Advanced stochastic methods in mathematical finance and related fields

该项目的重点是发展与数理金融、数理统计和精算科学密切相关的随机分析综合方法。该项目的一个坚实的部分是致力于长期依赖过程（分数布朗运动，分数列维过程）。发展这些过程的理论，我们将其应用到包含长程相关分量的统计模型中的参数估计问题。我们推导出漂移参数的似然估计（一致性，渐近正态性等），以及所谓的赫斯特指数的自相似性的一些属性。我们还提供了一个系统的研究金融市场的长期相关性。发展这些市场的期权定价理论，我们推导出标准，过去依赖和障碍期权的定价和套期保值公式。该项目的一部分涉及股票挂钩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。将部分套期保值方法和技术扩展到可违约市场和具有长期相关性的市场，我们将这些方法应用于此类长期金融/保险政策的定价。该项目的另一部分致力于多维回归模型和多维收益模型。我们提出了一个非常一般的半鞅模型，其中包括许多模型研究过。为了提供一个充分的研究最小二乘估计，我们开发了一个技术的算子值随机指数。我们还开发了多维概率分布的多项式扩展方法，以获得更好的拟合收益。该项目提出研究可选半鞅作为可能的技术适用于数学金融和过滤理论。该项目的特点是基础研究，旨在产生创新思想，综合技术和成果，这将是重要的理论和实践。

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.