Topics in Stochastic Control, Portfolio Optimization and Credit Risk Analysis

随机控制、投资组合优化和信用风险分析主题

• 批准号: 0202851
• 负责人: Lidia Filus
• 金额: $ 8.5万
• 依托单位: Northeastern Illinois University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202851&HistoricalAwards=false
• 关键词: Topics; Stochastic; Control; Portfolio; Optimization

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.

0202851 Bielecki拟议的研究将组织成两个主要的研究领域：1）风险调整的投资组合优化和固定收益投资管理的数学理论的发展，纳入因素建模的安全回报，和2）数学理论的发展信用风险评估和对冲。第一个研究领域有三个基本方面：1A）有限和无限规划视野的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。这类控制问题是吸引金融业，因为它扩展到更现实的动态框架，哈里马科维茨（1990年诺贝尔经济学奖）的经典方法。 1B）发展有限规划期与基础债券到期日一致的固定收益投资问题的控制方法。我们将试图克服在这种情况下出现的某些奇异性问题，这将导致适用的固定收益投资策略的推导。1C）明确考虑统计估计问题。最优控制方法很少用于金融行业，主要是因为与估计单个证券的扩散过程模型中的常数漂移系数相关的统计困难。从业者通常将精力集中在基于利率和公司特定会计指标等外生因素的证券回报预测上。拟议的研究将有助于减少理论和实践之间的差距，通过开发优化模型，明确纳入外源因素。通过对资产对因素的依赖性进行明确建模，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：2A）明确考虑可违约或有债权的信贷迁移的可能性。这将在条件马尔可夫链的背景下，无论是关于一个单一的违约索赔，以及关于几个这样的索赔。后一种情况对于一揽子信用衍生品的应用非常重要。2B）发展数学理论，用于对冲对金融业至关重要的某些类别的信用衍生品。这将在与条件马尔可夫链相关的鞅表示的上下文中完成。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Feynman-Kac特征，并将发现一些基本（篮子）信用衍生品，如违约互换的估值和对冲的应用。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的根本性进展，预计拟议的研究将导致新的和实用的工具，最终成为广泛应用于金融业，并可能在保险业以及。这一期望的主要原因是，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管理某些类别的风险。在金融/保险决策中，风险发生率的主要含义是财务损失的可能性。一般来说，无论是经济损失的可能性，还是偶尔的严重性，都不能完全消除。然而，正在寻求可行的工具，如定价和对冲战略，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。该研究将为此类工具的开发提供数学基础。