Parameter Estimation With Small Noise Perturbation Of The Two-dimensional Diffusions

Differential equations are widely applied in,such as electronic communication-s,biology,financial option pricing and molecular motion in physics and other field-s.However,with the development of economy and technology,we need higher require-ments for the accurate description of the problem,the influence of random factor in stochastic differential equations has attracted more and more attention of mathe-maticians,physicist,and scientists to study.This thesis mainly study on parameter estimation for two-dimensional diffusions under small noise perturbation conditions,for this we consider the fast and slow system equationsFirst,we used the Girsanov theorem to obtain the maximum likelihood function on two-dimensional diffusions.Then,obtained the maximum likelihood estimation and its unbiased,which the estimator was strong consistency,asymptotic normality and convergence in probability.Second,we used Euler method to obtain the difference equation between two-dimensional diffusions,then calculated the transfer density function.The joint proba-bility density function and the log likelihood function of the parameters was obtained,calculated the estimate parameters and proved the unbiased and strong consistency.At last,according to numerical simulation,the validity of the estimation value obtain by Euler estimation method was proved.The more the number of simulations,the more close to the true value of the estimate value.In addition,when the small perturbation? approaches to zero,the estimate value of the parameter is close to the true value,numerical simulation has verified the theoretical results.Finally,we illustrated the application of Euler approximate estimation in popula-tion prediction.First,according to the birth rate from 1986 to 2013,with the mortality to calculate the real natural growth rate;Second,the natural growth rate mathematical model was constructed to predict the natural growth rate from 2014 to 2023;Third,through the total population of the country from 1986 to 2014,to create the differential equation,use Euler approximate estimation method to calculate the total population expression.The final forecasting from 2014 to 2023 years of natural growth rate every year worth to predict the total population number values,which obtained the total pop-ulation remained at around 14.07 million in 2030,make some scientific and reasonable reference for statistical decision.