On the Dependence Structure of Certain Multi-dimensional Ito Processes and Corresponding Hitting Times

A direct approach to derive dependence properties among components of multi-dimensional stochastic processes has been discussed by N. Ebrahimi (1994, J. Multivariate Anal.50, 55-67). Dependence properties among hitting times involving multi-dimensional stochastic processes has been initiated by N. Ebrahimi (1987, J. Appl. Probab.24, 115-122) and explored further by N. Ebrahimi and T. Ramalingham (1988, J. Appl. Probab.25, 355-362; 1989, J. Appl. Probab.26, 287-295). Let X(t)=(X1(t), ..., Xk(t)) be an Ito process assuming values in Rk. In this article, under certain conditions, we show that the process X(t) has a certain dependence structure. We also consider the first passage problem involving X(t) and discuss the dependence structure among hitting times of X1(t), ...,  and Xk(t).