本文選題：非利普希茨 + 強(qiáng)解��； 參考：《中國(guó)科學(xué)技術(shù)大學(xué)》2017年碩士論文

【摘要】：隨機(jī)微分方程在統(tǒng)計(jì)物理、控制論、種群遺傳學(xué)、電路理論、金融期權(quán)等諸多領(lǐng)域均有重要應(yīng)用。研究一個(gè)隨機(jī)微分方程的首要課題是其解的存在唯一性問題(包括解是否爆炸的問題)。人們對(duì)隨機(jī)微分方程定義了多種類型的解,又對(duì)它的解定義了多種意義下的唯一性。有利普希茨系數(shù)項(xiàng)的隨機(jī)微分方程解的存在唯一性問題均已經(jīng)有了明確的結(jié)論。而利普希茨連續(xù)屬于較強(qiáng)的條件,故在實(shí)際問題中,需要考慮的方程的系數(shù)并不總是符合此條件。故有非利普希茨系數(shù)項(xiàng)的方程一直是隨機(jī)分析領(lǐng)域中研究的熱點(diǎn)。本文主要研究的問題就是有非利普希茨系數(shù)項(xiàng)的隨機(jī)微分方程解的存在唯一性問題(包括解是否爆炸的問題)。本文結(jié)構(gòu)如下:第一章是本文的緒論部分。第二章簡(jiǎn)要地介紹布朗運(yùn)動(dòng)的隨機(jī)積分與It6公式,以便于后面引出隨機(jī)微分方程這一概念。第三章我們給出了本文所針對(duì)的隨機(jī)微分方程的形式,并嚴(yán)格定義了其弱解、強(qiáng)解以及解的各種唯一性,還有解的爆炸時(shí),并且還說明了一下解的各種存在性與唯一性的內(nèi)在聯(lián)系。第四章,我們主要分類總結(jié)了現(xiàn)有的關(guān)于非利普希茨系數(shù)項(xiàng)方程解的存在唯一性的重要結(jié)論。第五章,我們主要介紹了近年來的一些關(guān)于非利普希茨系數(shù)項(xiàng)的方程解的存在唯一性的重要結(jié)論。在第六章,我們首先回顧了在第四、五章中綜述過的三個(gè)重要結(jié)論,對(duì)它們稍作了一點(diǎn)改進(jìn)推廣,并證明了它們;而后,我們基于前人的研究思想與成果,提出了兩個(gè)關(guān)于非利普希茨系數(shù)項(xiàng)的方程解的唯一性的新的定理,并證明了它們,以此解決了兩種條件下的非時(shí)齊類型的方程解的唯一性問題。
[Abstract]:Stochastic differential equations have important applications in the fields of statistical physics, cybernetics, population genetics, circuit theory, financial options and so on.The most important problem in the study of a stochastic differential equation is the existence and uniqueness of its solution (including the problem of whether the solution is exploded or not).Many types of solutions are defined for stochastic differential equations, and uniqueness is defined for their solutions.The existence and uniqueness of solutions for stochastic differential equations with Lipschitz coefficients have been clearly concluded.Lipschitz continuity is a strong condition, so the coefficients of equations that need to be considered in practical problems do not always meet this condition.Therefore, the equation with non-Lipschitz coefficient term has always been a hot topic in the field of stochastic analysis.The main problem of this paper is the existence and uniqueness of the solution of stochastic differential equations with non-Lipschitz coefficients (including the problem of whether the solution is exploded or not).The structure of this paper is as follows: the first chapter is the introduction of this paper.In chapter 2, the stochastic integral and It6 formula of Brownian motion are introduced briefly, so as to introduce the concept of stochastic differential equation.In chapter 3, we give the form of stochastic differential equation, and define strictly its weak solution, strong solution and all kinds of uniqueness of solution, and the explosion of solution.The existence and uniqueness of the solution are also explained.In chapter 4, we classify and summarize the existing important conclusions on the existence and uniqueness of the solutions of the non-Lipschitz coefficient equation.In chapter 5, we mainly introduce some important conclusions about the existence and uniqueness of the solution of the non-Lipschitz coefficient equation in recent years.In the sixth chapter, we first review the three important conclusions summarized in chapters 4 and 5, improve and generalize them a little bit, and prove them, and then, based on the previous research ideas and results,In this paper, two new theorems on the uniqueness of solutions of equations with non-Lipschitz coefficients are presented, and they are proved to solve the uniqueness problem of solutions of non-homogeneous equations under two conditions.
【學(xué)位授予單位】：中國(guó)科學(xué)技術(shù)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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相關(guān)期刊論文 前1條

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本文編號(hào)：1757574