Convergence Of Numerical Solutions To Stochastic Population Equations

The theory of stochastic differential equations(SDE) has been widely applied to control the-ory, biology, communication, ?nancial, etc. In the practical problem, due to the in?uenceof stochastic factors, the model with stochastic parameter was structured by stochastic dif-ferential equation could preferably re?ect the phenomenon's nature. But it was dif?cult toget explicit solution of stochastic differential equations, so it is important to research thenumerical method.This paper mainly studies the convergence of the numerical solutions to stochastic pop-ulation equations which was structured by stochastic differential equation. Deterministicpopulation equation only consider the in?uence of fertility,mortality and migration. But,infact,the population system also subject to many random factors. Therefore, the model in-cluding random factors of external environment is more perfect and reasonable.In the paper, Poisson process and Brown motion are used to represent stochastic fac-tors of the stochastic population equations. The Euler-Maruyama method is then used tode?ne the numerical solutions and the convergence of the numerical solutions to stochasticpopulation equations is proved. The main contents are following:1.The research history and present situation of the stochastic differential equation andpopulation equations are introduced.2.The de?nition,concept,nature,lemma and inequalities which will be used in the pa-per are given,including:Brown motion, Poisson jumps,stochastic differential equations,Ito?equation,Ho¨lder inequalities,Doob inequalities, Gronwall lemma.3.The convergence of the numerical solutions to stochastic population equations withBrown motion, Poisson jumps and delay is discussed and an example for numerical simulat-ing is given. Firstly, Euler-Maruyama method is used to de?ne the numerical solutions andthen the convergence of the numerical solutions to stochastic population equations is provedunder Lipschitz condition.