Random observations of marked Cox processes. Time insensitive functionals

This article has as an objective to analyze the behavior of multivariate, delayed stationary marked Cox processes with mutually dependent components about some critical levels. The original problems arise in biology, computer engineering, computer networks, software reliability testing, and stock market. The process under investigation can describe the evolution of stocks, indexes, cancer cells, proliferation of bacteria, inventories, military conflicts, in which the process is being observed only restrictively, i.e., at some specified random epochs. Given this (sometimes limited) information, it is possible to predict the first passage time when the process crosses the critical level (or levels) and see the main probability characteristics (such as distribution) of the components of the process upon the first passage time that occurs at one of the observation times. Among various questions to arise, one is how to choose the frequency of observations to provide more accurate information but not to exceed the budget (a quint essence of reliability analysis). On the other hand, there are ways to scrutinize the available information, as to making it analytically more time sensitive, without any additional efforts, which is one of the primary goals of this investigation. We formalize and provide preliminary results for the work to be continued in [J. Math. Anal. Appl. 293 (2004) 14-27] (about time sensitive functionals) and give closed-form expressions. Many examples from science and technology are presented. (C) 2004 Elsevier Inc. All rights reserved.