On the hyperplane crossing problem for processes generated by state space systems

The problem of estimating the probability Ps(T) that a stochastic processes x ?? IRn generated by a finite dimensional state space system will stay within a given region S over a time interval [0,T] is considered. The following simplifying assumptionsare made: (i) the stochastic state space system is subject to conditions that make x mean square differentiable, (ii) the exterior of S is approximated by a half space specified by a hyperplane H ?? Rn. (iii) The probability Ps(T) is replaced by the sequence of moments of the integer random variable C[0,T] which is defined to be the number of crossings of H by x over [0,T]. Explicit formulae are given for E Cn[0,T], n=1, 2,... and some extensions are described.