Topics in Stochastic Control, Portfolio Optimization and Credit Risk Analysis

本文拟开展的研究将分为两个主要研究领域：1)发展风险调整组合优化和固定收益投资管理的数学理论，纳入证券收益的因子模型；2)发展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：1A)有限和无限规划视野下投资问题的风险调整控制（优化）标准的概念。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。这类控制问题正吸引着金融业，因为它将哈里·马科维茨（1990年诺贝尔经济学奖）的经典方法扩展到更现实的动态框架。1B)针对与标的债券到期日一致的有限规划期限的固定收益投资问题开发控制方法。我们将尝试克服在这种背景下出现的某些奇点问题，这将导致推导适用的固定收益投资策略。1C)明确考虑统计估计问题。最优控制方法很少在金融行业中使用，主要是因为在单个证券的扩散过程模型中估计常漂移系数存在统计困难。从业人员通常将他们的精力投入到基于外部因素（如利率和公司特定会计措施）的证券回报预测上。本研究将通过建立明确考虑外生因素的优化模型来缩小理论与实践之间的差距。通过明确地建模资产对因素的依赖性，将有可能获得更真实的模型，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：2A)明确考虑违约或有债权信用迁移的可能性。这将在条件马尔可夫链的背景下完成，既涉及单个可违约索赔，也涉及多个此类索赔。后一种情况对于篮子信用衍生品的应用非常重要。2B)发展对金融行业至关重要的某些信用衍生品的套期保值数学理论。这将在与条件马尔可夫链相关的鞅表示上下文中完成。2C)条件马尔可夫过程某类泛函计算的分析工具的发展。这将建立在经典的费曼-卡茨特征的基础上，并将找到一些基本（一篮子）信用衍生品（如违约掉期）的估值和对冲应用。虽然拟议的研究需要应用数学、概率论和金融经济学领域的基础进展，但预计拟议的研究将产生新的实用工具，最终广泛应用于金融行业，也可能应用于保险业。这一预期的主要原因是，拟议的研究解决了这两个行业对定量方法的需求，这些方法将使金融和保险决策者能够适当地管理某些类别的风险。金融/保险决策中风险发生率的主要含义是经济损失的可能性。一般来说，既不能完全消除经济损失的可能性，也不能完全消除其偶尔的严重程度。然而，我们寻求可行的工具，例如定价和套期保值策略，以控制金融和保险业的一些潜在风险，从而将损失的可能性和严重程度保持在最低限度。建议的研究将为开发此类工具提供数学基础。

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.