Nonlinear Functionals of Fractional Brownian Motion

分数布朗运动既不是马尔可夫过程，也不是半鞅。因此，它可以并且最近已经被应用于描述这两种主要随机过程无法描述的现象。为了扩大应用范围，需要研究分数阶布朗运动的非线性泛函。一类重要而自然的泛函是由随机微分方程给出的泛函。在研究自交点地方时的基础上，提出研究分数阶布朗运动驱动的一般随机微分方程。这种研究的困难是由于强大的皮卡德迭代方法不起作用。作者建议将分数阶微积分、预期随机微积分和特征理论结合起来研究这类方程。为了用数学方法描述自然或社会现象，人们通常使用马尔可夫性质（尽管直到今天的整个历史都是已知的，但未来只取决于今天）。这是一个合理的简化，特别是如果考虑到处理整个过去所需要的复杂性。然而，吸收所有可用的信息以更好地预测未来的要求越来越高。分数布朗运动是捕捉这种长记忆特征的最简单的统计模型之一。它已经找到了许多应用。为了更充分地将数学模型拟合到所考虑的现象中，人们应该使用分数布朗运动作为构建块来获得更复杂的随机量。首席研究员多年来一直专注于这个统计模型，并取得了显著的成功。这项研究将大大推动这一进展，并有望对许多其他领域产生影响。直接应用于金融和生物信息学。

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.