三重马氏过程泛函的一些统计特性

Some Statistical Properties of Functionals of the 3-pie Markov Processes

假设{X(t).t∈R_+}是由随机积分(t—r)~2dB(r)所确定的三重马氏过程,这里{B(t)}是规范化的布朗运动过程.记 f 为有界 Borel 可测函数,若令 Y(t)=f(X(t)),则得三重马氏过程泛函 Y(t).在本文中,作者首先较详细地研讨了三重马氏过程 X(t)及随机泛函 Y(t)的统计特性.然后,又研讨了随机泛函 Y(t)的非线性预测问题,并给出了一些计算 Y(t+λ),λ>0的最佳非线性预测量y(t,λ)的显式公式.

Let{X(t),t∈R+}be a 3-ple Markov Gaussiam process given by X(t)=(t-u)~2dB (u),where{B(t),t∈R_+}is a normalized Brownian motion Process having E{X(t)}=0 and E{B(s)B(t)}=min{s,t}.And let f be a bounded Borel function defined on a mea- surable space(R_1,F).In this paper,the author deals with some statistical properties of the 3-ple Markov Gaussian process X(t)and the stochasfic functionals f(X(t)).The nonlin- ear predictions of fhe stochastic functionals Y(t)have also been discussed and(some)ex- plicit formulas for computing the optimum nonlinear preditor(t,λ)of Y(t+λ), λ>0 has been presented by the author.