【摘要】：近幾十年來,隨機(jī)微分方程在許多領(lǐng)域應(yīng)用廣泛.本文著重考慮了幾類帶有小擾動(dòng)項(xiàng)的二階非線性隨機(jī)微分方程的參數(shù)估計(jì)問題,并應(yīng)用極大似然估計(jì)方法對(duì)參數(shù)α進(jìn)行了估計(jì),討論了小擾動(dòng)項(xiàng)趨向零或時(shí)間T趨向無(wú)窮時(shí)估計(jì)量的性質(zhì),具體得到了:當(dāng)ε→ 0時(shí),未知參數(shù)的估計(jì)量具有無(wú)偏性以及漸近一致性;在ε取固定值和ε → 0的情況下,分別給出了估計(jì)量在T趨向無(wú)窮時(shí)的漸近分布;最后給出數(shù)值模擬結(jié)果,說明了估計(jì)量的無(wú)偏性及其漸近正態(tài)性.本文分為五個(gè)部分.第1章為引言,主要包括研究背景、研究意義等內(nèi)容.第2章為預(yù)備知識(shí),包括文中所涉及到的基本概念,并回顧了隨機(jī)分析的一些基礎(chǔ)知識(shí).第3章介紹了一類非線性隨機(jī)微分方程,并對(duì)參數(shù)α進(jìn)行了估計(jì),討論了時(shí)間T對(duì)估計(jì)量性質(zhì)的影響.第4章主要討論帶有小擾動(dòng)項(xiàng)的隨機(jī)微分方程的參數(shù)估計(jì),利用極大似然估計(jì)方法給出了參數(shù)的估計(jì)量表達(dá)式,并研究了小參數(shù)ε和時(shí)間T對(duì)估計(jì)量的統(tǒng)計(jì)特征的影響.最后應(yīng)用MATLAB進(jìn)行了數(shù)值模擬得出小參數(shù)ε越小,α的估計(jì)值越接近于真實(shí)值的結(jié)論.在最后一章中,介紹了一種改進(jìn)的和更加復(fù)雜的模型,它不僅受標(biāo)準(zhǔn)Brown運(yùn)動(dòng)B_t的影響,而且與之獨(dú)立的標(biāo)準(zhǔn)布朗Wt也對(duì)其造成了影響.在此基礎(chǔ)上我們討論了參數(shù)估計(jì)量的無(wú)偏性及其漸近分布.本章的最后通過數(shù)值模擬加深對(duì)估計(jì)量性質(zhì)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 偽 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 偽 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 偽 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.
【學(xué)位授予單位】：南京理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O211.63

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本文編號(hào)：2215163