Parameter Estimation For O-U Process Driven By The Gaussian Moving Average Process

This paper mainly study the estimation for Ornstein-Uhlenbeck process (O-U pro-cess) driven by the Gaussian Moving Average Process.The Gaussian Moving Average Process has the next stochastic expression: whereφ:R→R is a Borel function that equals zero on (∞,0), and W={Wt, t∈R}is a two-sided Brownian motion on (Ω,(?),P).In this paper we consider a linear stochastic differential equation of the type dXt=-θXtdt+εdGt, t≥0 X0=x∈R, where G is the Gaussian Moving Average process.,The only unknown quantity in SDE is the parameterθ.θ0 represents real value of parameter and{Xt,0≤t≤T} can be observed. We establish the maximum likelihood estimation and least squares estimation for SDE when G is a semartingale.We obtain the log-likelihood estimatorθThen obtain the strong consistencyθT→θa.s.[Pθ] as T→∞, and asymptotic normality of (MLE)θT(XT), then (?)(θT-θ0)(?)N(0,1), as T→∞. Then,we can represent the LSEθn,εasLast,we proof the strong consistencyθn,ε(?)θ0,n→∞,ε→0And the asymptotic normality.When n→∞andε→0,where N～(0,1)...