Stochastic Differential Systems Driven by Fractional Brownian Motion

[00:46 . 13]众所周知，分数布朗运动不是半鞅。半鞅的强大的随机演算并不适用于它们。由于应用的迫切需要，首席研究员和他的合作者基于Wick产品开发了一种新的Ito型随机演算。他建议继续这一研究课题，研究分数布朗运动驱动的随机微分系统。首先，他将研究分数布朗运动驱动的随机微分系统全局解的存在性、唯一性和逼近性。许多研究者试图在这些方面取得成果，但收效甚微。首席研究员发现了分数布朗运动驱动的随机微分系统和准线性双曲方程（无限多变量）之间的关系。众所周知，后一种方程也很难解。然而，有一些结果是有用的。这种联系将导致更好地理解随机微分系统和PI计划探索这种关系。其次，在分数布朗运动驱动的随机系统的应用中，还需要识别系数和赫斯特参数。PI提出了一个这样的识别问题，并将其应用到金融市场随机波动模型的研究中。为了从一个物理或社会系统中获得最大的利益，人们需要以尽可能精确的方式理解这个系统。这就需要为系统的动态演化建立一个数学模型。当系统受到某些不确定因素的影响时，应采用随机过程对系统进行建模。迄今为止，最受关注和研究最多的随机过程之一是基于布朗运动的随机微分方程。布朗运动有一些很好的性质，比如马尔可夫性：它的未来状态只取决于现在的状态，而不取决于过去。这种简单性使得数学变得简单，并取得了非常深刻的成果。事实上，在过去的一个世纪里，人们对它进行了大量的研究。然而，这种优雅的性质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未来不仅取决于现在而且取决于过去历史的系统！分数布朗运动是具有这种长距离依赖的随机过程，可以用来描述这样的系统。本提案旨在构建分数布朗运动的数学工具，分数布朗运动已经在水文学、气候学、网络流量分析和金融中得到了应用。这项研究将对这些领域以及生命科学产生影响。

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.