Stochastic Analysis of Gaussian Fractional Noises

该项目的目的是在随机分析的不同主题中建立新的结果。首先，介绍一种证明随机偏微分方程解的密度光滑性的新方法。该方法基于解的Malliavin导数的Feynman-Kac公式的随机版本。此外，将应用Malliavin微积分技术来推导解密度的上、下高斯估计。该项目的第二个目标是获得关于分数布朗运动驱动的随机微分方程欧拉型数值逼近格式的收敛速度的结果。应用Malliavin演算技术分析倒向随机微分方程的数值逼近格式也是该项目的主题之一。另一个研究方向是关于高斯过程的各种泛函的中心和非中心极限定理的证明，包括多重随机积分和加权幂变分。该项目还旨在建立分数乘性高斯噪声驱动的一维随机热方程的Feynman-Kac公式。随机分析是数学中的一个活跃领域，其动机是研究受随机噪声扰动的常微分方程和偏微分方程。这些方程在物理学和经济学的许多领域中扮演着模型的核心角色。这些方程在具体问题中的应用需要合适的数值逼近格式和解的概率分布的方便估计。该项目旨在通过开发和应用强大的数学技术，如Malliavin微积分，为这些问题做出新的相关贡献。另一方面，由于在水文学、电信和数学金融中的一些应用，最近人们对具有长记忆性质的输入噪声产生了兴趣，例如分数布朗运动。关于这些长记忆过程的随机演算的发展也是这个项目的目标之一。

The project aims to establish new results in different topics of stochastic analysis. First a new approach for proving the smoothness of the density for solutions to stochastic partial differential equations will be introduced. This method is based on a stochastic version of Feynman-Kac's formula for the Malliavin derivative of the solution. Moreover, the techniques of Malliavin calculus will be applied to derive upper and lower Gaussian estimates for the density of the solution. A second objective of the project is to obtain results on the rate of convergence of Euler-type numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion. The application of the techniques of Malliavin calculus to analyze numerical approximation schemes for backward stochastic differential equation is also one of the topics of the project. Another research direction deals with the proof of central and noncentral limit theorems for a large variety of functionals of a Gaussian process, including multiple stochastic integrals, and weighted power variations. The project also aims to establish Feynman-Kac's formulas for the one-dimensional stochastic heat equation driven by a fractional multiplicative Gaussian noise.Stochastic analysis is an active area in mathematics which is motivated by the study of ordinary and partial differential equations perturbed by a random noise. These equations play a central role as models in many areas of physics and economics. The application of these equations in concrete problems requires suitable numerical approximation schemes, and convenient estimates for the probability distribution of the solution. This project aims to make new relevant contributions to these problems, by developing and applying powerful mathematical techniques such as the Malliavin calculus. On the other hand, motivated by some applications in hydrology, telecommunications and mathematical finance, there has been a recent interest in input noises possessing a long memory property such as the fractional Brownian motion. The development of a stochastic calculus with respect to these long memory processes is also one of the aims of this project.