Optimal Portfolio and Model Selection in Financial Markets

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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.