Stochastic calculus for fractional Lévy processes and related processes

分数阶 Lévy 过程及相关过程的随机微积分

• 批准号: 192622538
• 负责人: Professor Dr. Christian Bender
• 金额: --
• 依托单位: Fachrichtung Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/192622538?language=en
• 关键词: Stochastic; calculus; fractional; vy; processes

Stochastic processes can be considered as models for time dependent systems which are subjectto random infuences. A statistical analysis of many real world phenomena in such diverse fields as finance, econometrics, hydrology, or internet traffic, reveals long memory effects. Arguably, the best-studied stochastic processes with long range dependence are fractional Brownian motions (with Hurst parameter H > 1=2). Fractional Lévy processes are generalizations of fractional Brownian motions which capture the memory effects in a similar fashion but provide more flexibility concerning the modeling of the distribution. The aim of the project is to establish a stochastic calculus for fractional Lévy processes and related processes which can be obtained by a convolution of a deterministic kernel with a Lévy process. We will focus on stochastic integrals in the Skorokhod sense, whose zero expectation property is important for its interpretation as a model for additive noise. In particular, we plan to study the change of variables formula (Itô formula) for these integrals and their behaviour under change of measure. As an application we will try to derive an integral representation formula for generalized fractional Ornstein-Uhlenbeck processes, which are promising models for volatility in financial markets.

随机过程可以看作是受随机影响的时变系统的模型。对金融、计量经济学、水文学或互联网流量等不同领域的许多真实的世界现象进行统计分析，就会发现长期记忆效应。可以说，研究得最好的长程相关随机过程是分数布朗运动（赫斯特参数H > 1=2）。分数维过程是分数布朗运动的推广，它以类似的方式捕捉记忆效应，但在分布建模方面提供了更多的灵活性。该项目的目的是建立一个随机演算分数Lévy过程和相关的过程，可以通过卷积的确定性内核与Lévy过程。我们将专注于随机积分的Skorokhod意义上，其零期望属性是重要的解释作为一个模型的加性噪声。特别是，我们计划研究变量的变化公式（伊藤公式），这些积分和他们的行为下的变化措施。作为一个应用，我们将尝试导出一个积分表示公式的广义分数Ornstein-Uhlenbeck过程，这是有前途的模型，在金融市场的波动。