Advanced stochastic methods in mathematical finance and related fields

数学金融及相关领域的高级随机方法

• 批准号: RGPIN-2014-05901
• 负责人: Melnikov, Alexander
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=659532
• 关键词: Advanced; stochastic; methods; mathematical; finance

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.

该项目着重于与数学金融，数学统计和精算科学密切相关的随机分析方法的发展。该项目的坚实部分致力于具有远距离依赖性的过程（分数布朗运动，分数征收过程）。开发这些过程的理论我们将其应用于包含长期依赖分量的统计模型中的参数估计问题。我们为漂移参数（一致性，渐近正态性等）以及所谓的自相似性Hurst索引的可能性估算得出了许多属性。我们还提供了对具有长期依赖性的金融市场的系统研究。为这些市场开发期权定价理论，我们为标准，依赖和障碍选项提供了定价和对冲公式。该项目的一部分涉及与股权相关的人寿保险合同的定价，这是数学金融和精算科学的发展方面。将部分对冲方法和技术扩展到具有长期依赖性的可默认市场和市场，我们将这些方法应用于此类长期财务/保险单的价格。该项目的另一部分专门用于多维回归模型和回报的多维模型。我们为此类建模提出了一个非常通用的Semimartingale模型，其中包括以前研究的许多模型。为了提供对最小二乘估计值的足够研究，我们开发了一种运算符值随机指数的技术。我们还开发了一种多维概率分布的多项式扩展的方法，以使回报更好。该项目建议将可选的半明星作为可能适用于数学金融和过滤理论的技术。该项目可以被描述为旨在产生创新思想，全面技术和结果对理论和实践至关重要的基础研究。