分数布朗运动以其简单的结构、精美的性质以及在通信、湍流、图像处理、金融等各种科学领域的广泛应用，近些年来备受关注。然而几乎所有研究都集中在长记忆(Hurst指数大于二分之一)情形，而对Hurst指数小于二分之一(短记忆)情形研究非常少，出现这种情况的主要原因是短记忆分数布朗运动没有正向积分以及方便易用的再生核希尔伯特空间，而且其样本轨道也更加奇异。即使对于长记忆的分数布朗运动，人们研究最多的是方程问题，很少关心其驱动的各类泛函问题。. 本项目将从建立短记忆分数布朗运动的广义正、倒向积分入手研究分数布朗运动的一些随机分析问题，包括：一、短记忆分数布朗运动与积分过程的广义正向与倒向积分、广义二次协变差、局部时--空间分析等；二、分数布朗运动驱动的一些积分泛函，特别是奇异与指数积分泛函；三、作为相关问题我们也考虑分数布朗运动驱动的随机微分方程，特别是短记忆情形。

In recent years fractional Brownian motion has become an object of intense study, due to its interesting properties, simple structures and its applications in various scientific areas including telecommunications, turbulence, image processing and finance. However, nearly all research has focused on the case of long memory (Hurst index 1/2<H<1), and there has been little systematic investigation on the case of short memory (0<H<1/2). The main reasons for this, in our opinion, are that fractional Brownian motion with short memory has no forward integral and the convenient reproducing kernel Hilbert space, and moreover, its sample paths are more irregular than the long memory case. Even for fractional Brownian motion with long memory, much research has been focused on stochastic equations and there has been little systematic investigation on some integral functionals driven by it...In this research, our starting point is to structure the generalized forward and backward integrals of fractional Brownian motion with short memory. Our objectives are to study some of the stochastic analysis topics within fractional Brownian motion, including: 1. The generalized forward and backward integrals, the generalized quadratic co-variations, local time-space calculus about fractional Brownian motion with short memory and some integrated processes. 2. Some integral functionals driven by fractional Brownian motion, in particular, the exponential and singular integral functionals. 3. As some related questions, we also consider the stochastic differential equations driven by fractional Brownian motion, especially the case of short memory.