Stochastic Differential Systems Driven by Fractional Brownian Motion

分数布朗运动驱动的随机微分系统

• 批准号: 0204613
• 负责人: Yaozhong Hu
• 金额: $ 9.33万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-07-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204613&HistoricalAwards=false
• 关键词: Stochastic; Differential; Systems; Driven; Fractional

0204613Hu It is well-known that the fractional Brownian motions are not semimartingales. The powerful stochastic calculus for semimartingales are not applicable to them. Motivated by the urgent need from applications the principal investigator and his collaborators have developed a new stochastic calculus of Ito type based on the Wick product. He proposes to continue this research topic and to study the stochastic differential systems driven by fractional Brownian motions. First he shall study the existence, uniqueness and approximation of global solutions to stochastic differential systems driven by fractional Brownian motions. Many researchers have attempted to obtain result on these aspects with little success. The principal investigator has discovered a relationship between stochastic differential systems driven by fractional Brownian motions and quasilinear hyperbolic equations (of infinitely many variables). It is well-known that the latter equations are also difficult to solve. However, there are a number of results which are useful. This connection will lead to a better understanding of stochastic differential systems and the PI plans to explore this relation. Secondly, in application of the stochastic systems driven by fractional Brownian motions, one also needs to identify the coefficients and the Hurst parameter. The PI proposes to study one such identification problem and apply it to the investigation of the stochastic volatility model in financial market. To obtain the maximum benefit from a physical or social system, one needs to understand the system in the most precise way possible. This requires building a mathematical model for the dynamic evolution of the system. When the system is under the influence of some uncertain factors, the system should be modeled by a random process. Up to now one of the random processes which has received the most attention and has been studied the most is stochastic differential equations based on the so-called Brownian motion. Brownian motion has some nice properties such as Markovian: Its future state depends only on the present state and does not depend on the past. This simplicity makes the mathematics for it easy and very profound results have been achieved. In fact there have been enormous work on it over the past century. However, this elegant property also limits the applicability of such a random process, since it cannot be used to describe those systems whose future depend not only the present but also on past history! Fractional Brownian motions are random processes having this long range dependence and may be used to describe such systems. This proposal aims to construct mathematical tools for the fractional Brownian motions which have already found applications in hydrology, climatology, network traffic analysis, and finance. This research will have impact on these areas as well as in life science.

0204613HU众所周知，分数布朗尼动作不是半明星。半明星的强大随机演算不适用于它们。受应用程序迫切需要的激励，首席研究员及其合作者基于Wick产品开发了一种新的ITO类型的随机演算。他建议继续这个研究主题，并研究由分数布朗动作驱动的随机差异系统。首先，他应研究由分数布朗尼动作驱动的随机差分系统的全球解决方案的存在，独特性和近似。许多研究人员试图获得这些方面的结果，几乎没有成功。首席研究者发现，由分数布朗运动驱动的随机差分系统与准线性双曲方程（无限多变量）之间的关系。众所周知，后一个方程也很难解决。但是，有许多有用的结果。这种联系将使人们对随机差异系统有更好的了解，并计划探索这种关系。其次，在应用由分数布朗尼动作驱动的随机系统时，还需要识别系数和Hurst参数。 PI建议研究一个这样的识别问题，并将其应用于金融市场中随机波动模型的调查。 为了从物理或社会系统中获得最大收益，需要以最精确的方式理解系统。这需要建立一个数学模型，以进行系统的动态演变。当系统受到某些不确定因素的影响时，该系统应通过随机过程进行建模。到目前为止，最受关注并已研究最多的随机过程之一是基于所谓的布朗运动的随机微分方程。布朗运动具有一些不错的特性，例如马尔可夫人：其未来状态仅取决于当前状态，并且不取决于过去。这种简单性使数学变得容易而非常深刻。实际上，在过去的一个世纪中，它已经进行了巨大的工作。但是，这种优雅的属性也限制了这种随机过程的适用性，因为它不能用来描述那些不仅依赖当前而且还依赖过去历史的系统！分数布朗运动是具有这种长距离依赖性的随机过程，可用于描述此类系统。该建议旨在为已经在水文学，气候学，网络交通分析和金融中发现的分数布朗动作构建数学工具。这项研究将对这些领域以及生活科学产生影响。