设｛X(t),t>=0｝是正交增量过程,X(0)=0,V是标准正态随机变量,若对任意的t>=0,X(t)与V相互独立,令Y(t)=X(t)+V,求随机过程{Y(t),t>=0}的协方差函数.

Question

设｛X(t),t>=0｝是正交增量过程,X(0)=0,V是标准正态随机变量,若对任意的t>=0,X(t)与V相互独立,令Y(t)=X(t)+V,求随机过程{Y(t),t>=0}的协方差函数.

Answer 1

cov(y(t_1),y(t_2))=cov(Y(t_1),Y(t_2)-y(t_1)+y(t_1))
=cov(Y(t_1),Y(t_2)-y(t_1))+cov(y(t_1)),y(t_1))
=cov(X(t_1)+V,X(t_2)-X(t_1))+D(y(t_1))
=0+D(X(t_1)+DV
=DX(t_1)+1 (t_2>t_1)