【摘要】：目前,在金融、生物、化學(xué)、通訊等多個(gè)研究領(lǐng)域中隨機(jī)微分方程理論都已被普遍地應(yīng)用.但是在實(shí)際生活中,任何領(lǐng)域中都將會(huì)出現(xiàn)各種各樣隨機(jī)因素的影響.因此,借助隨機(jī)擾動(dòng)參數(shù)對(duì)微分方程的研究更具有說(shuō)服力,更符合真實(shí)反映.本文在Brown運(yùn)動(dòng)及Poisson過(guò)程產(chǎn)生擾動(dòng)情況下對(duì)隨機(jī)微分系統(tǒng)的散逸性進(jìn)行了研究.另一方面,由于隨機(jī)系統(tǒng)自身的復(fù)雜性,通常情況下隨機(jī)微分方程大都無(wú)精確解或精確解難以解出,帶Poisson跳的方程更是這般.因而,借助數(shù)值方法對(duì)隨機(jī)微分方程的解以及其性質(zhì)的分析就顯得更為重要.本文的主要工作是探究了與年齡相關(guān)的隨機(jī)種群模型數(shù)值解的散逸性問(wèn)題.內(nèi)容主要包括下面三方面:(1)討論了一類(lèi)基于倒向Euler法的隨機(jī)種群模型數(shù)值解的均方散逸性.利用倒向Euler法以及根據(jù)其步長(zhǎng)h受限制和無(wú)限制的兩種條件下,對(duì)該隨機(jī)種群模型數(shù)值解的均方散逸性進(jìn)行研究并加以證明.最后通過(guò)數(shù)值例子以及結(jié)合MATLAB軟件包演示了結(jié)果的有效性.(2)利用 Ito 公式、Cauchy-Schwarz 不等式和 Bellman-Gronwall-Type 估計(jì)式,在滿(mǎn)足假設(shè)條件的情況下討論了隨機(jī)種群模型數(shù)值解的均方散逸性.并利用分步倒向Euler法和補(bǔ)償?shù)姆植降瓜駿uler法證明了此系統(tǒng)數(shù)值解的均方散逸性,最后借助數(shù)值實(shí)例對(duì)本章重要的結(jié)論加以驗(yàn)證.(3)借助Lyapunov函數(shù)、Barbashin-Krasovskii定理及Ito公式討論了基于年齡結(jié)構(gòu)的隨機(jī)種群模型數(shù)值解的全局穩(wěn)定性問(wèn)題.并且對(duì)該模型強(qiáng)解的存在性加以分析驗(yàn)證,從而獲得該模型零解依概率全局穩(wěn)定的充分條件;最后利用數(shù)值例子結(jié)合MATLAB軟件包對(duì)結(jié)論的有效性進(jìn)行演示.
[Abstract]:At present, stochastic differential equation theory has been widely used in finance, biology, chemistry, communication and so on. However, in real life, there will be a variety of random factors in any field. Therefore, the study of differential equations with stochastic perturbation parameters is more persuasive and more consistent with the real reflection. In this paper, the escapes of stochastic differential systems are studied in the case of Brown motion and perturbation of Poisson processes. On the other hand, due to the complexity of the stochastic system itself, most stochastic differential equations have no exact solution or exact solution, especially the equation with Poisson jump. Therefore, it is more important to analyze the solutions and properties of stochastic differential equations by numerical method. The main work of this paper is to investigate the problem of the numerical solution of the Age-related stochastic population model. The main contents are as follows: (1) the mean-square escape of numerical solutions for a class of stochastic population models based on backward Euler method is discussed. By using the backward Euler method and under the condition that the step size h is restricted and unrestricted, the mean square escape of the numerical solution of the stochastic population model is studied and proved. Finally, numerical examples and MATLAB software package are used to demonstrate the validity of the results. (2) using Ito formula, Cauchy-Schwarz inequality and Bellman-Gronwall-Type estimator, The mean square escape of the numerical solution of the stochastic population model is discussed under the condition that the assumption is satisfied. The mean-square escape of the numerical solution of the system is proved by the stepwise backward Euler method and the compensated stepwise backward Euler method. Finally, the important conclusions of this chapter are verified by numerical examples. (3) with the help of the Lyapunov function, Barbashin-Krasovskii theorem and Ito formula discuss the global stability of numerical solution of stochastic population model based on age structure. The existence of strong solutions of the model is analyzed and verified, and a sufficient condition for the global stability of the model with probability of zero solution is obtained. Finally, the validity of the conclusion is demonstrated by a numerical example combined with MATLAB software package.
【學(xué)位授予單位】：寧夏大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.8

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本文編號(hào)：2341177