Mathematical Sciences: Stochastic Analysis in Nonlinear Financial Markets

数学科学：非线性金融市场中的随机分析

• 批准号: 9503582
• 负责人: Jaksa Cvitanic
• 金额: $ 7.49万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503582&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Stochastic; Analysis; Nonlinear

9503582 Cvitanic This project, which is funded through the Applied Mathematics Program, the Statistics and Probability Program, and the Economics Program, will investigate various mathematical problems of stochastic analysis and control that arise in the context of modern theory and practice of nonlinear financial markets. These include (i) questions of optimization, hedging and pricing of contingent claims in markets with transaction costs, as well as in markets in which the prices can depend in a nonlinear fashion on the investment strategy and wealth of the agent; (ii) nonstandard models for asset prices. Mathematical questions related to (i) include existence and uniqueness of certain forward-backward stochastic differential equations, viscosity solutions to certain partial differential equations and variational inequalities, as well as a non-standard stochastic control problem in which the terminal loss function has a random component. With regard to (ii), it is expected that a new kind of stochastic calculus will have to be developed, one which goes beyond the semimartingales framework. One of the reasons for the present gap between the theory of finance and the reality of the stock market is the prevailing assumption in theoretical work that the market is perfect in the sense that every financial contract can be priced by calculating exactly its present value. This is, however, not the case in reality due to different types of market friction" such as transaction costs, different interest rates, presence of large investors who can influence the asset prices, and so forth Thus, it is very important to develop new methods for pricing financial instruments in imperfect markets, some of which will be the focus of this research. In a similar vein, the risk one undertakes when buying or selling financial instruments is typically much larger than predicted by the perfect-market theory (as recent scandals involving the trading of options confirm). This project will develop a theory of risk-hedging in imperfect markets that is more general and at the same time more applicable to realistic financial markets than current theory. ***

小行星9503582 这个项目，这是通过应用数学计划，统计和概率计划，经济学计划资助，将调查随机分析和控制中出现的现代理论和非线性金融市场的实践背景下的各种数学问题。 这些问题包括：（一）或有债权的最优化、套期保值和定价问题， 市场与交易成本，以及在市场中的价格可以依赖于在一个非线性的方式对投资策略和财富的代理;（ii）非标准模型的资产价格。与（i）有关的数学问题包括某些正倒向随机微分方程的存在性和唯一性，某些偏微分方程和变分不等式的粘性解，以及终端损失函数具有随机分量的非标准随机控制问题。 关于（二），它是 预计，一种新的随机演算将不得不发展，一个超越半鞅框架。 目前金融理论与股票市场现实之间存在差距的原因之一，是理论工作中流行的假设，即市场是完美的，在这个意义上，每一个金融合约都可以通过精确计算其现值来定价。然而，由于交易成本、不同的利率、能够影响资产价格的大投资者的存在等不同类型的市场摩擦，实际情况并非如此。因此，开发新的方法来为不完全市场中的金融工具定价是非常重要的，其中一些方法将是本文研究的重点。 以类似的 然而，人们在买卖金融工具时所承担的风险通常比完美市场理论所预测的要大得多（最近涉及期权交易的丑闻证实了这一点）。 该项目将开发一个 不完全市场中的风险对冲理论，比现有理论更普遍，同时更适用于现实的金融市场。 ***