N参数d维Ornstein-Uhlenbeck过程样本轨道的一个性质

A PROPERTY OF THE N-PARAMETER d-DIMENSIONAL ORNSTEIN-UHLENBECK PROCESS

对于N参数d维Ornstein-Uhlenbeek过程,本文借助于[2]中的方法,证明了在d<2N时,{X_(N, d)(t):t∈B_+~N}的d维Lebesgue测度大于零。

A stochastic process X={X_t, t∈ R_+~N} is called N-parameter Ornstein-Uhlenbeck process (OUP), if (?) Where X_0~N(0, 1), W is the N-parameter Wiener process and X_0 is independent of W. X^(N,d)={(X_t^1,…, X_t^d):t∈R_+~N} is defind ad the N-parameter d dimensional OUP. We prove that when d<2N, the range of X^(N,d) has a positive d-dimensional volume a.s. The fact is a foundation for further discussion of sample paths of X^(N,d).