Topics in Stochastic Control, Portfolio Optimization and Credit Risk Analysis

这项拟议的研究将分为两个主要研究领域：1)发展用于风险调整投资组合优化和固定收益投资管理的数学理论，纳入证券收益的因素建模；以及2)发展用于信用风险评估和对冲的数学理论。第一个研究领域有三个基本方面：1)有限和无限规划视野下投资问题的风险调整控制(优化)准则概念。更具体地说，我们将继续发展均值-方差随机控制的理论和应用。这种控制问题正吸引着金融业，因为它延伸到了更现实的动态框架，哈里·马科维茨(Harry Markowitz)(1990年诺贝尔经济学奖)的经典方法。1b)开发固定收益投资问题的控制方法，计划期限有限，标的债券的到期日一致。我们将试图克服在这种情况下产生的某些奇点问题，这将导致适用的固定收益投资策略的派生。1c)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相关的统计困难。从业者通常会根据利率和特定于公司的会计措施等外生因素，将精力投入到预测安全回报上。拟议的研究将有助于通过开发明确纳入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，将有可能获得更现实的模型，更好地理解统计估计的困难，并能够应用自适应控制方法。第二个研究领域的基本方面是：2)明确考虑可违约或有债权的信用转移的可能性。这将在有条件的马尔科夫链的情况下进行，既涉及单一的可违约索赔，也涉及若干此类索赔。后一种情况对于将信用衍生品纳入篮子的应用非常重要。2b)为对金融行业至关重要的某些类别的信用衍生品进行对冲的数学理论的发展。这将在与条件马尔可夫链相关联的鞅表示的背景下完成。2C)开发用于计算条件马尔可夫过程的某类泛函的分析工具。这将建立在经典的费曼-卡克特征的基础上，并将应用于一些基本(篮子)信用衍生品的估值和对冲，如违约互换。虽然拟议的研究将需要在应用数学、概率和金融经济学领域取得根本性进展，但预计拟议的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，可能还会在保险行业广泛使用。这种预期的主要原因是，拟议的研究解决了这两个行业对量化方法的需求，使金融和保险决策者能够适当地管理某些类别的风险。风险发生率在金融/保险决策中的主要含义是财务损失的可能性。总体而言，无论是财务损失的可能性，还是偶尔出现的严重程度，都无法完全消除。然而，人们寻求可行的工具，如定价和对冲策略，以控制金融和保险行业的一些潜在风险，以便将损失的可能性和严重性保持在最低限度。拟议的研究将为此类工具的开发提供数学基础。

0202851BieleckiThe proposed research will be organized into two principal research areas: 1) Development of mathematical theory for risk adjusted portfolio optimization and fixed income investment management, incorporating factor modeling of security returns, and 2) Development of mathematical theory for credit risk valuation and hedging. There are three fundamental aspects of the first research area: 1A) The concept of risk adjusted control (optimization) criteria for investment problems with finite and infinite planning horizons. More specifically, we shall continue to develop theory and applications for mean-variance stochastic control. This kind of control problems is appealing to financial industry as it extends to more realistic dynamic framework the classical approach of Harry Markowitz (1990 Nobel Prize in Economics). 1B) Development of control methodologies for the fixed income investment problems with finite planning horizon coinciding with the maturity of the underlying bond. We shall attempt to overcome certain singularity problems arising in this context, which will lead to derivation of applicable fixed income investment strategies. 1C) Explicit consideration given to statistical estimation issues. Optimal control methodologies are rarely used in the financial industry, largely because of statistical difficulties associated with the estimation of constant drift coefficients in diffusion process models of individual securities. Practitioners typically channel their energies into forecasting security returns based upon exogenous factors such as interest rates and firm-specific accounting measures. The proposed research will serve to reduce the gap between theory and practice by developing optimization models, which explicitly incorporate exogenous factors. By explicitly modeling the dependence of the assets on factors, it will be possible to obtain more realistic models, to better understand the statistical estimation difficulties, and to be in a position to apply adaptive control methods. The fundamental aspects of the second research area are: 2A) Explicit consideration given to possibility of credit migrations of defaultable contingent claims. This will be done in the context of conditionally Markov chains both with regard to a single defaultable claim as well as with regard to several such claims. The latter situation is very important for applications to basket credit derivatives. 2B) Development of mathematical theory for hedging of certain classes of credit derivatives that are vital for financial industry. This will be done in the context of martingale representations associated with conditionally Markov chains. 2C) Development of analytical tools for computation of certain class of functionals of conditional Markov processes. This will build upon the classical Feynman-Kac characterizations, and will find applications for valuation and hedging of some fundamental (basket) credit derivatives such as default swaps. Although the proposed research will require fundamental advances within the areas of applied mathematics, probability, and financial economics, it is anticipated that the proposed research will lead to new and practical tools that eventually become widely used in the financial industry, and possibly in the insurance industry as well. The main reason for this expectation is that the proposed research addresses the need of the two industries for quantitative methodologies that would enable financial and insurance decision makers to properly manage certain categories of risks. The major implication of incidence of risks in financial/insurance decision-making is the possibility of financial losses. In general, neither the possibility of financial losses, nor their occasional severity, can be completely eliminated. However, workable tools, such as pricing and hedging strategies, are sought for controlling some of the risks underlying financial and insurance industries, so that both the possibility and severity of losses are kept to minimum. The proposed research will provide mathematical basis for development of such tools.