Risk Sensitive Control Theory and Financial Decision Making

该研究项目的目的是继续发展涉及数学控制理论、随机分析、随机过程统计和金融数学的新方法，以了解和解决金融决策中的复杂问题。拟议研究的核心涉及开发连续时间、风险敏感的控制模型，以做出最优投资决策。这些模型的特点是潜在的经济因素被明确地建模为随机过程，最大化投资组合的风险调整增长率的无限范围准则，以及对交易策略的约束。这一基本理论将至少向三个方向延伸。第一个是离散时间的相似模型，因此导致了额外的计算方法。二是建立了包含交易费用的连续时间模型，从而引出了拟变分不等式的研究。第三个扩展将扩展到一个或多个基本因素不可观察的情况，从而在离散时间的情况下涉及隐马尔可夫模型理论。这项研究之所以重要，主要有两个原因。首先，新的数学方法可能会增强美国金融机构的竞争力。其次，这项研究将使人们更好地理解风险管理，以及投资者的决策如何影响金融市场和经济。例如，基本模型的一个重要特例是资产是债券，潜在因素是宏观经济变量。预期结果将阐明投资者的买卖决策、利率水平和失业率等其他宏观经济变量之间的关系。通过观察基础因素与金融衍生品相对应的情况，这项研究的另一个可能的副产品是更好地理解如何通过交易期货和期权来管理风险。该模型的其他版本可以用来研究宏观经济变量和股市之间的相互作用。事实上，这个项目可能会出现对商业周期和股市价格长期相关性的新解释，这些都已经在金融文献中得到了报道。

The purpose of the research project is to continue the development ofnew methodologies involving mathematical control theory, stochasticanalysis, statistics of random processes, and financial mathematics for thepurpose of understanding and solving complex problems in financial decisionmaking. The heart of the proposed research involves the development ofcontinuous time, risk sensitive control models for making optimal investmentdecisions. Such models will feature underlying economic factors that aremodeled explicitly as stochastic processes, the infinite horizon criterionof maximizing the portfolio's risk adjusted growth rate, and constraints onthe trading strategies. This basic theory will be extended in at least threedirections. The first is to analogous models in discrete time, therebyleading to additional computational methods. The second is to continuoustime models which include transaction costs, thereby leading to the study ofquasi-variational inequalities. And a third extension will be to cases whereone or more of the underlying factors are not observable, thereby involvingin the case of discrete time the theory of hidden Markov models. The research is important for two principal reasons. First, the newmathematical methodologies are likely to enhance the competitiveness of U.S.financial institutions. Second, the research will lead to a betterunderstanding about risk management and how investor decisions affect andare affected by financial markets and the economy. For example, an importantspecial case of the basic model is where the assets are bonds and theunderlying factors are macroeconomic variables. The anticipated results willelucidate the relationships between investors' buying and selling decisions,the level of interest rates, and other macroeconomic variables such asunemployment rates. By looking at cases where the underlying factorscorrespond to financial derivatives, another likely by-product of thisresearch is a better understanding of how to manage risk by trading futuresand options. Additional versions of the model can be used to study theinterplay between macroeconomic variables and the stock market. Indeed,likely to emerge from this project are new explanations of business cyclesand the long range dependence of stock market prices that have been reportedin the financial literature.