Stochastic Analysis and Applications

分数布朗运动是一类以Hurst参数为指标的区间（0，1）内的高斯随机过程，已被经验验证为适用于许多物理现象的模型。 对尼罗河流域沿着降雨的发生进行了初步的经验验证。随后对经济数据（如股票价格）、电信（如ATM流量）和医学（如癫痫发作的发生）进行了经验验证。本文利用H在（1/2，1）中的分数布朗运动的随机微积分来研究具有分数布朗运动的随机微分方程的解。 物理现象通常用随机微分方程来模拟。双线性随机微分方程在建模中有着广泛的应用，因此本文研究了具有分数布朗运动的有限维双线性方程的解，并利用随机微积分和一些Lie理论的方法，确定了方程中存在非对易线性算子的各种情况下的显式解.此外，在无限维希尔伯特空间中的双线性随机微分方程将被研究，因为它们作为一些重要的随机偏微分方程的模型。随机系统的参数辨识是建模问题的一个基本组成部分。对于具有分数布朗运动的线性随机微分方程，研究了估计的加权伪最小二乘方法，以验证估计族的收敛性。线性随机系统的参数的连续时间最小二乘估计算法的时间离散化的效果是重要的，因为通常对系统状态的观测进行采样。将研究具有分数布朗运动的线性系统的这种效应，以确定当采样间隔接近零时偏差是否持续存在。分数布朗运动的测度的绝对连续性问题的确定和应用将通过作为分数布朗运动的随机积分获得的鞅方法来解决。这种绝对连续性的Radon-Nikodym导数将应用于随机控制、滤波和互信息的计算等问题。随机模型提供了对物理现象的有用描述。随机模型用于描述系统的随机或未知扰动或未建模动态。分数布朗运动是一类随机过程，其有用性已被经验验证的许多物理现象发生在各种各样的领域，如水文，经济数据，电信和医学。 具有分数布朗运动的随机微分方程是一类形式上具有可加分数布朗运动的微分方程，是一类重要的物理现象的随机模型。这些随机微分方程的一些集合将被调查。由于随机模型的参数估计对于建模的重要性，本文将研究随机模型的参数估计。本文还将研究这类随机微分方程的其它一些问题。

Fractional Brownian motion is a family of Gaussian stochastic processes indexed by the Hurst parameter in the interval (0, 1) that have been empirically verified as suitable for models for many physical phenomena. The initial empirical verification was made for the occurrence of rainfall along the Nile River. Subsequent empirical verifications have been made for economic data (e.g. stock prices), telecommunications (e.g. ATM traffic), and medicine (e.g. the occurrence of epileptic seizures). In this project a stochastic calculus for fractional Brownian motion with H in (1/2, 1) is used to investigate the solutions of stochastic differential equations with a fractional Brownian motion. Physical phenomena are often modeled by stochastic differential equations. Bilinear stochastic differential equations have been used extensively in modeling so the solutions of finite dimensional bilinear equations with a fractional Brownian motion will be investigated to determine explicit solutions in a variety of cases with noncommuting linear operators appearing in the equations by using a stochastic calculus and some Lie theoretic methods. Furthermore, bilinear stochastic differential equations in an infinite dimensional Hilbert space will be investigated because they serve as models for some important stochastic partial differential equations. Parameter identification for stochastic systems is a basic component of the modeling problem. For linear stochastic differential equations with a fractional Brownian motion, a weighted pseudo least squares method for estimation will be investigated to verify the convergence of the family of estimators. The effect of time discretizations of the continuous time least squares estimation algorithm for the parameters of a linear stochastic system is important because typically the observations of the system state are sampled. This effect will be investigated for linear systems with a fractional Brownian motion to determine if biases persist as the sampling intervals approach zero. The determination and the application of the question of absolute continuity for the measure of a fractional Brownian motion will be addressed by a method of martingales that are obtained as stochastic integrals of a fractional Brownian motion. The Radon-Nikodym derivative for this absolute continuity will be applied to problems of stochastic control, filtering, and the calculation of mutual information.Stochastic models provide useful descriptions of physical phenomena. The stochastic models are used to describe random or unknown perturbations of a system or unmodeled dynamics. Fractional Brownian motion is a class of stochastic processes whose usefulness has been empirically verified for many physical phenomena that occur in a wide variety of fields, such as, hydrology, economic data, telecommunications and medicine. Stochastic differential equations with a fractional Brownian motion that are formally differential equations with an additive fractional Brownian motion, are an important class of stochastic models for physical phenomena. Some collections of these stochastic differential equations will be investigated. The estimation of parameters of a stochastic model will be investigated because of its importance for modeling. Some other questions for these stochastic differential equations will also be investigated.