Problems Of Estimation And Hypothesis Testing Of Parameters In Stochastic Differential Equations

In 21 century, the study of financial mathematics and biological mathematics is very important, financial mathematics and biological mathematics and the related region are the main aim to be dealt with. Therefore, this paper deals with the problem for the estimation and hypothesis testing of parameters of a continuous-time stochastic process such as biological mathematics models, applying the statistical method and the theory of stochastic differential equation. First, we study the theory results of the property of dynamics for a continuous-time stochastic process such as biological mathematics models. Next, the estimation of parameters in the models is studied and the classical Euler method to estimate a continuous-time stochastic model from discrete-time date is proposed. At last, statistical decision problems of the estimation of the parameters will be explored for randomized Logistic equations, asymptotic properties, consistency and hypothesis testing of MLEs of parameters are studied. The discussion and solvability of the above problems will supply the basic theory of mathematics to recognize and understand the action of statistics to the estimation and hypothesis testing of parameters for stochastic differential equation.This paper is composed of four parts. In the first chapter, we introduce some definitions and state some preminary results which will be used in our paper.In the second chapter, we study randomized Logistic equations dN(t) — N(t) [1 —N(t)/K ](rdt+ αdB(t)) and dN(t) = N(t)[(a - bN(t))dt + αdB(t)] with initial value N(0) = No and No is a random variable, where B(t) is the 1-dimensional standard Brownian motion. The existence, uniqueness of positive solutions and Maximum Likelihood Estimates (MLEs) of the parameters of the equations are studied. In addition, we obtain asymptotic properties and consistency of MLEs of parameters. At last, by martingale large number theorem and central limit theorem, we study the hypothesis testing of parameters. Simulation results show that the performance of MLEs is fit well.In the third chapter, we study the existence, uniqueness, global stability of positive solution and MLEs of parameters of Lotka-Volterra competition system, mutualism system and predator-prey system with random perturbation. We study the three system one by one in order. First, we show the positive solution exits in global. Next, since there is asymptotic stability equilibruim state for the determinis-tic system under certain conditions, we also consider the global asymptotic stability of average in time of the solution to the perturbation system. At last, since the parameters are usually unknown, so we give the MLEs of parameters. Simulation results show that the performance of MLEs is fit well.In the forth chapter, we shall consider the existence and uniqueness and asymptotic behavior of mild solutions to stochastic partial functional differential equations with infinite delay in LP(Vi,C^) space (p > 2) :dX(t) = \-AX{t) + f(t, Xt))dt + g(t, Xt)dW(t),where — A is a closed, densely denned linear operator and the generator of a certain analytic semigroup and W(t) is a given K-va\ue Wiener process. First, the Banach spaces Cvh and 1/(0.,CD are studied which is fundamental for the subsequent developments. Second, we shall discuss the existence and the uniqueness of solutions to stochastic partial functional equations with infinite delay by semigroup method. In Section 4.3, we devote to the study of p-th moment and almost sure Lyapunov exponential stability properties of mild solutions by using an estimate for stochastic convolution. Finally, we shall present in section 4.4 some applications about Volterra stochastic integro-differential equation with infinite delay. In addition, we shall present an example for Volterra stochastic integro-differential reaction-diffusion equation which illustrates our main theorems.