AMC-SS: Research on Dependence of Stochastic Processes and on Mathematical Aspects of Credit Derivatives and Convertible Bonds

该研究项目将继续在随机分析领域开发新的方法和模型，以及金融和金融工程中的随机方法，以解决金融决策和风险管理中的复杂问题。 重点将放在信用衍生工具的估值和对冲应用上，重点是信用违约掉期和篮子信用衍生工具，以及可转换债券的估值和对冲应用。 因此，该项目的一个目标是发展一篮子信用衍生品和可转换债券以及信用风险和可转换风险的对冲、估值和管理等相关问题的可靠数学理论。这特别需要新的结果表示随机过程，以及随机过程之间的随机依赖建模的新结果。 因此，该项目的另一个目标将是使用有限和无限维的随机分析来研究随机过程之间的相关性。特别地，我们将建立和应用一个新的半鞅Copula和Markov Copula理论。 此外，将开发随机分析的新应用，特别是关于鞅表示和具有反射的向后随机微分方程（也由跳鞅驱动）。这个研究项目将具有根本的重要性，原因有几个，无论是从应用的角度来看，以及从纯理论的角度来看。首先，蓬勃发展的信用衍生品行业将从中受益，因为用于评估和管理篮子信用衍生品（如篮子互换、债务抵押债券和信用指数）的易于处理的数学工具的发展将为该行业提供新的方法论上合理的程序。同样，可转换债券行业将受益于这项研究，因为我们的新的分解结果指定的制度转换市场模型应该提供一个更好的定量工具，这个行业遭受了巨大的损失，在2005年春季-可能是因为这样的复杂的混合衍生品的性质，可转换债券是不是真的很好地理解。 此外，将特别强调对金融业至关重要的信用违约掉期的估值和对冲，并将为此开发新的分析工具。在理论方面，项目也将具有根本的重要性，原因有几个。 首先，如果是真的，那么对于随机过程的标准空间上的概率测度的情形，Sklar定理的一个类似将是A. Sklar（1959），它是对真实的值随机变量证明的。 也许，在这个过程中会推导出一个类似于Sklar定理的关于某些一般向量空间（如波兰空间）上概率测度的定理。其次，对于那些半鞅过程，其局部特征决定其法律将是重要的研究以下问题：什么是类的多元（向量值）半鞅与给定的单变量局部特征。考虑到篮子产品对金融业的战略重要性，从篮子衍生品（篮子期权、篮子信用衍生品等）的估值和套期保值等潜在应用的角度研究上述问题具有实际意义。

This research project will continue development of new methodologies and models in the areas of stochastic analysis, and stochastic methods in finance and financial engineering, for the purpose of solving complex problems in financial decision making and risk management. Particular emphasis will be put on applications to valuation and hedging of credit derivatives, with focus on credit default swaps and basket credit derivatives, as well on applications to valuation and hedging of convertible bonds. One goal of the project is therefore development of a sound mathematical theory of basket credit derivatives and convertible bonds and related issues of hedging, valuation, and management of credit risk and convertible risk. This in particular will require new results in representation for stochastic processes, as well as new results for modeling of stochastic dependence between random processes. Thus, another goal of the project will be to use stochastic analysis in finite and infinite dimensions for studying of dependence between stochastic processes. In particular, a new theory of semimartingale copulae and Markov copulae will be worked out and applied. Moreover, new applications of stochastic analysis will be developed, in particular with regard to martingale representations and backward stochastic differential equations with reflections (also driven by jump martingales).This research project will be of fundamental importance for several reasons, both from the applications point of view as well as from the purely theoretical perspective. First, the booming credit derivative industry will benefit from it, as development of tractable mathematical tools for the purpose of valuing and managing of basket credit derivatives, such as basket swaps, collateralized debt obligations, and credit indices, will provide the industry with new methodologically sound procedures. Likewise, the convertible bond industry will benefit from this research as our new decomposition results specified for the regime switching market model should provide a better quantitative tools for this industry, which suffered great losses in the Spring of 2005 -- possibly because the nature of such complex hybrid derivatives as convertible bonds was not really well understood. In addition, valuation and hedging of credit default swaps, which is essential for the finance industry, will be specifically emphasized and new analytical tools will be developed for this purpose. On theoretical side, project will also be of fundamental importance for several reasons. First, if true, then an analog of Sklar's theorem for the case of probability measures on canonical spaces of stochastic processes will be an important extension of the classical theorem of A. Sklar (1959), which was proved for real valued random variables. Perhaps, an analog of Sklar's theorem for probability measures on some general vector spaces (such as Polish spaces) will be derived in the process. Second, for those semimartingale processes for which their local characteristics determine their laws it will be important to study the following question: what is the class of multivariate (vector valued) semimartingales with given univariate local characteristics. Given the strategic importance of basket products for financial industry there will be a practical importance of studying of the above problems in view of potential applications, such as valuation and hedging of basket derivatives (basket options, basket credit derivatives, etc.).