Mathematical Sciences: Stochastic Analysis in Nonlinear Financial Markets

9503582本项目由应用数学计划、统计和概率计划和经济学计划资助，将研究在现代非线性金融市场理论和实践背景下出现的各种随机分析和控制的数学问题。这些问题包括(1)在有交易成本的市场中以及在价格可能以非线性方式取决于代理人的投资策略和财富的市场中的或有债权的最优化、对冲和定价问题；(2)资产价格的非标准模型。与(I)有关的数学问题包括：某些正倒向随机微分方程解的存在唯一性，某些偏微分方程粘性解和变分不等式，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(Ii)，预计将不得不发展一种新的随机微积分，一种超越半鞅框架的随机微积分。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的普遍假设，即市场是完美的，即每一份金融合同都可以通过准确计算其现值来定价。然而，由于不同类型的市场摩擦，如交易成本、不同的利率、可能影响资产价格的大投资者的存在等，现实中的情况并非如此。因此，在不完全市场中开发新的金融工具定价方法是非常重要的，其中一些将是本研究的重点。同样，人们在买卖金融工具时承担的风险通常比完美市场理论所预测的要大得多(最近有关期权交易的丑闻证实了这一点)。该项目将发展一种在不完全市场中的风险对冲理论，该理论比目前的理论更具普遍性，同时更适用于现实的金融市场。***

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. ***