几类随机微分方程的参数估计问题
发布时间：2018-08-31 13:43
【摘要】：近几十年来,随机微分方程在许多领域应用广泛.本文着重考虑了几类带有小扰动项的二阶非线性随机微分方程的参数估计问题,并应用极大似然估计方法对参数α进行了估计,讨论了小扰动项趋向零或时间T趋向无穷时估计量的性质,具体得到了:当ε→ 0时,未知参数的估计量具有无偏性以及渐近一致性;在ε取固定值和ε → 0的情况下,分别给出了估计量在T趋向无穷时的渐近分布;最后给出数值模拟结果,说明了估计量的无偏性及其渐近正态性.本文分为五个部分.第1章为引言,主要包括研究背景、研究意义等内容.第2章为预备知识,包括文中所涉及到的基本概念,并回顾了随机分析的一些基础知识.第3章介绍了一类非线性随机微分方程,并对参数α进行了估计,讨论了时间T对估计量性质的影响.第4章主要讨论带有小扰动项的随机微分方程的参数估计,利用极大似然估计方法给出了参数的估计量表达式,并研究了小参数ε和时间T对估计量的统计特征的影响.最后应用MATLAB进行了数值模拟得出小参数ε越小,α的估计值越接近于真实值的结论.在最后一章中,介绍了一种改进的和更加复杂的模型,它不仅受标准Brown运动B_t的影响,而且与之独立的标准布朗Wt也对其造成了影响.在此基础上我们讨论了参数估计量的无偏性及其渐近分布.本章的最后通过数值模拟加深对估计量性质的理解。
[Abstract]:In recent decades, stochastic differential equations have been widely used in many fields. In this paper, we focus on the parameter estimation of some second order nonlinear stochastic differential equations with small perturbations, and estimate the parameter 伪 by using the maximum likelihood estimation method. In this paper, we discuss the properties of the estimator when the small disturbance term tends to zero or time T to infinity. It is obtained that the estimator of the unknown parameter is unbiased and asymptotically consistent when 蔚 is zero, and when 蔚 takes the fixed value and 蔚 _ (0), The asymptotic distribution of the estimator when T tends to infinity is given, and the numerical simulation results are given to illustrate the unbiased property and asymptotic normality of the estimator. This paper is divided into five parts. The first chapter is the introduction, mainly includes the research background, the research significance and so on. Chapter 2 is the preparatory knowledge, including the basic concepts involved in this paper, and reviews some basic knowledge of stochastic analysis. In chapter 3, a class of nonlinear stochastic differential equations is introduced, and the parameter 伪 is estimated. The influence of time T on the properties of estimator is discussed. In chapter 4, the parameter estimation of stochastic differential equation with small perturbation term is discussed. The expression of parameter estimator is given by using the method of maximum likelihood estimation, and the influence of small parameter 蔚 and time T on the statistical characteristics of estimator is studied. Finally, the numerical simulation with MATLAB shows that the smaller the parameter 蔚 is, the closer the estimated value of 伪 is to the real value. In the last chapter, an improved and more complex model is introduced, which is influenced not only by the standard Brown motion, but also by the independent standard Brownian Wt. On this basis, we discuss the unbiased property of parameter estimator and its asymptotic distribution. At the end of this chapter, the properties of the estimator are further understood by numerical simulation.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
本文编号：2215163
[Abstract]:In recent decades, stochastic differential equations have been widely used in many fields. In this paper, we focus on the parameter estimation of some second order nonlinear stochastic differential equations with small perturbations, and estimate the parameter 伪 by using the maximum likelihood estimation method. In this paper, we discuss the properties of the estimator when the small disturbance term tends to zero or time T to infinity. It is obtained that the estimator of the unknown parameter is unbiased and asymptotically consistent when 蔚 is zero, and when 蔚 takes the fixed value and 蔚 _ (0), The asymptotic distribution of the estimator when T tends to infinity is given, and the numerical simulation results are given to illustrate the unbiased property and asymptotic normality of the estimator. This paper is divided into five parts. The first chapter is the introduction, mainly includes the research background, the research significance and so on. Chapter 2 is the preparatory knowledge, including the basic concepts involved in this paper, and reviews some basic knowledge of stochastic analysis. In chapter 3, a class of nonlinear stochastic differential equations is introduced, and the parameter 伪 is estimated. The influence of time T on the properties of estimator is discussed. In chapter 4, the parameter estimation of stochastic differential equation with small perturbation term is discussed. The expression of parameter estimator is given by using the method of maximum likelihood estimation, and the influence of small parameter 蔚 and time T on the statistical characteristics of estimator is studied. Finally, the numerical simulation with MATLAB shows that the smaller the parameter 蔚 is, the closer the estimated value of 伪 is to the real value. In the last chapter, an improved and more complex model is introduced, which is influenced not only by the standard Brown motion, but also by the independent standard Brownian Wt. On this basis, we discuss the unbiased property of parameter estimator and its asymptotic distribution. At the end of this chapter, the properties of the estimator are further understood by numerical simulation.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63
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