本文選題：隨機(jī)微分方程　切入點(diǎn)：單調(diào)性條件　出處：《廣西師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：一般情況下,在研究隨機(jī)微分方程數(shù)值方法的收斂性時,需要方程的漂移項(xiàng)與擴(kuò)散項(xiàng)同時滿足全局Lipschitz條件和線性增長條件.然而由于線性增長條件太強(qiáng),現(xiàn)實(shí)生活中的絕大多數(shù)SDEs模型并不滿足此條件,因此本文在局部Lipschitz條件及單調(diào)性條件下為隨機(jī)微分方程構(gòu)造了一種新的半隱式數(shù)值方法,即截斷θ方法,并建立了相關(guān)的收斂性理論.本文結(jié)構(gòu)如下:第1章為緒論.主要介紹隨機(jī)微分方程的相關(guān)背景,研究現(xiàn)狀,本文的創(chuàng)新點(diǎn)和主要內(nèi)容.第2章為預(yù)備知識.主要介紹本文的相關(guān)基礎(chǔ)知識和本文所用符號的含義.第3章構(gòu)造了半隱式的截斷θ方法,并在局部Lipschitz條件,單調(diào)性條件及擴(kuò)散項(xiàng)的多項(xiàng)式條件下,證明了截斷θ方法的兩種連續(xù)類型的數(shù)值解是強(qiáng)收斂的,最后用數(shù)值實(shí)驗(yàn)驗(yàn)證了本章的理論結(jié)果.第4章討論了構(gòu)造的截斷θ方法在給定條件下的收斂速度,并且證明了該算法q階矩的收斂階近似于1/2,且用數(shù)值實(shí)驗(yàn)驗(yàn)證了本章的結(jié)論.最后對本文做了總結(jié)和展望.
[Abstract]:In general, in studying the convergence of numerical methods for stochastic differential equations, it is necessary to satisfy both the global Lipschitz condition and the linear growth condition for both the drift term and the diffusion term of the equation. However, the linear growth condition is too strong. Most SDEs models in real life do not satisfy this condition. In this paper, a new semi-implicit numerical method, truncated 胃 method, is constructed for stochastic differential equations under local Lipschitz condition and monotonicity condition. The structure of this paper is as follows: chapter 1 is the introduction. The innovation and main contents of this paper. Chapter 2 is the preparatory knowledge. It mainly introduces the basic knowledge of this paper and the meaning of the symbols used in this paper. In chapter 3, the semi-implicit truncation 胃 method is constructed, and the local Lipschitz condition is obtained. Under the monotonicity condition and the polynomial condition of diffusion term, it is proved that the numerical solutions of two continuous types of truncation 胃 method are strongly convergent. Finally, numerical experiments are used to verify the theoretical results of this chapter. Chapter 4 discusses the convergence rate of the constructed truncation 胃 method under given conditions. It is proved that the convergence order of the qth-order moment of the algorithm is approximately 1 / 2, and the conclusion of this chapter is verified by numerical experiments. Finally, the conclusion of this paper is summarized and prospected.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O211.63

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本文編號：1597171