一问题。我们的重点是基于最小化未知概率和贝叶斯先验之间的Kullback-Leibler熵距离的模型选择准则。在扩散过程的特殊情况下，这导致一个约束随机控制问题，可以通过拉格朗日乘子来解决。在蒙特卡罗模拟的情况下，必须构造适当的加权度量，这是一种让人想起“重要性抽样”的技术。这些问题及其推广到非线性约束将在应用数学和数值分析的方法统一的方式进行研究。目的是为了更好地理解金融经济学中的模型选择问题，这对于用定量方法管理金融风险至关重要。非技术描述本研究计划涉及称为衍生证券的复杂金融工具的定价和对冲。这里衍生的技术基于概率、统计和数值分析的数学领域，用于开发精确的工具来定价和管理复杂的金融工具。这项研究的动机来自于这样一个事实，即定量金融提供了许多具有挑战性的数学和计算机相关问题。这是将全球不同的金融市场和经济体联系起来的所谓“全球化”现象的结果。目前的建议涉及微调这些模型的新数学方法。我们的目标是更好地理解它们是如何工作的，以及它们是如何代表金融风险的。通过采用稳健的统计方法和强大的数学技术，我们希望揭示定价和风险管理系统，并开发出可以与金融业共享的更好的模型。这是数学在一个新的研究领域的重要应用：定量金融。这项提议是纽约大学（New York University）科朗研究所（Courant Institute）在金融和市场领域更大努力的一部分。到目前为止，我们已经成功地培养了具有独特专业技能进入商业领域的年轻科学家。这表明，这种研究直接或间接地适用于国民经济的大局。

Technical DescriptionThis project explores information-theoretic ideas for solving modelselection problems in Mathematical Finance. Typically, these problemsconsist in the specification of a diffusion measure, or more generally, ameasure on path-space, which describes the future states of the market.The data used in the inversion consists of expected values of functionals,which correspond to observed prices of "benchmark" securities. Thisproblem is studied from the point of view of partial differentialequations and Monte-Carlo simulation. We focus on model selectioncriteria based on minimizing the Kullback-Leibler entropy distance betweenthe unknown probability and a Bayesian prior. In the special case ofdiffusion processes, this leads to a constrained stochastic controlproblem that can be solved via Lagrange multipliers. In the case of MonteCarlo simulation, one must construct appropriate weighted measures, in atechnique which is reminiscent of "importance sampling." These problemsand their generalizations to non-linear constraints will be studied in aunified way using methods of Applied Mathematics and Numerical Analysis.The goal is to achieve a better understanding of the question of modelselection in Financial Economics, which is crucial for the management offinancial risk by quantitative methods.Non-Technical DescriptionThis research proposal deals with the pricing and hedging of complexfinancial instruments called derivative securities. The technology derivedhere, which is based on the mathematical fields of probability, statisticsand numerical analysis, is used to develop accurate tools for pricing andmanaging complex financial instruments. The motivation for this researchcomes from the fact that quantitative finance offers many challengingmathematical and computer-related problems. This is a consequence of theso-called "globalization'' phenomenon that links different financialmarkets and economies throughout the world. The current proposal dealswith new mathematical methods for fine-tuning these models. Our aim is tobetter understand how they work and how they represent financial risk. Bybringing to bear robust statistical methods and powerful mathematicaltechniques, we expect to shed light on pricing and risk-management systemsand to develop better models that can be shared with the financialindustry. This is an important application of Mathematics to a new areaof research: quantitative finance. The proposal is part of a greatereffort at New York University's Courant Institute in the field of financeand markets. So far, we have been successful in training young scientiststhat enter the business arena with a unique set of professional skills.This suggests that such research is both directly and indirectly suitablein terms of the larger picture of the national economy.