小噪声扰动的二维扩散的参数估计

发布时间：2018-03-17 18:02

本文选题：二维扩散　切入点：参数估计　出处：《南京理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：微分方程应用广泛,如电子通讯、生物学、金融期权定价和物理学中分子运动等领域.然而,经济与科技的发展伴随着对实际问题的描述的更高要求,此时随机微分方程中随机因素的影响也渐渐引起数学家、物理学家等的重视,于是科学家转而研究随机微分方程.本文章主要研究,在小噪声扰动条件下二维扩散的参数估计问题,研究了快慢系统方程首先,使用了Girsanov定理计算出关于二维过程的对数似然函数,关于参数求导得到极大似然估计,紧接着证明了极大似然估计的无偏性、强相合性、渐近正态性和依概率收敛性.其次,运用Euler方法得到二维扩散的差分方程,计算得到转移密度函数.随后得到联合概率密度函数和参数的对数似然函数,计算出参数的估计值,并证明了参数估计值的无偏性和强相合性.最后进行了数值模拟,证明了利用Euler估计方法得到估计值的有效性.随着模拟次数的增多,估计值更加接近真实值,此外当小扰动ε趋近于0,参数的估计值更加接近真实值,数值模拟验证了理论结果.最后,举例说明Euler近似估计在人口预测方面的应用.首先根据1986至2013-年人口出生率与死亡率计算出真实的自然增长率;其次构造了自然增长率的数学模型,预测2014至2023年的自然增长率;再次通过1986至2014年全国总人口的数量,建立了微分方程,利用Euler近似估计方式计算出总人口数量的表达式;最后通过2014至2023年每一年的自然增长率的预测值得到2014至2023年全国总人口数目的预测值,得出到2030年全国总人口维持在14.07亿左右,为统计决策作出一些科学合理的参考。
[Abstract]:Differential equations are widely used in areas such as electronic communications, biology, financial options pricing, and molecular motion in physics. However, the development of economics and technology has been accompanied by higher requirements for the description of practical problems. At this time, the influence of stochastic factors in stochastic differential equations gradually attracted the attention of mathematicians, physicists, and so on, so scientists turned to study stochastic differential equations. In this paper, the problem of parameter estimation for two-dimensional diffusion under the condition of small noise disturbance is studied. Firstly, the logarithmic likelihood function for two-dimensional process is calculated by using Girsanov theorem, and the maximum likelihood estimate is obtained for parameter derivation. Then, the unbiased, strong consistency, asymptotic normality and probabilistic convergence of the maximum likelihood estimation are proved. Secondly, the Euler method is used to obtain the two-dimensional diffusive difference equation. Then the joint probability density function and the logarithmic likelihood function of the parameters are obtained, the estimated values of the parameters are calculated, and the unbiased and strong consistency of the estimated values of the parameters are proved. Finally, the numerical simulation is carried out. The validity of using Euler estimation method to obtain the estimated value is proved. With the increase of the number of simulations, the estimated value is closer to the real value, and when the small perturbation 蔚 approaches zero, the estimated value of the parameter is closer to the real value. The numerical simulation verifies the theoretical results. Finally, an example is given to illustrate the application of Euler approximate estimation in population prediction. Firstly, the real natural growth rate is calculated according to the birth rate and death rate from 1986 to 2013. Secondly, the mathematical model of natural growth rate is constructed to predict the natural growth rate from 2014 to 2023, and then the differential equation is established through the number of the total population from 1986 to 2014, and the expression of the total population quantity is calculated by using the Euler approximate estimation method. Finally, through the forecast value of the natural growth rate of each year from 2014 to 2023, the forecast value of the national population number from 2014 to 2023 is obtained, and the total population of the whole country is maintained at about one billion four hundred and seven million by 2030, which makes some scientific and reasonable reference for the statistical decision.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63

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