Conditional and joint failure surface crossing of stochastic processes

In an unpublished 1990 work, H. O. Madsen interpreted the upcrossing rate of scalar stochastic processes as a particular sensitivity measure of the failure probability associated with a suitably modeled parallel system. Here this idea is generalized, and the outcrossing rate of vector processes for which updating information is available, the nth order joint crossing rate, and the nth order joint distribution of local extremes are expressed as parallel system sensitivity measures. The vector processes may be Gaussian, non-Gaussian, stationary, or nonstationary, and the failure function defining the boundary of the safe domain may be time-dependent. In the Gaussian case, several closed-form expressions are derived; e.g. for crossing into convex polyhedral sets of processes restricted to the safe set at an initial time, crossing through a linear surface of processes with position and velocity restriction at an initial time, joint crossing, t1 and t2 through a linear failure surface, and for a generalized Slepian process crossing a linear surface. The method is demonstrated for a scalar Gaussian process upcrossing a time-dependent threshold.