Topics in Risk Sensitive Control and Financial Mathematics

这项研究将有助于在数学控制理论、随机分析和金融数学领域发展新的方法和模型，以解决金融决策和金融工程中的复杂问题。预计这项研究将对数学在最优投资组合管理和资产配置领域以及在信用风险管理领域的应用产生最大影响。这项研究的一个基本方面是风险敏感随机控制理论的概念，该理论适用于无限规划范围内的投资问题。这种优化目标很有吸引力，因为固定交易策略可能是最优的，因此有很好的潜力能够计算出有意义的实际问题的最优解。此外，几类有限规划水平问题可以合理地用无限规划水平的问题来近似。拟议研究的第二个基本方面是明确考虑统计估计问题。最优控制方法在金融行业中很少使用，这主要是因为在单个证券的扩散过程模型中，与恒定漂移系数的估计有关的统计困难。这项研究将通过开发包含外生因素的优化模型来缩小理论和实践之间的差距。通过对资产对因素的依赖关系进行显式建模，可以得到更符合实际的模型，更好地理解统计估计的困难，并能够应用自适应控制方法。此外，拟议的研究将促进我们对违约信用风险的量化方面的理解，违约信用风险是金融决策中的关键要素。管理大量美元的名义资本与管理全球通信网络或全球制造企业一样复杂。因此，仅凭常识、直觉和经验通常不足以安全有效地管理可用的金融工具。本研究旨在为财务管理提供形式化的数学支持。虽然这项研究需要在数学控制理论、随机分析和金融数学领域取得根本性进展，但预计这项研究将导致新的实用工具，这些工具最终将在金融行业广泛使用，以提高其竞争力和效率。鉴于美国金融业对全球经济的影响，这一点尤为重要。这项研究的重点将放在开发解决大规模问题的计算机算法上，以便有效地利用高性能计算。

This research will contribute to the development of new methodologies and models in the areas of mathematical control theory, stochasticanalysis, and financial mathematics for the purpose of solving complexproblems in financial decision making and financial engineering. It isexpected that this research will have the greatest impact on applicationof mathematics in the areas of optimal portfolio management and assetallocation, as well as in the area managing of credit risk. A fundamentalaspect of the research is the concept of risk sensitive stochastic controltheory for investment problems with an infinite planning horizon. Thiskind of optimization objective is appealing because stationary tradingstrategies are likely to be optimal and thus there is good potential forbeing able to compute optimal solutions for meaningful, practicalproblems. In addition, several classes of finite planning horizonproblems can be reasonably well approximated by problems with an infiniteplanning horizon. A second fundamental aspect of the proposed research isthe explicit consideration given to statistical estimation issues. Optimalcontrol methodologies are rarely used in the financial industry, largelybecause of statistical difficulties associated with the estimation ofconstant drift coefficients in diffusion process models of individualsecurities. This research will serve to reduce the gap between theory andpractice by developing optimization models that incorporate exogenousfactors. By explicitly modeling the dependence of the assets on factors,it will be possible to obtain more realistic models, to better understandthe statistical estimation difficulties, and to be in a position to applyadaptive control methods. In addition, the proposed research will advanceour understanding of quantitative aspects of default credit risk,which is a vital element in financial decision making.Managing large sums of dollars of notional capital is as complex asmanaging global communication networks or as managing globalmanufacturing enterprises. Thus common sense, intuition and experiencealone will typically not suffice to safely and efficiently manageavailable financial instruments. This research is aimed at providingformal mathematical support for financial management. Although theresearch will require fundamental advances within the areas ofmathematical control theory, stochastic analysis, and financialmathematics, it is anticipated that this research will lead to new andpractical tools that eventually become widely used in the financialindustry in order to enhance its competitiveness and efficiency. This isparticularly important given the impact the U.S. financial industry has onthe global economy. The big emphasis will be placed in this research onthe development of computer algorithms for solving large-scale problems sothat the effective use can be made of high-performance computing. ----------------------