Properties Of A Type Of Gaussian Processes And Parameter Estimation Of Non-Ergodic Vasicek Models Driven By Them

In this paper,we study the problem of statistical inference that the parameters θ and μ of the non-ergoerical Vasicek model are equal to 0 and not equal to 0 driven by a class of Gaussian processes(Gt)t∈[0,T],and the limit theorems of two independent functions of a class of Gaussian processes.For the Vasicek model(that is,Ornstein-Uhlenbeck(O-U)process with parameter μ=0),if the covariance function R(s,t)=E[GtGs]of the Gaussian process(Gt)t∈[0,T],the second order mixed partial derivative can be decomposed into two parts(see Chen,Zhou(2021)[1]),one of which coincides with that of fractional Brownian motion(fBm)and the other of which is bounded by C(st)H-1 where H ∈(0,1/2)U(1/2,1),and C≥ 0 are absolute constants independent of T,then the four types of moment conditions of Machkouri,Es-Sebaiy(2016)[2]on Gt and ζt=∫0t e-θ(t-s)dGs(where θ>0 is constant)and the three types of moment conditions of Es-Sebaiy,Alazemi(2019)[3]on Gt andξc==∫0t e-θ(t-s)dGs can be directly obtained by the inner product representation formula of the Hilbert space h associated with(Gt),thus,it simplifies,unifies and generalizes the verification process of various Gaussian processes satisfying these moment conditions in Machkouri,Es-Sebaiy(2016)[2],and the strong consistency and asymptotic distribution of the parameter θ of the non-ergodic O-U process based on the least squares class estimators of continuous and discrete sample orbits when T→∞are obtained.For the Vasicek model with parameter μ≠0,if the second order mixed partial derivative of the covariance function of a class of Gaussian processes is a slightly stronger control condition C1(t+s)2H’K-2+C2(s2H’+t2H’)K-2(st)2H’-1 than the above control condition,where H’ ∈(0,1),K ∈(0,2),and H’K ∈(0,1),and C1,C2≥ 0 are absolute constants independent on T,then the five types of moment conditions of Es-Sebaiy,Es-Sebaiy(2021)[4]on Gt and ζt=∫0t e-θ(t-s)dGs(where θ>0 is constant)can be directly obtained by the inner product representation formula of the Hilbert space h associated with(Gt),thus,it simplifies,unifies and generalizes the verification process of various Gaussian processes satisfying these moment conditions in Es-Sebaiy,Es-Sebaiy(2021)[4],and the joint asymptotic distributions of the parameters(θ,μ)and(θ,α)of the non-ergodic Vasicek model based on the least squares class estimators of continuous sample orbits when T→∞ are obtained.For a class of Gaussian processes,on the basis of the above assumptions,the control conditions of its second derivative are given in this paper.Taking these two assumptions as corollations,we unifies and extend the strong local nondeterminism of fBm,the sub-fractional Brownian motion,the bi-fractional Brownian and the sub-bifraction Brownian motion studied by Xiao(2007)[5],Tudor,Xiao(2007)[6],Yan,Shen(2010)[7]and Kuang(2019)[8],respectively.And simplifies,unifies and generalizes the separate verification of the moment conditions concerning the fBm,the sub-fractional Brownian motion,the bi-fractional Brownian by Song,Xu(2019)[9],and as a corollary,the asymptotic behavior of 1/h(n)∫0t ent1 ∫0ent2 f(Gu-Gv)dudv when n→∞ are obtained.At the end of this paper,we also give some examples of fBm,sub-fractional Brownian motion and other Gaussian processes that satisfy the above assumptions,as well as some examples that do not.