發(fā)布時間：2018-11-28 07:54

【摘要】：G-布朗運動是在次線性期望(或者說非線性期望)理論框架下構(gòu)造出來的一類新型的布朗運動。G-布朗運動及對應(yīng)的G-隨機分析理論是分析金融市場中風險測度及更廣泛的不確定決策理論的基本工具。因此,G-期望在金融世界中得到越來越廣泛的應(yīng)用。G-布朗運動的引入很好的彌補了以往布朗運動研究中的不足。在至今形成的分析方法中,隨機平均法是一個較為常用并且很重要的近似分析方法。隨機平均原理作為一類近似解析方法的理論基礎(chǔ),為研究復雜微分方程的性質(zhì)提供了一種方便而又簡單的途徑。次線性期望框架下的G-布朗運動驅(qū)動的隨機微分方程的隨機平均原理還鮮有人研究。同時,考慮到全局或局部Lipschitz條件在實際的模型中是相當苛刻的,因此研究非Lipschitz條件下(比Lipschitz條件更弱的條件)G-布朗運動作用下隨機微分方程的隨機平均原理具有重要的理論和實際意義。論文主要工作如下:1、我們首先考慮Lipschitz條件下G-布朗運動作用的隨機微分方程的隨機平均原理。在適合的條件下,證明了經(jīng)過近似處理的方程的解與原始方程的解在均方意義下和依容度意義下的收斂性。平均原理作為平均法的理論基礎(chǔ)給了我們簡化方程復雜性的理論依據(jù)。通過我們的近似處理,我們可以忽略原始復雜的系統(tǒng),只考慮平均后的系統(tǒng),因為它們在均方意義下是等價的。2、考慮到全局或局部Lipschitz條件在實際的模型中是相當苛刻的,因此我們研究了非Lipschitz條件下(比Lipschitz條件更弱的條件)G-布朗運動作用的隨機微分方程的隨機平均原理具有重要的理論和實際意義。證明了經(jīng)過近似處理的方程的解與原始方程的解在均方意義下和依容度意義下的收斂性。
[Abstract]:G- Brownian motion is a new type of Brownian motion constructed under the framework of sublinear expectation (or nonlinear expectation) theory. G- Brownian motion and corresponding G- stochastic analysis theory is an analysis of stroke in financial market. Risk measures and the broader theory of uncertain decision-making basic tools. Therefore, G- expectation is more and more widely used in the world of finance. The introduction of G- Brownian motion makes up for the shortcomings of previous researches on Brownian motion. Among the analytical methods developed so far, the stochastic averaging method is a relatively common and important approximate analysis method. As the theoretical basis of a class of approximate analytical methods, the stochastic averaging principle provides a convenient and simple way to study the properties of complex differential equations. The stochastic averaging principle of G- Brownian motion driven stochastic differential equations under the framework of sublinear expectation is seldom studied. At the same time, considering that the global or local Lipschitz conditions are quite harsh in the actual model, Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations under non-Lipschitz conditions (weaker than Lipschitz conditions) under the action of G-Brownian motion. The main work of this paper is as follows: 1. We first consider the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under Lipschitz condition. Under suitable conditions, the convergence of the solution of the approximate equation and the solution of the original equation in the sense of mean square and tolerance is proved. As the theoretical basis of the averaging method, the averaging principle gives us a theoretical basis for simplifying the complexity of the equation. Through our approximation, we can ignore the original complex systems and only consider the average systems, because they are equivalent in the mean square sense. 2, considering that the global or local Lipschitz conditions are quite harsh in the actual model. Therefore, it is of great theoretical and practical significance to study the stochastic averaging principle of stochastic differential equations acting on G-Brownian motion under non-Lipschitz conditions (weaker than the Lipschitz condition). It is proved that the solution of the approximate equation and the solution of the original equation are convergent in the sense of mean square and tolerance.
【學位授予單位】：西北農(nóng)林科技大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O211.63

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相關(guān)期刊論文 前3條

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本文編號：2362272