Nonlinear Functionals of Fractional Brownian Motion

分数布朗运动的非线性泛函

• 批准号: 0504783
• 负责人: Yaozhong Hu
• 金额: --
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504783&HistoricalAwards=false
• 关键词: Nonlinear; Functionals; Fractional; Brownian; Motion

Fractional Brownian motion is neither Markov process nor semimartingale. Thus it can be and has recently been applied to describe phenomena that cannot be described by these two major stochastic processes. To enlarge the scope of application one needs to study the nonlinear functionals of fractional Brownian motion. An important and natural class of such functionals are those given by stochastic differential equations. In addition to studying self-intersection local time, the principal investigator proposes to study general stochastic differential equations driven by fractional Brownian motion. The difficulty in such study is caused by the fact that the powerful Picard's iteration approach fails to work. The principal investigator proposes to combine fractional calculus, anticipative stochastic calculus, and the characteristic theory to investigate such equations.To describe natural or social phenomena mathematically, people usually use Markov property (the future depends only on today although the whole history until today is known). This is a reasonable simplification, particularly if one considers the sophistication needed to deal with the entire past. However, it becomes more and more demanding to assimilate all the information available to better predict the future. Fractional Brownian motion is among the simplest statistical model that captures this long memory character. It has found many applications. To more adequately fit mathematical models to the phenomena under consideration, one should use fractional Brownian motion as building blocks to obtain more sophisticated random quantities. The principal investigator has focused on this statistical model for a number of years and has achieved significant success. This research will considerably further this progress and is expected to have impact on many other fields. Immediate applications are to be found in finance and bio-informatics.

布朗运动的分数既不是马尔可夫的过程，也不是半明星。因此，它可以并且最近应用于描述这两个主要随机过程无法描述的现象。为了扩大应用的范围，需要研究分数布朗运动的非线性功能。 重要和自然的此类功能是随机微分方程给出的功能。除了研究当地时间的自我交流外，主要研究者还建议研究由布朗尼分数运动驱动的一般随机微分方程。 这项研究的困难是由于强大的PICARD迭代方法无法正常工作的事实引起的。 主要研究者建议将分数演算，预期的随机演算以及研究这种方程式的特征理论结合。要数学上描述自然或社会现象，人们通常使用Markov属性（尽管今天已经知道了整个历史，但未来仅取决于今天）。这是一个合理的简化，尤其是如果人们认为整个过去所需的复杂性。但是，吸收所有可用的信息以更好地预测未来，这变得越来越苛刻。 分数布朗运动是捕获这种长期记忆特征的最简单统计模型之一。 它找到了许多应用程序。为了使数学模型更适合所考虑的现象，应将布朗尼运动作为构建块以获得更复杂的随机数量。主要研究人员多年来一直关注该统计模型，并取得了巨大的成功。这项研究将大大进一步发展，并有望对许多其他领域产生影响。即时应用，可以在金融和生物信息学中找到。