随着对分数布朗运动研究的深入，很多学者建议使用更一般的自相似过程作为一些现象的随机模型，由于这些自相似过程具有很多优美的性质及其广泛的应用，因此对其研究具有重要的理论意义和应用价值。本课题旨在结合Stein方法和Malliavin 分析深入研究几类自相似过程与相关泛函的渐近行为与相关问题。首先拟研究几类自相似高斯过程和Hermite过程与相关泛函（如赋权幂变差、局部时、随机流等）的渐近行为及相关问题；其次研究由自相似高斯噪声驱动的典型类随机偏微分方程，内容主要包括适度解的存在唯一性、密度的存在性和光滑性及其上下界的精确估计等精细概率分析性质；最后研究与自相似过程相关联的线性过程和自正则化和的渐近行为及相关问题，为金融时间序列与计量经济学的实际数据建模提供重要的理论基础。

In recent years, there has been considerable interest in studying fractional Brownian motion due to its applications in various scientific areas including telecommunications, turbulence, image processing and finance. Many authors have also proposed using more general sef-similar processes as stochastic models. However, the complexity of dependence structures has brought so many difficulties and challenges in studying them. This project aims to establish some new results in different topics of stochastic calculus for some selfsimilar processes. First, by combining the Stein’s method and Malliavin calculus, the asymptotic behavior of some functionals of self-similar processes (such as self-similar Gaussian processes and Hermite process), including weighted power variation,local time, stochastic currents and etc, will be investigated. A second objective of the project is to obtain results for the stochastic partial differential equation with self-similar Gaussian noises. The techniques of Malliavin calculus will be applied to prove the existence and smoothness of the density, to derive the upper and lower Gaussian estimates for the density and to study the asymptotic behavior of the solutions to stochastic partial differential equations. On the other hand, motivated by some applications in financial time series and econometrics, the development of a stochastic calculus with respect to linear process and self-normalized sums associated with self-similar processes is also one of the aims of this project.