Stochastic Analysis and Applications

本文主要研究具有分数布朗运动的系统的随机分析，特别是在控制问题上的应用。针对分数布朗运动驱动的随机微分方程解的现有结果非常有限的情况，本文研究了多维双线性方程的显式解以及非线性方程解的存在唯一性。双线性方程的显式解需要结合李理论和随机分析，它为解决双线性方程的各种随机问题提供了一种重要的方法。这些随机微分方程应该为许多物理现象提供有用的模型。提出了由分数布朗运动和有限时间区间二次代价泛函驱动的多维线性系统的随机最优控制问题。此外，还提出了具有遍历代价的受控系统的研究。在这两种情况下，最优控制都使用对分数布朗运动增量的预测。遍历控制问题是这些线性系统的自适应控制问题的自然背景。自适应控制问题需要对线性系统的未知参数进行辨识，并构造自寻优自适应控制。提出了一些参数识别方案，如加权伪最小二乘算法，以获得参数的强相容估计。该方案描述了一些随机模型的研究，这些模型使用了一族被称为分数布朗运动的随机过程，这些随机过程出现在尼罗河沿岸的一个降雨模型中。这些过程的潜在用途已被证明用于经济数据、电信、设备噪音和医学。为了得到具有分数布朗运动的有用的随机模型，需要了解方程的解的信息。这是该提案的目标之一。许多随机模型也被控制，并在一个代价准则下寻求最优控制。在这项研究中，将研究具有二次代价泛函的线性系统的一些控制问题。通常系统的某些参数是未知的，也需要对系统进行控制。这些问题要求识别参数，并基于参数的估计确定控制。这类问题的受控线性系统和分数布朗运动将被研究。

The research focuses on stochastic analysis for systems with fractional Brownian motions with particular application to control problems. Since the available results for the solutions of stochastic differential equations driven by fractional Brownian motions are very limited, the investigation of explicit solutions of multidimensional bilinear equations and the existence and uniqueness of solutions of nonlinear equations is proposed. The explicit solutions for bilinear equations requires a combination of Lie theory and stochastic analysis and the explicit solutions provide an important method for solving various stochastic problems for bilinear equations. These stochastic differential equations should provide useful models for many physical phenomena. The stochastic optimal control of multidimensional linear systems driven by fractional Brownian motions and a quadratic cost functional for a finite time interval is proposed. Furthermore this controlled system with an ergodic cost is also proposed for study. In both cases the optimal control uses a prediction of the increments of a fractional Brownian motion. The ergodic control problem is the natural setting for an adaptive control problem for these linear systems. The adaptive control problem requires the identification of the unknown parameters of the linear system and the construction of a self-optimizing adaptive control. Some parameter identification schemes are proposed such as a weighted pseudo least squares algorithm to obtain strongly consistent estimators of the parameters.The proposal describes the investigation of some stochastic models that use a family of stochastic processes called fractional Brownian motions which arose empirically in a model for the rainfall along the Nile River. The potential usage of these processes has been demonstrated for economic data, telecommunications, device noise, and medicine. To have useful stochastic models with fractional Brownian motions it is necessary to have information about the solutions of the equations. This is one goal of the proposal. Many stochastic models are also controlled and with a cost criterion an optimal control is sought. In the research some control problems for linear systems with a quadratic cost functional will be studied. Often some parameters of the system are unknown and it is also required to control the system. These problems require that the parameters are identified and a control is determined based on the estimates of the parameters. Such problems with controlled linear systems and fractional Brownian motion will be investigated.