Stochastic Analysis and Applications

分数布朗运动是以区间(0，1)中的Hurst参数为指标的一族高斯随机过程，已被经验证明适用于许多物理现象的模型。对尼罗河沿岸降雨的发生进行了初步的经验验证。随后对经济数据(例如股票价格)、电信(例如自动取款机流量)和药物(例如癫痫发作的发生)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演算，研究了具有分数布朗运动的随机微分方程解。物理现象通常用随机微分方程组来模拟。双线性随机微分方程在建模中得到了广泛的应用，因此我们将利用随机微积分和一些Lie理论方法来研究具有分数布朗运动的有限维双线性方程的解，以确定方程中出现非对易线性算子的各种情况下的显式解。此外，我们还将研究无限维Hilbert空间中的双线性随机微分方程解，因为它们是一些重要的随机偏微分方程组的模型。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组，我们将研究一种估计的加权伪最小二乘方法，以验证估计族的收敛。连续时间最小二乘估计算法的时间离散化对线性随机系统参数的影响是重要的，因为系统状态的观测通常是抽样的。对于具有分数布朗运动的线性系统，将研究这种影响，以确定当采样间隔接近于零时，偏差是否持续存在。分数布朗运动的绝对连续性问题的确定和应用将通过一种作为分数布朗运动的随机积分得到的鞅方法来解决。这种绝对连续性的Radon-Nikodym导数将被应用于随机控制、滤波和互信息的计算问题。随机模型为物理现象提供了有用的描述。随机模型用于描述系统或未建模动态的随机或未知扰动。分数布朗运动是一类随机过程，在水文、经济数据、电信和医学等领域的许多物理现象中都得到了实证验证。分数布朗运动随机微分方程是一类重要的物理现象的随机模型，它是一类形式上具有加性分数布朗运动的微分方程。这些随机微分方程组的一些集合将被研究。由于随机模型的参数估计对于建模的重要性，因此将对其进行研究。本文还将对这些随机微分方程解的其它问题进行研究。

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.