Advanced stochastic methods in mathematical finance and related fields

该项目侧重于发展与数理金融、数理统计和精算科学密切相关的随机分析综合方法。该项目的坚实部分致力于具有长期依赖性的过程（分数布朗运动，分数利维过程）。发展这些过程的理论，我们将其应用于包含远程依赖成分的统计模型中的参数估计问题。我们为漂移参数的似然估计（一致性，渐近正态性等）以及所谓的自相似的赫斯特指数导出了一些性质。我们还提供了长期依赖的金融市场的系统研究。针对这些市场发展期权定价理论，我们推导出标准期权、过去依赖期权和障碍期权的定价和套期保值公式。该项目的一部分涉及与股票挂钩的人寿保险合同的定价，这是数学金融和精算科学的一个发展良好的领域。将部分对冲方法和技术扩展到可违约市场和具有长期依赖性的市场，我们将这些方法应用于此类长期金融/保险政策的定价。项目的另一部分致力于多维回归模型和多维回报模型。我们提出了一个非常一般的半鞅模型，它包含了许多以前研究过的模型。为了提供对最小二乘估计的充分研究，我们开发了一种算子值随机指数技术。我们还开发了一种多维概率分布的多项式扩展方法，以获得更好的拟合收益。本课题拟研究可选择的半鞅，作为可能适用于数学金融和滤波理论的技术。该项目的特点是基础研究，旨在产生对理论和实践都很重要的创新思想、综合技术和结果。

The project focuses on developments of comprehensive methods of Stochastic Analysis which are closely related to Mathematical Finance, Mathematical Statistics and Actuarial Science. A solid part of the project is devoted to processes with long-range dependence (Fractional Brownian Motion, Fractional Levy Processes). Developing theory of these processes we apply it to parameter estimation problems in statistical models containing a long-range dependence component. We derive a number of properties for the likelihood estimates of the drift parameter (consistency, asymptotic normality, etc) as well as for the so-called Hurst index of self-similarity. We provide also a systematic study of financial markets with long-range dependence. Developing option pricing theory for these markets we derive pricing and hedging formulas for standard, past-dependent and barrier options. A part of the project deals with the pricing of equity-linked life insurance contracts, a well-developing area of Mathematical Finance and Actuarial Science. Extending partial hedging methods and techniques to defaultable markets and markets with long-range dependence we apply these methods to pricing of such long-term finance/insurance policies. Another part of the project is devoted to multidimensional regression models and multidimensional models for returns. We propose a very general semimartingale model for such modeling which includes many models studied before. To provide an adequate study of the least squares estimates we develop a techniques of operator-valued stochastic exponentials. We develop also a method of polynomial extensions of multidimensional probability distributions to get a better fitting for returns. The project proposes to study optional semimartingales as possible techniques applicable in Mathematical Finance and Filtering theory. The project can be characterized as fundamental research aimed at producing innovative ideas, comprehensive techniques and results that will be important for theory and practice.