Optimal Portfolio and Model Selection in Financial Markets

来自：Jaksa Cvitanic [cvitanic@math.usc.edu]发送：2001年7月2日星期一下午4点33分收件人：Pang， jong - shsubject：回复：您的邮件亲爱的彭博士，我在这里附上我的提案摘要。真诚的，Jaksa Cvitanic------首席研究员：Jaksa Cvitanic研究的是现代金融市场理论的各个方面以及随机分析，滤波和控制的相关数学问题。将研究的问题包括：(i)寻找算法来计算最优财富/对冲过程的扩散项，以及有关鞅表示性质和马利维微积分的相关问题；（ii）随机微分效用最大化问题、与正-倒向随机微分方程的联系以及不完全/不对称信息问题；（iii）在一般扩散模型中寻找最佳退休投资组合/消费投资的分析和数值方法；（iv）效用最大化/风险最小化理论。有摩擦市场的一般半鞅模型；(v)随机波动模型的滤波和校正；（六）高管薪酬优化设计。我们期望来自随机分析和鞅理论、凸对偶理论、泛函分析、随机控制、蒙特卡罗/模拟方法的工具在解决这些问题中证明是有价值的，有时需要开发新的工具，从而增强对这些领域的理论和应用方面的理解。最优投资组合选择和消费选择是一种理论，它提供了如何在金融市场上投资于不同资产之间分配资金，以及如何消费以购买各种商品的问题的答案。到目前为止，这一理论已发展得几乎具有全面性。然而，这取决于市场的数学模型，以及我们估计模型参数的能力。例如，复杂金融合约（如另类期权）的正确定价，取决于我们对期权标的股票的“波动性”（风险）估计得有多好。我们提出要研究的问题之一是如何通过观察到的股票价格来进行这种估计。同样，实际计算相应的最优交易策略的问题一般也没有得到解决。目前常用的算法通常不能很好地用于金融市场的更复杂和更现实的模型，由于市场建模者越来越复杂，这些模型正在成为一种标准。因此，探索新的分析和计算方法来寻找最优交易策略是很重要的，其中一些在本提案中提出。类似的方法被建议用于探索一个重要的问题，即公司应该如何补偿其高管，从而使高管的行为是最优的。

Pang, Jong-ShiFrom: Jaksa Cvitanic [cvitanic@math.usc.edu]Sent: Monday, July 02, 2001 4:33 AMTo: Pang, Jong-ShiSubject: Re: your mailDear Dr. Pang,I enclose here the abstract for my proposal.Let me know , please, if it's O.K.Sincerely,Jaksa Cvitanic------ Principal Investigator: JAKSA CVITANICResearch is proposed on various aspects of the modern theory of financial markets and related mathematical problems of stochastic analysis, filtering and control. Issues that will be studied involve:(i) finding algorithms to compute the diffusion term ofthe optimal wealth/hedging process, and related questionsabout Martingale Representation Property and MalliavinCalculus;(ii) questions on maximizing Stochastic DifferentialUtility and connections to Forward-Backward Stochastic Differential Equations and problems of incomplete/asymmetric information;(iii) analytical and numerical methods for finding optimalportfolio/consumption investment for retirement, in general diffusion models;(iv) theory of utility maximization/risk minimization ingeneral semimartingale models of markets with frictions;(v) filtering and calibration of stochastic volatility models;(vi) optimal design of executive compensation.It is expected that tools from stochastic analysis and martingaletheory, convex duality theory, functional analysis, stochastic control, Monte Carlo/simulation methods, will prove valuable in the resolution of these questions, sometimes requiring development of new tools, thus enhancing the understanding of both the theoretical and applied aspects of these fields.The optimal portfolio selection and consumption selection is the theory that provides answers to the question of how to allocate money between investing in different assets in financial markets, and consuming it in order to buy various goods. The theory has been developed in almost full generality by now. However, it depends on a mathematical model of the markets, and our ability to estimate the model parameters. For example, correct pricing of complex financial contracts, such as exotic options, depends on how well we can estimate the "volatility" (riskiness) of the stock on which the option is written.One of the problems we propose to study is how to do this estimation by using observed stock prices. Similarly, the problem of actually computing the corresponding optimal trading strategies has not been resolved in general. The algorithms that are commonly used today typically do not work wellin more complex and realistic models for financial markets, that are becoming a standard, due to the increased sophistication of market modelers. Thus, it is important to explore new analytical and computational methods for finding optimal trading strategies, some of which are suggested in this proposal. Similar methods are suggested for exploring important problem of how a firm should compensate its executive so that the resulting behavior of the executive is optimal.