The Stationary Distribution Problem Of Three Kinds Of Stochastic Model

Actually, stochastic population systems are a form of stochastic LV model, which comefrom the classical LV model disturbed by environmental noise. population systems aremainly subject to environmental noise, meanwhile how the noise affects the populationsystems causes the strong interest of Math workers. some experts reveal if the noise issufficiently large then the population may become extinctive; however population systemswould tend to the steady state when the noise is relatively small, which is the core content ofthis paper. Meanwhile the stationary distribution of stochastic R&D model,Solow model andthe interest rate model is researched.In the first part of this paper, the research background of the model, the researchfulcontent and my main jobs are introduced. And some related theorems and method which areused in the second part. In the third part, firstly the global positive solutions to equation1.5which is the key point of the later proof and its property of tending to0exponentially withprobability one, are obtained in certain conditions by using Lyapunov function; secondlyexistence and uniqueness of stationary distribution of the solution to equation1.5is provedbased on literature[10]; ultimately,the mean vector and covariance matrix of stationarydistribution are obtained by using the ergodic theory and some related theorems. The fourthpart of this paper introduce stochastic R&D model, because it is the two-dimensional formof stochastic population system,the related conclusions are directly applied to the model. Inthe fifth part, the proof of the existence and the uniqueness of stationary distribution ofequation1.5is applied to solow model, and then it is proved that Solow model exists thestationary distribution, and the moment estimator of it is given. Finally, the availability ofrelated conclusions is validated by special examples and computer simulations.